Quantifying inter-species differences in contractile function through biophysical modelling

This is an Accepted Article that has been peer-reviewed and approved for publication in the The Journal of

Physiology, but has yet to undergo copy-editing and proof correction. Please cite this article as an 'Accepted

Article'; doi: 10.1113/jphysiol.2014.279232.

This article is protected by copyright. All rights reserved. 1

Quantifying Inter-species Differences in Contractile Function

Through Biophysical Modelling

Authors and Affiliations

Kristin Tøndel1,2,*

, Sander Land1, Steven A. Niederer

1 and Nicolas P. Smith

1

1Department of Biomedical Engineering, King’s College London, St. Thomas’ Hospital, Westminster

Bridge Road, London SE1 7EH, United Kingdom 2Simula Research Laboratory, Martin Linges v. 17/25, Rolfsbukta 4B, Fornebu 1364, Norway

*Corresponding author:

Kristin Tøndel

Email: [email protected]

KT: Kristin Tøndel

SL: Sander Land

SAN: Steven A. Niederer

NPS: Nicolas P. Smith

Running Title: Quantifying Inter-species Differences in Contractile

Function

Keywords: Cardiac cell, Contraction, Inter-species differences.

Total number of words: 9469

Table of Contents category: Computational physiology and modelling

Key Points Summary:

To facilitate translation of data from animal models into clinical applications, it is important

to analyse and quantify the differences and relevance of specific physiological mechanisms

between species.

We propose a novel approach for quantification of inter-species differences in terms of

biophysical model parameters and apply this to elucidate the differences in cardiac

contraction mechanisms between mouse, rat and human.

Our results indicate that the parameters related to the sensitivity and cooperativity of calcium

binding to troponin C and the activation and relaxation rates of tropomyosin/crossbridge

binding kinetics differ most significantly between mouse, rat and human.


Our results predict crossbridge binding to be slowest in human, and fastest in mouse.

Abstract

Animal models and measurements are frequently used to guide and evaluate clinical interventions. In

this context, knowledge of inter-species differences in physiology is crucial for understanding the

limitations and relevance of animal experimental assays for informing clinical applications. Extensive

effort has been put into studying the structure and function of cardiac contractile proteins and how

differences in these translate into the functional properties of muscles. However, integrating this

knowledge into a quantitative description, formalising and highlighting inter-species differences both

in the kinetics and regulation of physiological mechanisms remains challenging. In this study we

propose and apply a novel approach for quantification of inter-species differences between mouse, rat

and human. Assuming conservation of the fundamental physiological mechanisms underpinning

contraction, biophysically based computational models are fitted to simulate experimentally recorded

phenotypes from multiple species. The phenotypic differences between species are then succinctly

quantified as differences in the biophysical model parameter values. This provides the potential of

quantitatively establishing the human relevance of both animal-based experimental and computational

models for application in a clinical context.

Our results indicate that the parameters related to the sensitivity and cooperativity of calcium

binding to troponin C and the activation and relaxation rates of tropomyosin/crossbridge binding

kinetics differ most significantly between mouse, rat and human, while for example the reference

tension, as expected, shows only minor differences between the species. Hence, while confirming

expected inter-species differences in the calcium sensitivity due to large differences in the observed

calcium transients, our results also indicated more unexpected differences in the cooperativity

mechanism. Specifically, the decrease in the unbinding-rate of calcium to troponin C with increasing

active tension was much lower for mouse than for rat and human. Our results also predicted

crossbridge binding to be slowest in human, and fastest in mouse.

Abbreviations

PCA, Principal Component Analysis; Ca

2+, Calcium ion; TnC, Troponin C; ATP, Adenosine

Triphosphate; FMM, fading memory model; RT50, Time to 50% relaxation; RT90, Time to 90%

relaxation; RT95, Time to 95% relaxation; TTP, Time to peak tension; Peak, peak tension; LHD,

Latin Hypercube design; Cad, diastolic Ca2+

; Camax, peak Ca2+

concentration; τCa1, time to peak Ca2+

concentration; τCa2, time constant for Ca2+

decline; PLSR, Partial Least Squares Regression; HC-

PLSR, Hierarchical Cluster-based Partial Least Squares Regression; PC, Principal Component; TnI,

Troponin I; HR, Heart rate.

Introduction Computational models of cardiac myocyte physiology provide a biophysical and quantitative

framework for integrating and simultaneously interpreting multiple experimental data sets. This

capacity to integrate data allows consistency of different data sources to be tested within a single

mechanistically based framework, with the potential to provide greater insight into the complex and

often multi-scale regulation that is crucial to physiological systems. However, this ability to integrate

data also exposes models to the risk of including irrelevant or inappropriate information, which can

significantly distort model predictions. To provide a specific set of examples of particular relevance to

this study, in models of cardiac myocyte electrophysiology and contraction, the majority of integrated

data has been recorded from small mammals, namely mouse, rat, guinea pig and rabbit. The increase

in both quality and quantity of this experimental data is now supporting a transition from more

generic mammalian based models (Luo & Rudy, 1994) to increasing species specific models

https://www.researchgate.net/publication/15683616_Luo_CH_Rudy_Y_A_dynamic_model_of_the_cardiac_ventricular_action_potential_ISimulations_of_ionic_currents_and_concentration_changes_Circ_Res_19947461071-1096?el=1_x_8&enrichId=rgreq-898b169b-fa20-412d-a13d-9700b9a992ef&enrichSource=Y292ZXJQYWdlOzI2OTIyMzk4MjtBUzoxODM3NDY0Njg3ODYxNzlAMTQyMDgxOTk4MDk2Mg==


parameterised mainly from data collected from a given species under consistent conditions (Smith et

al., 2007; Niederer et al., 2009; Li et al., 2010). However, in the majority of cases this transition

remains incomplete due to reuse of model components because of lack of experimental data, slowing

the transition to species-specific understandings.

To facilitate translation of data from animal models into clinical applications, it is important

to analyse and quantify the differences and relevance of specific physiological mechanisms between

species. Quantitative descriptions of this system have the capacity to formalise and highlight such

inter-species differences both in kinetics and in regulation. However, such a formalisation is not

trivial due to the complexity of the system and the many factors involved. To address this issue, in

this study we propose a novel method for characterising inter-species differences using biophysical

modelling. The biophysical models consist of sets of mathematical equations, representing the time-

dependent dynamics of various components of the modelled system (“model outputs”), which are

controlled by a number of input parameters. The underlying assumption of the approach applied in

this study is that the fundamental physiological mechanisms are conserved between species, meaning

a single set of mathematical model equations can be used to represent multiple species by tuning the

model parameters to enable the biophysical models to replicate experimentally measured data for each

species. Inter-species differences can then be represented entirely by differences in values of the

model input parameters. Analysing these resulting parameter sets then enables both quantification and

qualitative insight into the mechanistically based differences in physiological function between

species. This information can also be used to determine the relevance of experimental results obtained

in one species for another. This approach, in turn, ultimately provides the potential of quantitatively

establishing the human relevance of both animal-based experimental and computational models for

application in a clinical context.

However, even in mature models it is important to acknowledge that obtaining meaningful

results from computational models is challenging. In addition to the substantial variation observed in

measured values from experiments, the structure of the biophysical models is such that a variety of

different combinations of parameter values can generate approximately the same model output

(Gutenkunst et al., 2007). It is therefore crucial in this study to also take measurement error into

account when parameterising models, and to evaluate the robustness of the parameter estimates.

We recently presented a generalised framework for combined model parameterisation and

analysis of model mechanisms (Tøndel et al., 2014) (see Figure 1 and Figure 2), based on systematic

exploration of the effects of varying the values of the input parameters. Using this framework requires

two sets of data, one containing the input parameter values and one containing the model outputs

resulting from the simulations. These two datasets are used to generate a metamodel (Tøndel et al.,

2012, 2013) - a statistical regression-based approximation of the relationships between input

parameters and model output metrics. The metamodel can then be used to analyse the impact of

variations/noise in the model inputs on the simulation results. This procedure is repeated iteratively,

gradually focusing onto the most relevant ranges for the parameters by identifying sets of simulations

that give output metrics close to the measured data. Analysis of the resulting parameter estimates with

Principal Component Analysis (PCA) (Jolliffe, 2002), provides a route for evaluation of the variation

in parameter values replicating a set of measured data, and to define the degree to which different

parameters can be constrained by available measurements. Combined, this constitutes a powerful

approach to identify robust parameter estimates, which can be applied to generate new hypotheses

about the investigated biological system.

In this study, we apply this approach to fit the parameters of two models of cardiac

contraction to replicate the physiology of multiple species (mouse, rat and human). The models are

fitted to replicate routine experimental measurements and metrics that quantitatively describe

measurable cellular transients. Cardiac contraction is largely controlled by changes in the cytosolic

calcium (Ca2+

) concentration measured as Ca-transients (time series) that vary substantially between

species, and are prone to measurement noise. Hence, it is important to determine the sensitivity of the

parameter estimates to noise in the measured Ca-transients, and thereby analyse the robustness of the

parameter estimates. Accounting for these sensitivities, the resulting sets of species-specific parameter

values then form the basis for a quantification of inter-species differences in terms of physiological

parameters.


Methods

Ethical Approval

All direct data used in this study relates to model parameters which were fitted to measured data in

the referenced studies, each of which complied with National ethical requirements where

the experiments were performed.

Biophysical Models of Cardiac Contraction

Cardiac muscle fibres generate tension during the action of actin and myosin crossbridge cycling

(Gordon et al., 2000; Land, 2013). The muscle cells consist of many contractile sub-units, called

sarcomeres, each organised into thin and thick filaments. The thick filaments contain myosin

crossbridges that bind to the thin actin filament, generating force. This process is initiated by electrical

activation, which results in an increase in cytosolic calcium. Binding of calcium to the regulatory

calcium binding site on troponin C (TnC) within the sarcomeres initiates a cascade of conformational

changes in the associated tropomyosin complex, that make the thin filament actin sites available for

binding to the thick filament myosin crossbridges. A crossbridge cycle consists of a binding of the

myosin crossbridge to actin followed by a force-generating power stroke and a subsequent detachment

using Adenosine Triphosphate (ATP). The functional properties of TnC, including its ability to be

activated by Ca2+

, therefore have significant regulatory influence on the contractile reaction of the

myocyte. Myocyte contractility is also influenced by the strength of interaction between actin and

myosin, the rate of crossbridge cycling, and the rate of ATP hydrolysis by myosin ATPase (Gordon et

al., 2000).

Contractile protein isoforms differ in amino acid sequence between species. This translates

into functional differences in the sarcomere and gives rise to inter-species differences in the resulting

function (Gillis et al., 2007). While there is relatively high conservation of the amino acid sequences

of TnC homologs between species and tissue types, there is wide variation in the functional properties

of these proteins (Gillis et al., 2007). Moreover, regulatory proteins in the heart are potential targets

for phosphorylation (Scruggs et al., 2009). Such posttranslational modification results in changes in

the calcium sensitivity and kinetics of force development and leads to changes in the rate and strength

of cardiac contraction (Weisberg & Winegrad, 1996; Stelzer et al., 2007), and may differ between

species. In order to study inter-species differences, we fit the parameters of two biophysically based

models of cardiac cell contraction, the model developed by (Niederer et al., 2006) (the “Niederer-

model”) and the model developed by (Land et al., 2012a) (the “Land-model”), both consisting of

differential equations describing length-dependence and velocity-dependence of the contractile force,

using experimental data for mouse, rat and human Ca-transients and tension dynamics. The Land-

model was originally parameterised for mouse at 37 °C in a whole-organ context, while the Niederer-

model was originally parameterised for rat at 25 °C, and, as such, is unable to capture the fast

relaxation kinetics of mouse cardiac muscle at higher pacing frequencies with the default parameter

values.

Both models applied here have a relatively low level of detail compared to many other

contraction models, making them suitable for use with organ-scale simulations, but they include

enough biophysical mechanisms to enable the direct coupling of parameters to biological data and

exploration of different hypotheses. The two models represent two different frameworks for

simulating the generation of contractile force in cardiac cells as a consequence of calcium binding,

and were originally parameterised for different species and temperatures, using un-physiological data

sets that provided the necessary data to constrain the parameters. This data is not readily available for

humans at physiological temperatures, so we aimed to fit the models to other measurements

describing more physiological responses.

The rationale behind utilising two different models was to have the opportunity to evaluate

whether our results were consistent between the two models, even though they were based on two

https://www.researchgate.net/publication/225054556_An_analysis_of_deformation-dependent_electromechanical_coupling_in_the_mouse_heart?el=1_x_8&enrichId=rgreq-898b169b-fa20-412d-a13d-9700b9a992ef&enrichSource=Y292ZXJQYWdlOzI2OTIyMzk4MjtBUzoxODM3NDY0Njg3ODYxNzlAMTQyMDgxOTk4MDk2Mg==


different model frameworks and originally developed for different species and temperatures.

Consistency in the results for the two models provides confidence in the conclusions drawn from the

study. A description of the two contraction models including the differential equations is given in

Appendix 1.

In the Land-model, the calcium/TnC dynamics are represented by a standard cooperative

binding equation having a Hill curve where the binding sensitivity is length-dependent and assumed

to approximate the steady state solution, while the Niederer-model uses a simple molecular binding

model for troponin binding. Tropomyosin/crossbridge dynamics are represented in the Land-model by

a modified version of the 4-state Markov model crossbridge dynamics component from (Rice et al.,

2008), using only the so-called non-permissive and permissive (crossbridge cycling) states. In the

Niederer-model, tropomyosin/crossbridge dynamics are represented by a model of the transient

changes in the proportion of available actin sites which each have a length-dependent sensitivity for

binding myosin crossbridges. Both models utilise the fading memory model (FMM) (Hunter et al.,

1998) for the velocity-dependence. The FMM represents the velocity response as several strain-rate

dependent variables which all decay with time. An advantage of this model is that it is independent of

the contraction model, and can be added after modelling isometric tension and length-dependence.

The velocity-dependence parts of the two models were not considered in this study (the velocity was

set to zero for all simulations), due to lack of available data for human and rat at 37 °C.

Fitting of the Model Parameters

The model parameters of the two contraction models described above were fitted to experimental data

for mouse, rat and human found in the literature, and differences in the obtained parameter values

were used to quantify inter-species differences. The generation of simulated data for mouse, rat and

human, as well as the parameter fitting methodology are described in detail in Appendix 2.

Output Metrics Used to Fit the Model Parameters

Given an input Ca-transient, the model simulations result in output tension transients, with

characteristics depending on the input parameter values and initial conditions for the model (see

Appendix 1 for the model equations). The following output metrics were calculated from the tension

transients resulting from simulations with different model parameter values (and input Ca-transients

(see Figure 3)), and matched to measured data: Time to 50% and 90% relaxation (RT50 and RT90),

time to peak tension (TTP) and peak tension values (Peak). For human, data for 95% relaxation

(RT95) was used instead of RT90.

For mouse, additional output metrics calculated from the relationship between individual Ca2+

concentrations and the resulting force (described in more detail in Appendix 2) were included in the

parameter fitting. Since the Land-model had been parameterised for mouse at 37 °C, the force-pCa

relationship resulting from simulations with this model using the default parameter values was used as

reference data for these metrics in addition to measured tension transient metrics. In the mouse

parameterisation, simulations were run with both the Land- and the Niederer-model using sarcomere

lengths of 90, 100 and 110% of resting sarcomere length (i.e. extension ratio, λ, equal to 0.9, 1 and

1.1). In the rat and human parameterisations, only the tension transient metrics at resting sarcomere

length were included, due to lack of data on these additional metrics.

Parameter Fitting Procedure

The parameter fitting procedure is illustrated in Figure 2, and is based on varying the input parameters

in an experimental design (here consisting of 5000 different combinations of parameter values

sampled using Latin Hypercube Design (LHD) (McKay et al., 1979)), running the model to generate

corresponding output metrics, and using the resulting input-output data to generate a metamodel

approximation of the relationships between parameters and output metrics. This procedure is repeated

iteratively, the simulations generating output metrics closest to the measured data are identified in

each iteration, and new experimental designs in the parameters are generated based on the parameter


values of these close simulations (Tøndel et al., 2014). Thereby a zooming into the most biologically

relevant ranges for the parameters is achieved.

The initial ranges (i.e. the ranges within which the parameter values were varied in the first

experimental design of the parameter fitting pipeline) for the Land- and Niederer-model parameters

used in the parameter fitting to data for mouse, rat and human are given in Tables 1-2, while the

constraints used on some of the Niederer-model parameters during the parameter fitting are given in

Table 3. The measured data used to fit the model parameters are given in Table 4.

When fitting the Land- and Niederer-model parameters to the measured data in Table 4, the

mean values of the measurements were used as the target values, however, all resulting simulations

generating output metrics values within the error bars for the measurements (i.e. the span of the values

given in Table 4) were considered successful in replicating measured data. This was assessed in a

look-up of simulations within the ranges given in Table 4.

The possibility of a reduction of the Niederer-model complexity by setting the fast relaxation

rate parameter αr2=0 (omitting the second relaxation term of Equation (A1.11) in Appendix 1) while

keeping the replication of measured data was tested for rat and human, since this was found to be

possible for mouse in (Tøndel et al., 2014). This makes the relaxation parameters nr and Kz redundant,

so these were not varied during the parameter fitting.

Evaluation of the Sensitivity to the Ca-transient

In order to evaluate the importance of the Ca-transient characteristics (and possible noise in measured

Ca-transients used in the simulations) for the results, simulations were run with three different types

of Ca-transients: 1) Experimentally measured Ca-transients (Figure 3), 2) synthetic Ca-transients

where the Ca-transient characteristics were varied within a range around the measured Ca-transient

parameters and 3) Ca-transients modelled using previously published biophysical models of the

ventricular myocyte paced at room temperature (ten Tusscher & Panfilov, 2006; Grandi et al., 2010;

O’Hara et al., 2011) (Figure 3C). The Ca-transients modelled using the Grandi-Pasqualini-Bers, Ten

Tusscher and O’Hara Rudy models were used only in parameter fitting to human data, and were

included as supplementary analyses due to lack of a complete measured Ca-transient time series. The

“measured” Ca-transient for human was based on Equation (A2.1) in Appendix 2 using measured Ca-

transient parameters from (Beuckelmann et al., 1992), while the measured Ca-transients used for

mouse and rat were measured time series data (Ca-traces).

The synthetic Ca-transients were generated by varying the following parameters in an

experimental design consisting of 5000 different combinations of parameter values sampled using

LHD within the ranges given in Table 5: diastolic Ca2+

(Cad), peak Ca2+

concentration (Camax) and the

time constants τCa1 (time to peak Ca2+

concentration) and τCa2 (time constant for Ca2+

decline), and

formed the basis for a systematic analysis of the sensitivity of the model outputs to variations in the

Ca-transient characteristics compared to the sensitivity to the varied model parameters. A regression-

based sensitivity analysis (from a classical metamodel (see Figure 1)) was carried out to identify the

Ca-transient parameters and model input parameters most important for the model outputs. The

regression coefficients from the metamodel are direct measures of the impact of variations in the

various inputs on the simulation results. The generation of the synthetic Ca-transients and the

sensitivity analysis are described in more detail in Appendix 2.

Analysis of Inter-species Differences in Contraction Model Parameter Spaces

For each of the two contraction models, maps of the parameter spaces illustrating the distribution of

the parameter sets giving output metrics within the error bars for measured data for the three different

species, as well as the inter-species differences were generated using PCA (Jolliffe, 2002) of all

parameter sets resulting from fitting the parameters to data for mouse, rat and human (when using the

measured Ca-traces). More details of the PCA are given in Appendix 2: Section A2.6 and A2.7. These

maps show which ranges of parameter values that correspond to specific species, and illustrate the

separation of the results for the three different species as clusters/regions in the parameter spaces.

These regions correspond to different combinations of ranges of values for the various parameters,


giving simulations that replicate measured data for the different species, and can be related back to the

underlying physiology through the parts of the mathematical models controlled by the various

parameters.

The PCA score vectors represent the directions of highest variance in the data, as described in

Appendix 2. In our analysis, plots of the PCA score vectors consequently show the separation of

parameter values replicating measured data for the different species, while the corresponding loading

vectors represent the contribution of the individual model parameters to the species differences. The

size of the clusters of parameter values representing each species provides a quantitative indication of

how constrained the system is given the utilised set of output metrics. Similarly, a PCA of the output

metrics for the different species can indicate which metrics differ the most between species.

Results

We have analysed the sensitivity of the model results to variability in the input Ca-transients, and

investigated how this affects the parameter estimates. Building on these results, we have fitted the

model parameters for two contraction models to data for mouse, rat and human, respectively, using a

previously published parameter fitting method. The parameter fitting resulted in several alternative

parameter sets for each of the three species, replicating the parameterising data. Analysis of these

parameter sets with PCA showed the value ranges for the different input parameters within which

measured data for the different species could be replicated, i.e. the degree of identifiability of the

parameters. Comparison of the results obtained for the different species, taking the span within the

species into account, indicated inter-species differences in parameters directly linked to physiological

cell mechanisms.

Analysis of the Sensitivity to the Ca-transient

Figure 4 shows the regression coefficients from the metamodels made for the Land-model and the

Niederer-model, using synthetic Ca-transients with varying characteristics. Parameters having high

absolute values for the regression coefficients had high impact on the model outputs. This allows us to

elucidate the sensitivity of the simulation results to variations in the Ca-transient given as input.

Figure 4 shows that both models were quite sensitive to the Ca-transient parameters, since the

regression coefficients for the Ca-transient parameters had comparable absolute values to those for the

contraction model parameters. This indicates that variations in the Ca-transient can have a large

impact on the simulation outputs and consequently also on the parameter fitting results. This is

notable due to the significant variation in Ca-transients recorded experimentally, and the common use

of a single representative trace in contraction model fitting. In order to achieve robust parameter

estimates and indications of the uncertainty in the parameter values, it is therefore important to take

possible variations in the Ca-transients into account. Only the sensitivity analysis of the output metrics

from the mouse data obtained at 110% of resting sarcomere length are shown (taking length-

dependence into account), but the sensitivity patterns for 90% and 100% of resting sarcomere length

were very similar for both models.


Fitting of the Model Parameters to Measured Data

As described in the previous section, we have shown that Ca-transients can play a significant role in

parameter estimation. To account for these effects in our parameter fitting for the two contraction

models, we introduced three different sets of Ca-transient data:

1) for mouse and rat, we fitted the model parameters using representative experimentally

measured Ca-traces,

2) for all three species, we used synthetically generated Ca-transients based on measured

phenotypic data, where the Ca-transient characteristics were allowed to vary within a certain

range of the measured values (see Table 5), and

3) for human we used Ca-transients generated by three different biophysical models of the

human ventricular myocyte paced at room temperature (the Grandi-Pasqualini-Bers, Ten

Tusscher and O’Hara Rudy models).

Table 6 and Table 7 show the mean and standard deviations of the Land- and Niederer-model

parameter values producing simulated tension transients replicating measured data for the three

different species, using the three different sets of Ca-transients described above. The resulting tension

transients from the simulations with both models at resting sarcomere length, using measured and

synthetic Ca-transients are shown in Figures 5, 6 and 7, for mouse, rat and human, respectively.

Figure 8 shows the resulting human tension transients obtained using Ca-transients modelled with the

Grandi-Pasqualini-Bers, Ten Tusscher and O’Hara Rudy models. The span for the TTP and Peak

values for mouse, rat and human - which were used together with RT50 and RT90/RT95 to fit the

model parameters - are illustrated in Figures 5-8 together with the tension transients from simulations

replicating the measured data.

Similar to what has previously been found for mouse (Tøndel et al., 2014), our results show

that reduction of the Niederer-model by setting the fast relaxation rate parameter αr2 to zero was

possible while maintaining the fit to measured data also for both rat and human. This parameter was

found to be very poorly constrained both when using measured Ca-traces and when including

variations in the Ca-transients (seen from the results of the mouse parameterisation). Hence, for all

three species applied here, the equation system of the Niederer-model could be simplified. The results

for rat and human reported here are therefore produced using this reduced Niederer-model version.

More details of the results of the parameter fitting are given in Appendix 3.

For the Land-model, the mouse data could only be replicated when using the measured Ca-

transient, no succeeding parameter sets were found using the synthetic Ca-transients. Similarly, using

the Ca-transients modelled using the Ten Tusscher and O’Hara Rudy models did not result in any

simulations replicating the human data with the Land-model, but the human data could be replicated

using the Grandi-Pasqualini-Bers Ca-transient. However, for the Niederer-model, parameter sets

replicating measured data for all three species could be found using all three types of Ca-transients.

Hence, our results indicate that the Land-model cannot be fitted to replicate human contraction

behaviour using these modelled Ca-transients within physiologically feasible parameter ranges. This

can be attributed to either deficiencies of the Land-model equations or the Ca-transients generated by

the Ten Tusscher and O’Hara Rudy models. It should be noted that the time to peak values for these

Ca-transients are relatively low compared to experimental data.

The results in Table 6 and Table 7 show that, as expected due to the results from the

regression-based sensitivity analysis (Figure 4), the parameter estimates were quite sensitive to

introduction of variations to the Ca-transients around the measured Ca-traces. However, for most of

the parameter estimates, the mean value obtained using synthetic Ca-transients was within the error

bars (standard deviations) of the estimates obtained using the measured Ca-traces. Hence, the results

were relatively consistent between these two types of Ca-transients.

However, the human model parameter estimates found using the Grandi-Pasqualini-Bers, Ten

Tusscher and O’Hara Rudy Ca-transients showed relatively large differences in most model

parameters compared to the values obtained using the Beuckelmann transient. This was expected due

to the large difference in e.g. the peak values for the Ca-transients and the sensitivity of the model

outputs to the Ca-transients used, as seen in the sensitivity analysis. The maximum difference between

the mean Land-model parameter estimates obtained with the Beuckelmann transient and those


obtained when Ca-transient variation was included was found in the kTRPN parameter (30%), while the

maximum difference between the results obtained with the Beuckelmann-transient and those obtained

with the Grandi-Pasqualini-Bers Ca-transient was seen for the kxb parameter (76%). For the Niederer-

model, the maximum difference between the mean parameter estimates obtained with the

Beuckelmann transient and those obtained when Ca-transient variation was included was found in the

Ca50ref parameter (44%), while the maximum difference between the Beuckelmann transient results

and the results obtained with the modelled Ca-transients were seen for the αr1 parameter (80% for all

three modelled Ca-transients).

Correlations between Model Parameters and Ca-transient Characteristics

Using synthetic Ca-transients with varying characteristics allows us to identify possible correlations

between the obtained model parameter estimates and the Ca-transient parameter values giving

simulations that replicate measured data. For the Land-model, this analysis was not possible for

mouse, since only the measured Ca-transient gave simulations which replicated the mouse data.

Moreover, the high correlations between many of the Niederer-model parameters and the Ca-transient

parameters, indicated in Figure 5C, may be partly caused by the low number of parameter sets (only

3) included in this analysis. For the rat and human parameterisations, however, more parameter sets

replicating the measured data were obtained, providing more reliable indications of parameter

correlations.

The correlation patterns between the Land- and Niederer-model parameters and the Ca-

transient parameters giving rat model outputs are shown in Figure 6B and Figure 6D, respectively,

while the correlation patterns for the human parameters are shown in Figure 7B and Figure 7D,

respectively. The results are described in more detail in Appendix 3. In summary, the main correlation

patterns for both the Land- and Niederer-models were the same for the rat and human results.

Moreover, for both models, the correlation patterns observed between the model parameters when

using one representative Ca-trace (measured for rat and based on the Beuckelmann Ca-transient

parameters for human) showed the same main correlation patterns (results not shown).

For the Land-model, the main correlation patterns were a negative correlation between the

parameters nxb and nTRPN (the Hill coefficients for cooperative crossbridge action and cooperative

binding of Ca2+

to TnC, respectively), and a positive correlation between Ca50ref (the calcium

sensitivity at resting sarcomere length) and Camax (the peak value of the Ca-transient).

For the Niederer-model, the strongest correlations observed were a positive correlation

between the parameters kon (the binding rate of Ca2+

to TnC) and nH (the Hill coefficient in the

steady-state force-pCa curve), a negative correlation between αr1 (the slow relaxation rate) and γ (the

effect of tension on the unbinding rate of Ca2+

from TnC), a positive correlation between α0 (the

monoexponential activation rate) and Ca50ref. Moreover, the peak value of the Ca-transient had a high

positive correlation with krefoff (the unbinding rate of Ca2+

from TnC in the absence of tension) and a

negative correlation with α0. As seen for the Land-model, Ca50ref and Camax were positively correlated.

Analysis of Inter-species Differences in Contraction Model Parameter Spaces

PCA was carried out for the two models separately, on all parameter sets able to replicate the mouse,

rat and human measurements (using measured Ca-transients). The scores from the PCA represent the

spread in the parameter values, and are plotted in Figure 9A for the Land-model and Figure 9B for the

Niederer-model. Our results showed that the fitting procedure resulted in distinct clusters of parameter

values representing each of the three species in the first three Principal Components (PCs). Figure 9

shows that the spread of Land-model parameter values giving mouse output metrics is smaller for

human than for rat or mouse, while for the Niederer-model the spread is smallest for the mouse data.

The sparse data for mouse compared to rat and human was probably caused by application of output

metrics describing the force-pCa relationship in addition to tension transient metrics to constrain the

model parameters for this species (the force-pCa metrics were not available for rat and human). Using

additional metrics to constrain parameters gives the system less degrees of freedom, and generally

lowers the uncertainty in the parameter estimates, which was the motivation for including these

additional metrics for mouse. Hence, inclusion of these additional metrics also for rat and human


might decrease the sizes of these clusters. The fact that fewer data points represent mouse in the PCA

model implies that the “model centre”, i.e. the mean parameter values, are closer to rat and human

values than to mouse values. However, Figure 9 shows that the PCA model is still capable of

separating the data for the three species into clearly defined clusters, and thereby illustrate the inter-

species differences and the relative contributions of the various model parameters to these differences.

The three species were mostly separated along PC1 and PC2, which combined explained

about 60% of the variance in the data for both models. The PCA loadings shown in Figure 10 indicate

the relative contribution of each of the model parameters to the inter-species differences, while the

mean parameter estimates are shown together with the minimum and maximum values obtained for

each parameter for the three species in Figure 11. Since PCA accounts for the entire variation in the

dataset, not only between the different species, the span in parameter values within a single species

seen in Figure 11 has to be taken into account when making conclusions about the inter-species

differences. Figure 11 shows that for the Land-model, the parameters kTRPN, nxb and kxb were the ones

having the largest spans in the parameter estimates within the results for the individual species (kxb

was badly constrained only for mouse). For the Niederer-model, krefoff, kon, nH and α0 had the largest

spans. The other parameters were relatively well constrained by the measured data used to fit the

parameters.

Our results indicated that the parameter Ca50ref, the calcium concentration needed for 50%

bound TnC in steady state, differs most significantly between mouse, rat and human in both the Land-

and the Niederer-model. This parameter was among the highest ranked contributors to the first three

PCs for both models (Figure 10), and shows relatively high variability in mean value and span (Figure

11). The value of this parameter is highest for rat and lowest for mouse. A high calcium concentration

in the Ca-transients is linked to a lower calcium sensitivity, i.e. a high value of Ca50ref, a pattern that

was confirmed by the parameter correlation analysis above. This is consistent with the measured Ca-

transients, where the highest value for Camax was observed for rat, and the lowest for mouse.

Furthermore, the parameter nTRPN, which participates in controlling the dynamics of the

fraction of regulatory TnC sites with bound Ca2+

in the Land-model, differed significantly between

species; much higher values of nTRPN were obtained for human than for rat and mouse (Figure 11).

This parameter determines the cooperativity of calcium binding to troponin C. Figure 10 shows that

this parameter has a relatively large contribution to the first PC. There is presently not sufficient data

available to fully characterise this effect at physiological temperatures, but the proposed mechanism

involves changes in the unbinding-rate of calcium to troponin C when the thin filament is unblocked.

A similar effect was represented in the Niederer-model as a decrease in this off-rate with increasing

active tension, linked to the parameter γ, which was also among the most changing parameters and

had a large contribution to PC2 in the PCA for the Niederer-model (Figure 10B). The values of γ were

much higher for mouse than for rat and human, meaning that for mouse, the unbinding rate of Ca2+

from TnC decreases to a lower degree with increasing active tension. An explanation of this result

might be that a lower effect of tension on Ca2+

unbinding is required for mouse, since a high effect of

tension leads to a slower relaxation. The unbinding rate of Ca2+

from TnC in the absence of tension

(krefoff) showed slightly higher values for human than for rat and mouse, and had a large contribution

to PC1 in Figure 10B.

In the Land-model, the TnC sensitivity, TRPN50, and the unbinding rate of Ca2+

from TnC,

kTRPN, showed only minor differences taking the span of the parameter value estimates (shown in

Figure 11) into account. The small changes in kTRPN are consistent with the hypothesis that calcium

binding is fast and diffusion limited, and the similarities in TRPN50 are also expected physiologically,

considering the tight coupling between troponin C binding, thin filament activation and crossbridge

cycling.

The identifiability of the Land-model parameter kxb (scaling factor for the rate of crossbridge

binding, which participates in controlling the dynamics of the fraction of available crossbridges

cycling) was very low for mouse (causing a high contribution to PC2 in Figure 10A), probably caused

by the fast mouse kinetics giving this parameter an undefined upper limit (as was seen also in (Tøndel

et al., 2014)). The identifiability of this parameter was much higher for rat and human, which have

slower kinetics (see Figure 11). Our results thus demonstrated the importance of differences in

crossbridge binding and predicted that this is slowest in human, and fastest in mouse. This was also

confirmed by the results from the Niederer-model, where the tropomyosin/crossbridge parameters α0


and αr1 were among those that showed the largest differences between the species (and relatively large

loading values in Figure 10B). The results for human showed the highest activation rate and the

lowest relaxation rate, while the lowest activation rate and the highest relaxation rate was observed for

mouse.

For both models and all species, the reference tension Tref had approximately the same value,

which is consistent with the conservation of systolic pressure seen across mammals. Also nH, the Hill

coefficient in the steady-state force-pCa curve had values within the same range for all three species.

The mean values of kon, the binding rate of Ca2+

to TnC in the Niederer-model, differed only slightly

between the species, but the identifiability of this parameter was relatively low for rat and human.

Discussion

Models are increasingly moving from generic representations, with parameters fitted to data from a

wide variety of species, to focusing on representing specific species and using correspondingly

focused data. In parallel, these models have developed to a point where they can provide a framework

for simulating human physiology with a stronger link to clinical data (Ten Tusscher et al., 2009; Iyer

et al., 2010; Niederer et al., 2011; Smith et al., 2011; O’Hara & Rudy, 2012). However, due to the

inherent practical and ethical limitations on access to human tissue, the data for fitting the parameters

for human models are routinely augmented using animal measurements. To effectively inform our

interpretation of these models it is thus important for us to distinguish and quantify differences

between human cells and other species at the level of model parameters.

Assuming conservation of the underlying physiological mechanisms, parameter fitting can be

exploited to provide a method for quantifying differences in physiology between species identified by

distinct regions of the model parameter space corresponding to measurements from different species.

We show in this study that the same model frameworks can be used to represent physiological

function for three different species by adjusting the parameter values. Hence, a single set of model

equations can represent multiple species by tuning the model parameters to the experimental data set

from each species. This provides the opportunity to gain a deeper understanding of the differences in

e.g. ion conductivities, kinetics and binding affinities. Understanding such differences is important for

the use of animal models to guide clinical interventions.

Specifically, in our study, we have tuned the parameters of the Land- and Niederer-models of

contraction to represent differences between mouse, rat and human, and presented maps of the

parameter spaces of the two models generated with PCA. Our results showed distinct clusters in both

parameter spaces corresponding to the three different species-specific models. These results indicate

that the parameters related to calcium sensitivity, the cooperativity of Ca2+

binding to TnC and the

activation and relaxation rates of tropomyosin/crossbridge binding kinetics differ most significantly

between mouse, rat and human, while for example the Hill coefficient in the steady-state force-pCa

curve and the reference tension show only minor differences between the species. The latter was not

unexpected due to the similarities in the systolic pressure observed across these species.

Our results also identify that the sensitivity of calcium binding to troponin C is highest for

mouse and lowest for rat. This was potentially due to the large differences in the calcium transients

seen in Figure 3, with a higher calcium concentration in the transients being linked to a lower calcium

sensitivity. The relatively small inter-species differences in the unbinding rate of Ca2+

from TnC

(represented by the parameter kTRPN in the Land-model) seen here support the hypothesis that calcium

binding is fast and diffusion limited. However, on the other hand, tension dependent feedback on the

Ca2+

unbinding rate from TnC seems to be species dependent. In the Niederer-model, active tension

has a lower effect on the unbinding rate for mouse than for rat and human (seen from the differences

in the parameter γ). Furthermore, the dynamics of the fraction of actin sites available for crossbridge

binding showed large differences between the species, with the highest activation rate and the lowest

relaxation rate observed for human, and the lowest activation rate and the highest relaxation rate for

mouse. The large inter-species differences in crossbridge binding are consistent with the results of

(Palmer & Kentish, 1998), who estimated crossbridge cycling rate differences between rat and guinea

pig to be six times faster in rat. Also (Harding et al., 1990) showed that myocytes from larger animals

tend to contract and relax more slowly than those from smaller animals.


Rice et al. (Rice et al., 2008) previously published species-specific values for the rate

constants for unbinding of calcium to troponin C, as well as crossbridge cycling rates, according to

differences between rat and rabbit. In their study, the largest modification to adjust the model from rat

to rabbit, was a factor-of-5 decrease in the transition rates in the crossbridge cycle, to simulate the

changes in myosin isoforms. Additionally, a 10% lower unbinding rate of Ca from TnC, and a slightly

higher Ca2+

sensitivity was used for rabbit compared to rat. These modifications are consistent with

our findings with respect to differences between smaller and larger animals. Our results have also

demonstrated the importance of differences in crossbridge binding, and predicted this to be slowest in

human, and fastest in mouse. Moreover, expected inter-species differences in the sensitivity of

calcium binding to troponin C - caused by large differences in the calcium transients – have been

confirmed in our analysis.

By fitting contraction models to measured tension transient characteristics (time to peak

tension, peak tension and relaxation time), we have identified unexpected inter-species differences in

the cooperativity mechanism of Ca2+

binding to TnC. This is represented by the decrease in the

unbinding-rate of calcium to troponin C with increasing active tension, which was much lower for

mouse than for rat and human. This result was unexpected, since TnC sequences are known to be

relatively consistent between species (Gillis et al., 2007). This discrepancy between conserved TnC

sequence and differences in kinetics may be explained by differences in troponin I (TnI) properties or

phosphorylation level between species. Differences in TnI phosphorylation have previously been

shown to be important for contractile performance in myocytes (Westfall & Borton, 2003; Wijnker et

al., 2014). (Takimoto et al., 2004) showed that TnI phosphorylation plays an important role in the

rate-dependence of cardiac muscle. Thus, such differences in phosphorylation may also play an

important role in differences between species depending on their heart rate (HR).

The smaller decrease in the unbinding-rate of calcium to troponin C with increasing active

tension observed here for mouse, compared to rat and human, may be due to the higher HR in mice.

Additionally, with the smaller ventricular cavity within which flow is viscous rather than inertia

dominated, the need to maintain pressure for a longer period following the initial development of

tension will be smaller. This increased cooperativity may be due to that the need for the mouse to

sustain tension over this period is less than for rat and human. Specifically a rapid spike in tension that

decays quickly will have a lower effect in human due to a higher inertia. However, it is important to

note that a larger difference is here predicted between mouse and rat, than between rat and human.

This may be caused by the larger representative Ca-transient recorded in the rat in contrast to the

smaller transient in the mouse. This allows the rat model to accommodate a decrease in unbinding of

TnC without producing a long tension transient.

As discussed in (Tøndel et al., 2014), there is no guarantee that all possible clusters of

parameter values producing feasible model outputs have been found. However, a set of 5000 different

parameter value combinations in each iteration of the parameter fitting pipeline is quite substantial,

ensuring a dense sampling. Moreover, LHD (McKay et al., 1979), which was used here, is a sampling

method that is developed especially for generating even sampling in high-dimensional spaces. It is

therefore likely that all feasible regions of the parameter space are identified during the fitting

procedure. Moreover, the results in (Tøndel et al., 2014) indicated that the fitting procedure used in

this study is relatively robust, since two independent parameter fittings (one with the full Niederer-

model version and one with the reduced equation system) identified the same region of the Niederer-

model parameter space replicating mouse data.

In the present study, independent parameter identifications were run using first a single Ca-

transient, and subsequently with variations mimicking measurement error introduced in the Ca-

transients. The resulting parameter estimates from these independent fittings were relatively

consistent for most of the parameters, and for both models. However, the results obtained in the

human parameterisation showed that the differences in the obtained parameter estimates due to

differences between the modelled Ca-transients were larger than those caused by introducing

variations in the Ca-transient parameters around measured data. This result highlights the importance

of using replicated measurements of Ca-transients when parameterising models and careful

consideration of the model to use if modelled Ca-transients are applied.

Moreover, the resulting values of parameters for the two models representing the same

mechanisms are relatively consistent between the models, giving additional confidence in the


uniqueness of the reported parameter estimates. The lack of Land-model simulations replicating

mouse data when using synthetic Ca-transients was unexpected, especially given the dense sampling

used here, but indicates that the model was unable to replicate the tension transient with nominal noise

in the Ca-transient. This could reflect the fact that contraction parameters or proteins are tuned to the

Ca-transient given the need to generate similar tension in all cells. Another explanation might be that

some bias was introduced due to fitting the model parameters to the force-pCa relationship of the

default Land-model in addition to the tension transients for mouse.

In this study our combined parameter fitting and multivariate data analysis approach was used

to quantify species differences, but other interesting applications would be analysis of temperature-

differences or differences between patients, something that is highly significant for the application of

models for both understanding basic physiology and ultimately clinical application. Additionally, this

type of analysis could also be used to guide the identification of the value of specific experimentally

based assays for both understanding given physiological systems or subsystems, to delineate

mechanisms and/or to develop more models capable of providing increasingly relevant physiological

understandings in experimental contexts.

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Additional Information

Competing Interests

The author(s) declare that they have no competing interests.

Authors’ Contributions

KT contributed to conception of the study and design of the computer experiments, performed the

computer simulations, analysed the data and wrote the paper. SL participated in designing the

computer experiments, collecting the experimental data from the literature and to writing of the paper.

SAN contributed to conception of the study and to writing of the paper. NPS contributed to

conception and coordination of the study and to writing of the paper. All authors read and approved

the final manuscript.

Funding

The research leading to these results has received funding from the Seventh Framework Programme

(FP7/2007-2013) under grant agreement n° 611823; FP7 Marie Curie Actions Intra-European

Fellowship for Career Development (IEF) n° 298494; the Department of Health via the National

Institute for Health Research (NIHR) comprehensive Biomedical Research Centre award to Guy's &

St Thomas' NHS Foundation Trust in partnership with King's College London and King’s College

Hospital NHS Foundation Trust; the United Kingdom EPSRC (EP/G007527/2, EP/H02025X/1),

Welcome Trust (WT 088641/Z/09/Z) and Biotechnology and Biological Sciences Research Council

BBSRC (BB/J017272/1).

Acknowledgments

William E. Louch at Institute for Experimental Medical Research, Oslo University Hospital is thanked

for providing us with the measured Ca-transient for rat. Johan Hake at Simula Research Laboratory is

thanked for helpful discussions.


Appendix 1. Description of the Biophysical Models

A1.1. Length-dependence Equations of the Land-model

In the Land-model (Land et al., 2012a), the calcium sensitivity is given by

)) (A1.1)

where Ca50ref is the calcium sensitivity at resting sarcomere length, is the extension ratio, i.e. the

sarcomere length (SL) relative to the resting SL, and β1 is the magnitude of the length-dependent

activation effects.

The dynamics of the fraction of regulatory troponin C sites with bound calcium (represented by

TRPN) is given by

((

)

) ) (A1.2)

where kTRPN is the unbinding rate of Ca2+

from TnC, [Ca+2

]i is the concentration of free Ca2+

, and nTRPN

is the Hill coefficient for cooperative binding of Ca2+

to TnC.

The dynamics of the fraction of available crossbridges cycling (XB) is given by

( )

) (A1.3)

√(

)

(A1.4)

where kxb is the scaling factor for the rate of crossbridge binding, nxb is the Hill coefficient for

cooperative crossbridge action and TRPN50 is the troponin C sensitivity.

The influence of filament overlap on tension h( ) is given by

) ))) (A1.5)

and

) ) ) (A1.6)

where β0 is the magnitude of the filament overlap effects.

When the velocity dependence is not taken into account (velocity is set to zero), the normalised force

Fn then becomes

) (A1.7)

and the active tension Ta is given by

(A1.8)



A1.2. Length-dependence Equations of the Niederer-model

Also in the Niederer-model (Niederer et al., 2006), the Ca-sensitivity, is given by Equation

(A1.1). The dynamics of the concentration of Ca2+

bound to Troponin C site II, [Ca2+]Trpn , is given by

d[Ca2+]Trpn

dt= kon[Ca

2+ ]i [Ca2+]TrpnMax -[Ca2+]Trpn( ) - koff T( )[Ca2+]Trpn (A1.9)

where kon is the binding rate of Ca2+

to TnC, [Ca2+]TrpnMax is the maximum concentration of ions that

can bind to Troponin C site II, [Ca+2

]i is the concentration of free Ca2+

, T is the tension and

koff = krefoff 1-T

gTref

æ

è

çç

ö

ø

÷÷

(A1.10)

where krefoff is the unbinding rate of Ca

2+ from TnC in the absence of tension, γ is the effect of tension

on the unbinding rate of Ca2+

from TnC and Tref is the reference tension.

The dynamics of the fraction of actin sites available for crossbridge binding (z) is given by

dz

dt=a0

[Ca2+]Trpn

[Ca2+]Trpn50

æ

èçç

ö

ø÷÷

nH

1- z( ) -ar1z-ar2

znr

znr +KZnr

(A1.11)

where α0 is the monoexponential activation rate seen in caged Ca2+

experiments, [Ca2+

]Trpn50 is the

concentration of Ca2+

bound to TnC at half activation, nH is the Hill coefficient in the steady-state

force-pCa curve, αr1 is the slow relaxation rate, αr2 is the fast relaxation rate and nr and Kz are a

relaxation parameters.

In this model it is assumed that crossbridges bind rapidly relative to thin filament kinetics and that not

all actin sites are available at full activation. Thus, tension is proportional to z and the ratio of z to the

fraction of actin sites available at full activation, zMax, for a given SL (z/zMax) is equal to the ratio of the

isometric tension at a given [Ca2+

]i and SL, T0, to the maximum tension at full activation for the same

SL (T0/T0Max):

z

zmax

=T0

T0max

=Ca2+é

ëùûi

nH

Ca2+éë

ùûi

nH

+ Ca2+éë

ùû50

nH (A1.12)

with half activation [Ca2+

]50.

The fraction of actin sites available at full activation (zMax) is defined by solving dz/dt=0 with

[Ca2+

]Trpn=[Ca2+

]TrpnMax (full activation) from Equation (A1.11), giving


zMax =

a0

[Ca2+ ]Trpn50

[Ca2+ ]TrpnMax

æ

èçç

ö

ø÷÷

nH-K2

ar1 +K1 +a0

[Ca2+ ]Trpn50

[Ca2+ ]TrpnMax

æ

èçç

ö

ø÷÷

nHnH

(A1.13)

where

rrrr

r

n

Z

n

p

p

n

Z

n

p

n

p

r

Kz

z

Kz

zK 122 (A1.14)

21

2

1rr

rr

n

Z

n

p

n

Zr

n

pr

Kz

KnzK

(A1.15)

The maximum tension at full activation for a given SL, T0Max, is defined by

T0Max =Tref 1+ b0 l -1( )( ) (A1.16)

where Tref is the reference tension (the maximum tension at resting SL), β0 is the magnitude of the

filament overlap effects and is the extension ratio (relative to resting SL).

Combining Equation (A1.9) with Equation (A1.16) gives

[Ca2+]Trpn50 = [Ca2+ ]TrpnMax[Ca2+]50

[Ca2+ ]50 +krefoff

kon

1-1+ b0 l -1( )( ) × 0.5

g

æ

è

çç

ö

ø

÷÷

(A1.17)

Combining Equation (A1.16) with T0/T0Max=z/zmax gives isometric tension defined as

T0 =T0Max

z

zMax=Tref 1+ b0 l -1( )( ) ×

z

zMax (A1.18)


Appendix 2. Supplementary Descriptions of the Methods

A2.1. Generation of Simulated Mouse Data

Versions of the Land- and Niederer- models functioning for mouse at 37 °C have already been

published in (Land et al., 2012a) and (Tøndel et al., 2014), respectively. However, as shown in

(Tøndel et al., 2014), intrinsically coupled parameters exist, meaning that many combinations of

parameter values can generate the same model output, as long as a specific relationship between the

values of the coupled parameters is maintained. The original models may have been fully constrained

by detailed un-physiological measurements, but here we fit the parameters to a set of functional,

ensemble, physiological data. Hence, the published parameter sets may not represent a unique set for

replicating the observed metrics. This was shown to be the case for the Niederer-model in (Tøndel et

al., 2014), resulting in several different alternative parameterisations capable of reproducing measured

data. In order to evaluate whether this is the case also for the Land-model and to analyse the span of

different parameter values replicating mouse experimental data, a set of simulations was run with the

Land-model, using 5000 different length-dependence parameter value combinations varied in a Latin

Hypercube design (LHD) (McKay et al., 1979). The ranges used for the Land-model parameters are

given in Table 1 and the Ca-transient (Land et al., 2013) used in the simulations is shown in Figure

3A. The outputs from the simulations were compared to those obtained with the default parameter set

using the procedure described in (Tøndel et al., 2014).

For mouse, the default output from the Land-model (which had already been fitted to data for

mouse at 37°C) was used as the target values in fitting of the Niederer-model to mouse at 37°C, and in

searching for additional Land-parameter values replicating measured data. In addition to the output

metrics in Table 4 calculated from the tension transients resulting from the simulations, we also

matched the force-pCa (F-pCa) relationships of the two models to the Land-model default output,

using metrics from simulations run with fixed Ca2+

concentrations. The Ca2+

concentrations used were

a logarithmically spaced series of 82 different concentrations from 0.15 to 1 μM together with the

concentration 10 μM. The resulting steady state tensions were normalised by the maximal simulated

tension value.

Model and experimental steady state force-calcium curves are routinely approximated by a Hill-curve

that can be logarithmically transformed to be linear. The relationship between pCa and log(F/(1-F))

was therefore fitted to a straight line using Ordinary Least Squares (OLS) Regression (values of (1-

F)<10-3

were removed in order to avoid numerical errors), and the following metrics were calculated

to represent the properties of the force-pCa relationship (see also (Tøndel et al., 2014)):

Slope and intercept of the fitted line

Root Mean Square Error of prediction from fitting to a straight line (representing the

deviation from a straight line)

Correlation coefficient between the fitted line and the simulated force-pCa data (representing

the deviation from a straight line)

Maximum tension

RMSD between the simulated force values and the target Land-model force (in standardised

variables)

The F-pCa curves were simulated for 90, 100 and 110% of resting sarcomere length, and the resulting

F-pCa metrics used as additional output constraints (together with the tension transient characteristics

in Table 4) to fit the model parameters.

In order to analyse the sensitivity of the Land-model and the Niederer-model to the input Ca-

transient, and provide an indication of the spread in possible Ca-transients and parameter values that

would generate output metrics within the error bars for the measured metrics - thereby taking into

account that the measured Ca-transients can be noisy - we also carried out an analysis where the input

Ca-transient was varied using an extended version of the equation from Hunter et al. (Hunter et al.,

1998). Equation (A2.1) was used to calculate the Ca-transients based on four new metrics; diastolic

Ca2+

(Cad), peak Ca2+

concentration (Camax) and the time constants τCa1 (time to peak Ca2+



concentration) and τCa2 (time constant for Ca2+

decline). Here we used τCa1 = τCa2 = τCa, which was

equal to the time to peak Ca2+

concentration. As described below, for human, the values of τCa1 and

τCa2 were not set equal, which was the reason for separating the equation into two parts.

) )

(A2.1)

) )

Simultaneously, the model parameters of the two models were varied in a LHD of 5000 parameter

value combinations, in order to evaluate the sensitivity of the modelling results to the Ca-transient,

while varying the parameter values within a specific distance from the mouse parameter values found

in (Land et al., 2012a) and (Tøndel et al., 2014). Thereby the sensitivity of the model outputs to

variations in the model parameters and the Ca-transient could be analysed simultaneously, and

possible coupling between Ca-transient parameters and model parameters identified. The ranges used

for the model parameters of the Land-model and the Niederer-model are given in Table 1 and Table 2,

respectively. The ranges for the Niederer-model parameters were found using the mean of the

parameter sets found in (Tøndel et al., 2014) ± the mean divided by 10, keeping the parameter values

in the proximity of those previously determined (Tøndel et al., 2014).

The following values for the three Ca-transient parameters correspond to the measured Ca-

transient (shown in Figure 3A) used in (Tøndel et al., 2014): Cad=0.2, Camax=0.5 and τCa =21. Here,

Cad, Camax and τCa were varied in a LHD of 5000 combinations of values, using ranges according to

the precision levels for Cad and Camax reported in (Beuckelmann et al., 1992), i.e. 50% and 33%,

respectively. In addition, a 10% error was applied to the time constant τCa. This gave the ranges shown

in Table 5 for the Ca-transient parameters in Equation (A2.1). These are illustrated as error bars

Figure 3A. The same ranges for Cad, Camax and τCa were used in both the Land-model and the

Niederer-model simulations. Simulations were run with both models from the resulting Ca-transients

calculated using Equation (A2.1), for 90, 100 and 110% of resting sarcomere length. Based on the

simulation results, the output metrics in Table 4 were calculated, used in the sensitivity analysis

described below, and compared to the experimentally measured data. The resulting dataset formed the

first iteration of the parameter fitting pipeline illustrated in Figure 2, which was used to identify

additional parameter sets replicating mouse data.

A2.2. Generation of Simulated Rat Data

The Land- and Niederer-models were fitted to rat data, first using a measured Ca-transient (shown in

Figure 3B), and subsequently using Ca-transients generated from Equation (A2.1) with Ca-transient

parameters varied in an experimental design, in order to take measurement error into account. As for

the mouse parameterisation, τCa1 = τCa2 = τCa, and the error bars used on the Ca-transient parameters

were calculated to match the precision level reported in (Beuckelmann et al., 1992). A 10% error was

applied to τCa. This produced the ranges in Table 5 for the Ca-transient parameters in Equation (A2.1),

as illustrated in Figure 3B.

For the Land-model, the parameter ranges given in Table 1 were used in the initial

experimental design in the parameter fitting procedure (see Figure 2), with β1 and β0 at default values,

since only resting sarcomere length was used. These initial ranges were based on prior knowledge

about biologically feasible values for the different parameters. The parameters Tref, TRPN50, nTRPN,

kTRPN and nxb were only allowed to vary within the ranges given in Table 1, with the exception of

nTRPN, which was given an upper bound of 5. The other parameters were allowed to vary outside the

initial ranges during the parameter fitting.

For the Niederer-model, the initial parameter ranges given in Table 2 were used in the

parameter fitting. For all simulations, only resting sarcomere length was used. In addition, we set

αr2=0, causing the model component involving nr and Kz to be zero, and the values of nr and Kz



consequently redundant, in order to test whether we could make a reduced model version for rat, as

was found to be the case for mouse in (Tøndel et al., 2014). During the fitting procedure, the

constraints shown in Table 3 were used on the Niederer-model parameter values.

For both models, the experimental designs contained 5000 simulations based on LHD

(McKay et al., 1979) in each iteration of the parameter fitting pipeline, and the output metrics in

Table 4 were calculated from the resulting tension transients (for rat, only the metrics representing the

tension transient characteristics were included due to lack of data for the force-pCa relationship). The

experimentally measured data in Table 4 was used to fit the parameters for both contraction models.

A2.3. Generation of Simulated Human Data

As for the rat parameterisation, the initial ranges for the model parameters submitted to the fitting

pipeline (see Figure 2) when fitting to human data where according to Table 1 and Table 2 for the

Land-model and the Niederer-model, respectively. Only resting sarcomere length was used.

According to (Beuckelmann et al., 1992), Cad has been measured to be 95 ± 47 nM and Camax to 746

± 249 nM in normal cells. The time constants τCa1 and τCa2 were fitted from a measured Ca-transient

from (Beuckelmann et al., 1992), giving τCa1 = 120 ms and τCa2 = 237 ms. The human Ca-transient

data was generated using Equation (A2.1), resulting in the Ca-transient shown in Figure 3C.

The models were first fitted using the Ca-transient in Figure 3C, and subsequently fitted using

Ca-transient parameters varied in an experimental design using ranges according to the error bars

given above and illustrated in Figure 3C. A 10% error was put on the fitted time constants τCa1 and

τCa2. The ranges used for the Ca-transient parameters are shown in Table 5. For both contraction

models, the experimental designs contained 5000 simulations based on LHD (McKay et al., 1979) in

each fitting iteration, and the output metrics and measured data in Table 4 were used to fit the model

parameters (only the metrics representing the tension transient characteristics were included due to

lack of data for the force-pCa relationship). The same constraints were used for these parameter

values as in the rat parameterisation. In the same way as for the parameter fitting procedures for

mouse and rat, the feasibility of a reduced version of the Niederer-model was analysed by setting the

Niederer-model parameter αr2=0.

Since the human Ca-transient used above was not based on measured time series data, but

based only on measurements for Ca-transient parameters (and the Ca-transient calculated using

Equation (A2.1)), for comparison, the exact same procedure was repeated using Ca-transients

calculated using the following three models: Grandi-Pasqualini-Bers (2010) (Grandi et al., 2010), Ten

Tusscher (2006) (ten Tusscher & Panfilov, 2006) and O’Hara Rudy (2011) (O’Hara et al., 2011).

These Ca-transients are shown in together with the Beuckelmann transient in Figure 3C.

A2.4. Sensitivity Analysis Methodology

In order to evaluate the sensitivity of the modelling results to the Ca-transient, the output metrics from

the simulations were related to the Ca-transient parameters using metamodelling; an approach that has

been shown to be useful for sensitivity analysis and analysis of interactions between input parameters

and covariance patterns between model outputs (Tøndel et al., 2013). Here so-called classical metamodelling (see Figure 1) was applied for the sensitivity analysis,

predicting the model outputs as functions of the input parameters using regression methodology

(Kleijnen, 2007; Tøndel et al., 2010, 2011, 2012, 2013; Vik et al., 2011; Isaeva et al., 2012a, 2012b,

2012c; Martens et al., 2013) (when the input parameters are predicted as functions of the model

outputs, the approach is referred to as inverse metamodelling (Tøndel et al., 2012; Isaeva et al.,

2012b)). The regression coefficients can then be used as measures of the sensitivity of the model

outputs to variations in the input parameters and the Ca-transient. High absolute values for the

regression coefficients indicate high sensitivity. The approach described in (Tøndel et al., 2014)

(based on Partial Least Squares Regression (PLSR) (Wold et al., 1983; Martens & Martens, 1986,

2001; Martens & Næs, 1989; Tøndel et al., 2010; Vik et al., 2011)) was used for the sensitivity

analysis.


The sensitivity analysis was done on the results from the mouse simulations (with the

parameter value ranges given in Tables 1-2), since feasible parameter ranges were already known for

this species ((Land et al., 2012a) and (Tøndel et al., 2014)).

A2.5. Parameter Fitting Methodology

The approach presented in (Tøndel et al., 2014), and there applied to re-parameterise the Niederer-

model to match the output from the Land-model, was applied to fit the model parameters of both

models to measured data for rat and human at 37 °C. As described in (Tøndel et al., 2014), the fitting

of model parameters from measured metrics was based on a combination of inverse metamodelling

(illustrated in Figure 1) using the locally linear regression method Hierarchical Cluster-based Partial

Least Squares Regression (HC-PLSR) (Tøndel et al., 2011, 2012, 2013) and an iterative zooming into

regions of the parameter space that produce feasible model outputs using repeated experimental

designs in the parameter values. The same HC-PLSR settings as in (Tøndel et al., 2014) were used.

The parameter fitting pipeline is illustrated in Figure 2. Steps 2-8 were repeated iteratively until

parameter values generating output metric values within the error bars for the measured data from the

literature were found.

A2.6. Principal Component Analysis (PCA)

PCA (Jolliffe, 2002) decomposes the data into latent variables (linear combinations of the original

variables) represented by so-called scores and loadings, where the scores represent the coordinates of

each data point in the new coordinate system spanned by the principal components (PCs) from PCA

and the loadings are the coefficients in the linear combinations of the original parameters making up

the new coordinate system. The PCA score vectors therefore represent the spread of the simulations,

while the corresponding PCA loading vectors are the weights in the linear combination, and represent

the contribution of each of the model parameters to the PCA scores. Equation (A2.2) shows the PCA

decomposition of the data (X) into scores (T) and loadings (P).

(A2.2)

Using A PCs, this becomes

(A2.3)

where EA represents the error caused by excluding (Amax-A) PCs. In the present paper, the number of

included PCs was chosen so that they together explained 99% of the total variance in X.

A2.7. Data used to Analyse Inter-species Differences in Contraction Model Parameter

Spaces

Only the results achieved using the measured Ca-transients were used for mouse and rat, while for

human, only the results achieved using the Ca-transient from (Beuckelmann et al., 1992) in Figure 3C

were included in the PCA. For the Niederer-model, the parameter sets published in (Tøndel et al.,

2014) were used for mouse. For the Land-model, all parameter sets from the simulations with the

measured mouse Ca-transient (as described in Section A2.1) that were found to replicate the output

metrics from the default Land-model parameter set were used. The parameter values were scaled by

subtracting the mean and dividing by the standard deviation over the values for each parameter prior

to the PCA, in order to make the analysis independent on the absolute value ranges for the different

parameters.



Appendix 3: Details of the Results from the Parameter Fitting

A3.1. Fitting of the Model Parameters to Mouse Measurements

A3.1.1. Fitting of the Land-model Parameters to Measured Data for Mouse

To characterise the uncertainty in the parameter value estimates, 5000 simulations were run with the

Land-model, first using the measured mouse Ca-transient and subsequently using varying Ca-transient

parameters. For mouse, both the tension transients for 90, 100 and 110% of resting sarcomere length,

and the resulting force-pCa relationships were matched to the result obtained with the default

parameter set as described in (Tøndel et al., 2014). This resulted in 11 distinct parameters sets for the

Land-model that were capable of replicating the experimental data when the measured mouse Ca-

transient was used in the simulations. The tension transients from these 11 simulations at resting

sarcomere length are shown in Figure 5A and the mean and standard deviations for the mouse

parameter values are given in Table 6. When we used the measured mouse Ca-trace, the maximum

variation was found in the parameter kxb (±81%). No additional parameter sets replicating the

measured data were identified when Ca-transient variation was included in the parameter fitting.

A3.1.2. Fitting of the Niederer-model Parameters to Measured Data for Mouse

In (Tøndel et al., 2014), 7 Niederer-model parameter sets were found to replicate measured data for

mouse using the measured Ca-transient. When taking Ca-transient variations into account, 3

additional parameter sets were identified that gave tension transients for 90, 100 and 110% of resting

sarcomere length and force-pCa relationships matching the results from the default Land-model. The

tension transients at resting sarcomere length from the Niederer-model simulations that gave mouse

output metrics are shown in Figure 5B and the mean and standard deviations over the set of parameter

value combinations are given in Table 7.

Table 7 shows that most of the parameter estimates are sensitive to variations in the input Ca-

transient, but that a few parameters, e.g. the Ca2+

sensitivity, Ca50ref, the monoexponential activation

rate, α0, and the slow relaxation rate, αr1, showed good stability against variations in the Ca-transients.

When we used the measured mouse Ca-trace, the maximum variation was found in the αr2 (±100%)

parameter for the Niederer-model. When Ca-transient variation was included in the fitting, the

variation in the αr2 parameter changed to ±140%. Hence, the conclusion that the αr2 parameter was

redundant made in (Tøndel et al., 2014) was confirmed here. The second largest variation was found

in the parameter β0 for the Niederer-model (±90% when using the measured Ca-transient, and ±87%

when Ca-transient variation was included).


A3.2. Fitting of the Model Parameters to Rat Measurements

A3.2.1. Fitting of the Land-model Parameters to Measured Data for Rat

Fitting of the Land-model parameters to measured data for rat using only the measured Ca-transient

resulted in 72 parameter sets producing output metric values within the reported ranges of the

measured data. When a varying Ca-transient was used in the parameter fitting, 67 additional

parameter sets produced output metrics replicating the measurements. The mean and standard

deviations for the rat parameter values are given in Table 6, while the tension transients from

simulations at resting sarcomere length are shown in Figure 6A. When we used the measured rat Ca-

trace, the maximum variation was found in the parameter kxb (±67%) (the same parameter as for the

mouse results). When Ca-transient variation was included in the fitting, the variation in the kxb

parameter changed to ±75%.

A3.2.2. Fitting of the Niederer-model Parameters to Measured Data for Rat

For the Niederer-model, 63 simulations gave output metrics within the measured ranges for the rat

data when using only the measured Ca-transient. When a varying Ca-transient was used in the

parameter fitting, 46 additional parameter sets produced output metrics replicating the measurements.

The tension transients from simulations at resting sarcomere length with all rat parameter sets are

shown in Figure 6C, and the mean and standard deviations for the parameter values are given in Table

7.

When we used the measured rat Ca-trace, the maximum variation was found in the α0 (±50%)

parameter for the Niederer-model. When Ca-transient variation was included in the fitting, the

variation in the α0 parameter changed to ±75%.

Similar to what was found for mouse in (Tøndel et al., 2014), our results show that reduction

of the Niederer-model by setting the fast relaxation rate parameter αr2 to zero was possible while

maintaining the fit to measured data also for rat.

A3.3. Fitting of the Model Parameters to Human Measurements A3.3.1. Fitting of the Land-model Parameters to Measured Data for Human

When the Ca-transient from (Beuckelmann et al., 1992) was used, 15 parameter sets produced metrics

within the error bars for the human measurements. Varying the Ca-transient parameters within the

error bars reported in (Beuckelmann et al., 1992) resulted in 13 additional parameter combinations

replicating measured data for human. The mean and standard deviations over these human parameter

values are given in Table 6 and the tension transients resulting from all simulations at resting

sarcomere length replicating human data are shown in Figure 7A.

Fitting of the Land-model to the Grandi-Pasqualini-Bers Ca-transient resulted in 12 parameter

sets (the mean and standard deviations are given in Table 6), giving the tension transients at resting

sarcomere length shown in Figure 8A. The results obtained using the Grandi-Pasqualini-Bers Ca-

transient showed large deviations from the results obtained with the Beuckelmann transient,

something that was expected due to the large difference in e.g. the peak value for the two Ca-

transients. Especially the following parameters showed large differences from the results obtained

with the Beuckelmann transient: Tref, Ca50ref, nTRPN and kxb.

When we used the Beuckelmann Ca-transient for human, the maximum variation was found

in the parameter kTRPN (±35%). When Ca-transient variation around the Beuckelmann transient was

included in the fitting, the variation in the kTRPN parameter changed to ±38%. However, when the

Grandi-Pasqualini-Bers Ca-transient was used, the variation in the Land-model parameter kTRPN

changed to ±64%.

A3.3.2. Fitting of the Niederer-model Parameters to Measured Data for Human


Fitting of the Niederer-model parameters to measured data for human resulted in 52 parameter sets

replicating the measured data when the Ca-transient from (Beuckelmann et al., 1992) was applied,

and 31 additional parameter sets when the Ca-transient parameters were varied. The mean and

standard deviations for these parameter sets are given in Table 7, and the tension transients resulting

from simulations at resting sarcomere length are shown in Figure 7C. Our results showed that

reduction of the Niederer-model by setting αr2=0 is possible while keeping the replication of measured

data also for human. Hence, for all three species applied here, the equation system of the Niederer-

model could be simplified.

Fitting of the Niederer-model to the Grandi-Pasqualini-Bers, Ten Tusscher and O’Hara Rudy

Ca-transients resulted in 6, 99 and 122 parameter sets, respectively (the mean and standard deviations

are given in Table 7), giving the tension transients at resting sarcomere length shown in Figure 8B.

The relatively large differences in the parameter estimates reflect the large differences in the Ca-

transients and the sensitivity of the model outputs to the Ca-transients used. The following parameters

showed large differences in estimated values according to the Ca-transient applied: Ca50ref, kon, γ, nH,

α0 and αr1.

When we used the Beuckelmann Ca-transient for human, the maximum variation was found

in the parameter kon (±48%). When Ca-transient variation around the Beuckelmann transient was

included in the fitting, the variation in kon changed to ±34%. When we used the modelled Ca-

transients, the maximum variation was found in the krefoff (±41%) parameter when using the Grandi-

Pasqualini-Bers Ca-transient, the αr1 (±3.6%) parameter when using the Ten Tusscher Ca-transient,

and in the kon (±30%) parameter when using the O’Hara Rudy Ca-transient.

A3.4. Correlations between Model Parameters and Ca-transient Characteristics

A3.4.1. Parameter Correlation Patterns for the Land-model

The correlation patterns between the Land-model parameters and the Ca-transient parameters giving

rat model outputs are shown in Figure 6B, and indicate that the Hill coefficients for cooperative

crossbridge action and cooperative binding of Ca2+

to TnC (the parameters nxb and nTRPN) are highly

negatively correlated, the rate of crossbridge binding (represented by the scaling factor kxb) is

positively correlated to the troponin C sensitivity (TRPN50), and the calcium sensitivity at resting

sarcomere length (Ca50ref ) has a high positive correlation with the peak value of the Ca-transient

(Camax). Several of the other parameters were also correlated, but to a lower degree.

The correlation patterns between the Land-model parameters and the Ca-transient parameters

giving human model outputs shown in Figure 7B, show that the strong negative correlation between

nxb and nTRPN seen for rat was confirmed here, but here nxb was in addition negatively correlated to the

reference tension Tref. The parameters kxb (scaling factor for the rate of crossbridge binding) and kTRPN

(unbinding rate of Ca2+

from TnC) were also negatively correlated, while Ca50ref had a high positive

correlation with the troponin C sensitivity (TRPN50) and the peak value of the Ca-transient. The latter

two parameters were also positively correlated. Diastolic Ca2+

(Cad) was negatively correlated with

Tref.

A3.4.2. Parameter Correlation Patterns for the Niederer-model

The correlation patterns between the Niederer-model parameters and the Ca-transient parameters

giving rat model outputs are shown in Figure 6D, indicating that several of the parameters of the

Niederer-model are correlated, for example kon (the binding rate of Ca2+

to TnC) and nH (the Hill

coefficient in the steady-state force-pCa curve), which were positively correlated. nH was also

negatively correlated to the slow relaxation rate, αr1, which was again negatively correlated to the

effect of tension on the unbinding rate of Ca2+

from TnC (represented by the parameter γ). The

monoexponential activation rate, α0, was positively correlated with the Ca2+

sensitivity, Ca50ref. The

peak value of the Ca-transient had a positive correlation with krefoff (the unbinding rate of Ca2+

from

TnC in the absence of tension), and a negative correlation with the activation rate α0. The Ca-transient

metric τCa was positively correlated to Ca50ref and α0. The correlation patterns between the Niederer-model parameters and the Ca-transient

parameters giving human model outputs shown in Figure 7D, show that the strong positive correlation


between kon and nH seen for rat was confirmed here, as was the negative coupling between αr1 and γ.

Likewise, the activation rate parameter α0 was positively correlated with Ca50ref, and the peak value of

the Ca-transient had also for human a high positive correlation with krefoff and a negative correlation

with α0. The Ca-transient metric τCa1 was negatively correlated to Ca50ref and positively correlated to γ

and Tref. As seen for the Land-model, Ca50ref and Camax were positively correlated. Hence, also for the

Niederer-model, the correlation patterns for rat and human were relatively similar.

A3.5. Summary

The parameter sets provided here for the Land- and Niederer- models enable the spatial coupling of

common cellular models for use in human electromechanics simulations. Moreover, we previously

found that for mouse at 37 °C, a reduced version of the Niederer-model was sufficient to reproduce

the isometric twitch and steady state force-calcium experimental data (Tøndel et al., 2014), and in this

study we have shown that this reduced version applies for all three species. This reduced complexity

of the Niederer-model, achieved for all three species by identifying redundant model components

makes this model more suitable for integration with large-scale whole organ simulations. All

parameter sets were successfully tested for model stability.


Tables

Table 1. Initial ranges for the Land-model parameters used in the model parameterisation.

Parameter Description Parameter

set from

original

publication

Fitting to mouse data Fitting to rat and

human data

Minimum

value

Maximum

value

Minimum

value

Maximum

value

Tref Reference tension

(kPa) 120 100 140 90 140

Ca50ref

Calcium sensitivity

at resting sarcomere

length (M) 0.6-0.8 0.5 0.8 0.5 2

TRPN50 Troponin C

sensitivity 0.35 0.25 0.5 0.3 0.5

nTRPN

Hill coefficient for

cooperative binding

of Ca2+

to TnC 2 1 2.5 1 2

kTRPN Unbinding rate of

Ca2+

from TnC (ms-1

) 0.1 0 0.5 0.05 0.4

nxb

Hill coefficient for

cooperative

crossbridge action 5 3 7 2 7

kxb

Scaling factor for the

rate of crossbridge

binding (ms-1

) 0.1 0 0.6 0 0.1

β1

Magnitude of length-

dependent activation

effects -1.5 -2 -1 -1.5* -1.5

*

β0

Magnitude of

filament overlap

effects 1.65 1 5 1.65* 1.65

*

* β1 and β0 were not varied in the model parameterisation to rat and human data, since only resting

sarcomere length was used.


Table 2. Initial ranges for the Niederer-model parameters used in the model parameterisation.

Parameter Description Parameter

set from

original

publication

Fitting to mouse data Fitting to rat and

human data

Minimum

value

Maximum

value

Minimum

value

Maximum

value

Ca50ref

Calcium sensitivity

at resting sarcomere

length (mM) 0.3e-3 0.27e-3 0.41e-4 0.5e-3 2e-3

krefoff

Unbinding rate of

Ca2+

from TnC in the

absence of tension

(ms-1

) 0.2 0.07 0.15 0.05 0.4

kon

Binding rate of Ca2+

to TnC (M-1

s-1

) 100 140.1 317.0 50 500

nr Relaxation parameter 3 1.17 2.30 -* -

*

β0

Magnitude of

filament overlap

effects 4.9 0.02 0.96 -1.5**

-1.5**

β1

Magnitude of length-

dependent activation

effects -4 -1.60 -1.01 1.65**

1.65**

γ

Effect of tension on

the unbinding rate of

Ca2+

from TnC 2 3.29 5.00 1 5

nH

Hill coefficient in the

steady-state force-

pCa curve 5 9.0 15.0 4 15

Tref Reference tension

(kPa) 100 91.9 140.0 90 140

α0

Monoexponential

activation rate seen

in caged Ca2+

experiments (ms-1

) 0.008 0.02 0.06 0.01 0.5

αr1 Slow relaxation rate

(ms-1

) 0.002 0.24 0.52 0.01 0.5

αr2 Fast relaxation rate

(ms-1

) 0.00175 0 0.02 0* 0

*

Kz Relaxation parameter 0.15 0.04 0.12 -* -

*

* β1 and β0 were not varied in the model parameterisation to rat and human data, since only resting

sarcomere length was used. **

Since αr2 was set to zero, the parameters nr and Kz became redundant.


Table 3. Constraints used on some of the

Niederer-model parameters during the

parameter fitting.

Parameter Minimum

value

Maximum

value

krefoff 0.05 0.4

kon 50 500

γ 1 5

nH - 15

Tref 90 140


Table 4. Description of the output metrics used to describe the tension transients, together with

measured data for mouse, rat and human at 37 °C, used to fit the contraction model

parameters.

Metric Description Mouse

data1

Rat

data2

Human

data3

RT50 Time to 50%

relaxation (ms) 16-30 27-37 109-125

RT90 Time to 90%

relaxation (ms) 41-59 40-68 -

RT95 Time to 95%

relaxation (ms) - - 291-377

TTP Time to peak tension

(ms) 26-41 34-58 147-172

Peak Peak tension (kPa) 32-52 36-48 20-50 1Measured data for mouse at 37 °C from (Land et al., 2012a).

2Measured data for rat at 37 °C, based on data from (Hiranandani et al., 2006),

(Janssen et al., 2002), (Monasky et al., 2008) and (Monasky & Janssen, 2009). 3Measured data for human at 37 °C (Land et al., 2012b) used

to fit the contraction model parameters, based on data from (Land, 2013).

https://www.researchgate.net/publication/49819833_Efficient_Computational_Methods_for_Strongly_Coupled_Cardiac_Electromechanics?el=1_x_8&enrichId=rgreq-898b169b-fa20-412d-a13d-9700b9a992ef&enrichSource=Y292ZXJQYWdlOzI2OTIyMzk4MjtBUzoxODM3NDY0Njg3ODYxNzlAMTQyMDgxOTk4MDk2Mg==


https://www.researchgate.net/publication/11570877_Myofilament_properties_comprise_the_rate-limiting_step_for_cardiac_relaxation_at_body_temperature_in_the_rat_Am_J_Physiol_Heart_Circ_Physiol_282H499-H507?el=1_x_8&enrichId=rgreq-898b169b-fa20-412d-a13d-9700b9a992ef&enrichSource=Y292ZXJQYWdlOzI2OTIyMzk4MjtBUzoxODM3NDY0Njg3ODYxNzlAMTQyMDgxOTk4MDk2Mg==


Table 5. Ranges for the Ca-transient parameters used to generate the synthetic Ca-transients.

Parameter Description Mouse simulations Rat simulations Human simulations

Min Max Fitted

value*

Min Max Fitted

value*

Min Max Measured/ fitted

value**

Cad Diastolic Ca2+

(M) 0.1 0.3 0.2 0.12 0.36 0.24 0.048 0.142 0.095

Camax Peak Ca2+

concentration

(M) 0.33 0.67 0.50 1.21 2.41 1.81 0.497 0.995 0.746

τCa1 Time to peak

Ca2+

concentration

(ms) 18.9 23.1 21.0 23.4 28.6 26.0 108 132 120

τCa2 Time constant for

Ca2+

decline (ms) 18.9 23.1 21.0 23.4 28.6 26.0 213.3 260.7 237 * Values fitted from the measured Ca-traces.

** Measured values from (Beuckelmann et al., 1992) were used for Cad and Camax, while the values for τCa1 and τCa2

were fitted from the measured Ca-trace given in (Beuckelmann et al., 1992).


Table 6. Mean and standard deviations (SD) for the Land-model parameters giving mouse, rat and human

output metric values.

Parameter Mouse data Rat data Human data Using the

measured

Ca-transient

Using the

measured Ca-

transient

Using

synthetic Ca-

transients

Using the

Beuckelmann

Ca-transient

Using

synthetic Ca-

transients

Using the

Grandi-

Pasqualini-

Bers Ca-

transient

Mean ± SD

(N=11*)

Mean ± SD

(N=72)

Mean ± SD

(N=67) Mean ± SD

(N=15)

Mean ± SD

(N=13) Mean ± SD

(N=12)

Tref 118.4 ± 11.2 109.91 ± 13.12 112.8 ± 13.8 132.5 ± 5.8 128.2 ± 6.1 117.1 ± 14.8

Ca50ref 0.63 ± 0.06 1.92 ± 0.24 1.78 ± 0.23 1.19 ± 0.05 0.95 ± 0.17 0.52 ± 0.09

TRPN50 0.39 ± 0.04 0.40 ± 0.04 0.45 ± 0.04 0.34 ± 0.02 0.35 ± 0.04 0.37 ± 0.05

nTRPN 2.07 ± 0.24 1.55 ± 0.34 1.53 ± 0.34 2.91 ± 0.32 2.63 ± 0.38 1.54 ± 0.32

kTRPN 0.16 ± 0.09 0.21 ± 0.09 0.21 ± 0.10 0.23 ± 0.08 0.16 ± 0.06 0.14 ± 0.09

nxb 5.25 ± 0.86 3.6 ± 1.0 3.97 ± 1.18 3.25 ± 0.44 3.66 ± 0.69 3.38 ± 0.84

kxb 0.21 ± 0.17 0.03 ± 0.02 0.04 ± 0.03 0.02 ± 1.17e-3 0.02 ± 4.11e-3 4.9e-3 ± 8.2e-4

β1 -1.49 ± 0.27 - - - - -

β0 2.45 ± 0.81 - - - - -

Cad -**

- 0.24 ± 0.07 - 102.4 ± 24.2 -

Camax - - 1.53 ± 0.23 - 594.6 ± 140.6 -

τCa1 - - 26.15 ± 1.43 - 121.6 ± 7.5 -

τCa2 - - 26.15 ± 1.43 - 224.1 ± 5.8 - *N denotes the number of succeeding parameter sets.

**The parameters for which no results are given were not varied in the respective parameter fittings. The

parameters β0 and β1 were not varied for rat and human, since only resting sarcomere length was used. Ca-transient

parameters are not given where the measured Ca-transients were used.

T


Table 7. Mean and standard deviations (SD) for the Niederer-model parameters giving mouse, rat and human output metric values.

Parameter Mouse data Rat data Human data

Using the

measured

Ca-

transient

Using

synthetic

Ca-

transients

Using the

measured

Ca-

transient

Using

synthetic

Ca-

transients

Using the

Beuckel-

mann Ca-

transient

Using

synthetic

Ca-

transients

Using the

Grandi-

Pasqualini

-Bers Ca-

transient

Using the

Ten

Tusscher

Ca-

transient

Using the

O’Hara

Rudy Ca-

transient

Mean ±

SD

(N=7*)

Mean ±

SD

(N=3)

Mean ±

SD

(N=63)

Mean ±

SD

(N=46)

Mean ±

SD

(N=52)

Mean ±

SD

(N=31)

Mean ±

SD

(N=6)

Mean ±

SD

(N=99)

Mean ±

SD

(N=122)

Ca50ref 0.34e-3 ±

2.03e-5 0.34e-3 ±

2.4e-5 1.89e-3 ±

3.4e-4

1.78e-3 ±

3.79e-4

1.11e-3 ±

1.57e-4

1.60e-3 ±

2.53e-4

1.1e-3 ±

4.0e-4

2.0e-3 ±

6.4e-5

0.93e-3 ±

6.2e-5

krefoff 0.11 ±

0.03 0.07 ±

0.01 0.13 ±

0.05

0.17 ±

0.09

0.28 ±

0.08

0.21 ±

0.03

0.17 ±

0.07

0.11 ±

3.2e-3

0.29 ±

0.04

kon 230.10 ±

44.80 234.20 ±

34.90 194.6 ±

95.4

188.0 ±

91.9

271.1 ±

130.6

330.1 ±

113.2

299.9 ±

79.3

212.0 ±

6.20

240.9 ±

73.5

nr 1.78 ±

0.33 1.64 ±

0.81 - - - - - - -

β0 0.40 ±

0.36 0.60 ±

0.52 - - - - - - -

β1 -1.30 ±

0.10 -1.50 ±

0.26 - - - - - - -

γ 4.39 ±

0.40 3.82 ±

0.98 2.14 ±

0.40

2.75 ±

1.29

2.46 ±

0.82

1.68 ±

0.52

1.08 ±

0.05

1.02 ±

0.01

1.74 ±

0.26

nH 12.22 ±

1.34 14.05 ±

0.92 7.74 ±

2.18

8.27 ±

3.35

10.08 ±

2.07

11.22 ±

2.21

6.17 ±

1.60

7.32 ±

0.21

13.66 ±

0.80

Tref 122.00 ±

11.20 115.30 ±

13.00 111.7 ±

14.2

112.0 ±

15.2

118.9 ±

12.8

119.2 ±

13.2

107.0 ±

17.0

92.2 ±

1.79

104.2 ±

7.98

α0 0.04 ±

0.01 0.04 ±

0.02 0.08 ±

0.04

0.12 ±

0.09

0.26 ±

0.11

0.31 ±

0.06

0.41 ±

0.12

0.08 ±

2.4e-3

0.28 ±

0.03

αr1 0.38 ±

0.08 0.37 ±

0.11 0.11 ±

0.04

0.16 ±

0.09

0.05 ±

0.01

0.07 ±

0.02

0.01 ±

2.2e-3

0.01 ±

3.6e-4

0.01 ±

9.0e-4

αr2 0.01 ±

0.01 4.3e-3 ±

6.0e-3 - - - - - - -


Kz 0.07 ±

0.03 0.09 ±

0.04 - - - - - - -

Cad -**

0.22 ±

0.03 - 0.24 ±

0.07

- 103.2 ±

24.4

- - -

Camax - 0.51 ±

0.03 - 1.73 ±

0.32

- 869.0 ±

92.2

- - -

τCa1 - 21.15 ±

0.81 - 26.45 ±

1.46

- 120.2 ±

7.4

- - -

τCa2 - 21.15 ±

0.81

- 26.45 ±

1.46

- 232.7 ±

13.3

- - -

*N denotes the number of succeeding parameter sets.

**The parameters for which no results are given were not varied in the respective parameter fittings. The parameters β0 and β1 were not varied for rat and human, since only

resting sarcomere length was used. Ca-transient parameters are not given where the measured or fixed Ca-transients were used.


Figure 1. Illustration of classical and inverse metamodelling for sensitivity analysis and

parameter estimation.

The classical metamodelling was used for sensitivity analysis of the Land-model and the Niederer-

model, using the regression coefficients as sensitivity measures (Tøndel et al., 2014). The inverse

metamodelling was included in the parameter fitting pipeline shown in Figure 2. The figure is

reproduced from (Tøndel et al., 2014) with permission from BioMed Central.


Figure 2. Schematic representation of the parameter fitting pipeline.

Steps 2-8 were repeated in each iteration (Tøndel et al., 2014). The figure is reproduced from (Tøndel

et al., 2014) with permission from BioMed Central.


Figure 3. Measured Ca-transients at 37 °C and alternative, modelled human Ca-transients.

A) Measured mouse Ca-transient, B) measured rat Ca-transient and C) modelled human Ca-transients.

The human Ca-transient in grey was modelled using Equation (A2.1) in Appendix 2 with measured

data from (Beuckelmann et al., 1992) and used to represent a “measured” Ca-transient for human,

while the Ca-transient in red was modelled using the Grandi-Pasqualini-Bers (2010) model (Grandi et

al., 2010), the Ca-transient in green using the Ten Tusscher (2006) model (ten Tusscher & Panfilov,

2006) and the Ca-transient in black was modelled using the O’Hara Rudy (2011) model (O’Hara et

al., 2011). The error bars used on the Ca-transient parameters in the sensitivity analysis and

subsequent parameter fitting are illustrated in blue

.


Figure 4. Sensitivity analysis results.

Sensitivity patterns for A) the Land-model and B) the Niederer-model, represented by the regression

coefficients from a global PLSR metamodel, made using the output metrics calculated at 110% of

resting sarcomere length.


Figure 5. Results from parameter fitting to data for mouse.

Tension transients for A) the Land-model simulations and B) the Niederer-model simulations giving

mouse output metric values. The grey lines represent the parameter sets found using the measured Ca-

transient shown in Figure 3A, while the red lines represent the parameter sets found when the Ca-

transient parameters were varied according to the ranges in Table 5. C) Parameter correlation patterns

for the Niederer-model simulations giving mouse output metric values.


Figure 6. Results from parameter fitting to data for rat.

A) Tension transients and B) parameter correlation patterns for the Land-model simulations giving rat

output metric values. C) Tension transients and D) parameter correlation patterns for the Niederer-

model simulations giving rat output metric values. The grey lines represent the parameter sets found

using the measured Ca-transient shown in Figure 3B, while the red lines represent the parameter sets

found when the Ca-transient parameters were varied according to the ranges in Table 5.


Figure 7. Results from parameter fitting to data for human.

A) Tension transients and B) parameter correlation patterns for the Land-model simulations giving

human output metric values. C) Tension transients and D) parameter correlation patterns for the

Niederer-model simulations giving human output metric values. The grey lines represent the

parameter sets found using the Beuckelmann Ca-transient shown in Figure 3C, while the red lines

represent the parameter sets found when the Ca-transient parameters were varied according to the

ranges in Table 5.


Figure 8. Results from model parameter fitting using the alternative, modelled Ca-transients for

human.

A) Tension transients for the Land-model simulations giving human output metric values when using

the Grandi-Pasqualini-Bers Ca-transient. B) Tension transients for the Niederer-model simulations

giving human output metric values when using the Grandi-Pasqualini-Bers Ca-transient (grey), the

Ten Tusscher Ca-transient (red) and the O’Hara Rudy Ca-transient (green).


Figure 9. Maps of the inter-species differences in the model parameter spaces.

PCA-based maps of the parameter spaces of A) the Land-model and B) the Niederer-model. The three

first PCs from a PCA of the parameter sets found to replicate measured data for mouse, rat and

human, respectively, are shown. The percentage explained variance by each PC is shown in

parenthesis. The lines illustrate the distance from the different points to the plane spanned by PC1 and

PC2.


Figure 10. Contribution of the different parameters to the inter-species differences.

PCA loadings for the first three PCs from the analysis of the succeeding parameter sets for A) the

Land-model and B) the Niederer-model.


Figure 11. Land-model and Niederer-model parameter values for mouse, rat and human.

Mean parameter estimates obtained for A) the Land-model and B) the Niederer-model from parameter

fittings using the measured Ca-transients for mouse and rat, and the Beuckelmann Ca-transient

(Beuckelmann et al., 1992) for human. The error bars indicate the maximum and minimum values

obtained for each parameter.

Quantifying inter-species differences in contractile function through biophysical modelling

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