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V18 Stochastic simulations of cellular signalling Traditional
computational approach to chemical/biochemical kinetics:
(a) start with a set of coupled ODEs (reaction rate equations)
that describe the time-dependent concentration of chemical
species,(b) use some integrator to calculate the concentrations as
a function of time given the rate constants and a set of initial
concentrations.
Successful applications : studies of yeast cell cycle, metabolic
engineering, whole-cell scale models of metabolic pathways
(E-cell), ...
Major problem: cellular processes occur in very small volumes
and frequently involve very small number of molecules. E.g. in gene
expression processes a few TF molecules may interact with a single
gene regulatory region. E.coli cells contain on average only 10
molecules of Lac repressor.
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Include stochastic effects (Consequence1) modeling of reactions
as continuous fluxes of matter is no longer correct.(Consequence2)
Significant stochastic fluctuations occur.
To study the stochastic effects in biochemical reactions
stochastic formulations of chemical kinetics and Monte Carlo
computer simulations have been used.
Daniel Gillespie (J Comput Phys 22, 403 (1976); J Chem Phys 81,
2340 (1977)) introduced the exact Dynamic Monte Carlo (DMC) method
that connects the traditional chemical kinetics and stochastic
approaches.
Assuming that the system is well mixed, the rate constants
appearing in these two methods are related.
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Dynamic Monte Carlo In the usual implementation of DMC for
kinetic simulations, each reaction is considered as an event and
each event has an associated probability of occurring.
The probability P(Ei) that a certain chemical reaction Ei takes
place in a given time interval t is proportional to an effective
rate constant k and to the number of chemical species that can take
part in that event.
E.g. the probability of the first-order reaction X Y + Zwould be
k1Nx with Nx :number of species X, and k1 : rate constant of the
reaction
Similarly, the probability of the reverse second-order reactionY
+ Z Xwould be k2NYNZ.Resat et al., J.Phys.Chem. B 105, 11026
(2001)
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Dynamic Monte Carlo As the method is a probabilistic approach
based on events, reactions included in the DMC simulations do not
have to be solely chemical reactions.
Any process that can be associated with a probability can be
included as an event in the DMC simulations.
E.g. a substrate attaching to a solid surface can initiate a
series of chemical reactions. One can split the modelling into the
physical events of substrate arrival, of attaching the substrate,
followed by the chemical reaction steps. Resat et al., J.Phys.Chem.
B 105, 11026 (2001)
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Basic outline of the direct method of Gillespie(Step i) generate
a list of the components/species and define the initial
distribution at time t = 0.
(Step ii) generate a list of possible events Ei (chemical
reactions as well as physical processes).
(Step iii) using the current component/species distribution,
prepare a probability table P(Ei) of all the events that can take
place.Compute the total probability
P(Ei) : probability of event Ei .
(Step iv) Pick two random numbers r1 and r2 [0...1] to decide
which event E will occur next and the amount of time by which E
occurs later since the most recent event.Resat et al., J.Phys.Chem.
B 105, 11026 (2001)
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Basic outline of the direct method of GillespieUsing the random
number r1 and the probability table,the event E is determined by
finding the event that satisfies the relationResat et al.,
J.Phys.Chem. B 105, 11026 (2001)The second random number r2 is used
to obtain the amount of time between the reactionsAs the total
probability of the events changes in time, the time step between
occurring steps varies.
Steps (iii) and (iv) are repeated at each step of the
simulation.
The necessary number of runs depends on the inherent noise of
the system and on the desired statistical accuracy.
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Weighted SamplingIn the commonly used MC algorithm, the Markov
chain is generated using transition probabilities (i j) that are
based on the physical probability distribution:Resat et al.,
J.Phys.Chem. B 105, 11026 (2001)The ensemble average of any
physical quantity is obtained by taking the arithmetic average of
all the n simulation runs.
The individual averages i could e.g. be time-averages over the
simulation run.
This choice disfavors the transitions with low
probabilities.
If the system characteristics depend on the events that happen
less frequently, then the common implementation of MC requires
extremely lengthy simulations to acquire enough statistical
sampling.
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Weighted SamplingThis statistical sampling problem can be
avoided if the probability distribution is multiplied with a weight
function that adjusts the sampling probability distribution such
that the low probability parts of the sampling space are visited
more often.
In the case of weighted sampling, the Markov chain is generated
by using the modified probability distribution functionResat et
al., J.Phys.Chem. B 105, 11026 (2001)where Y is the biasing weight
function.
Since the probability of the transition i j is weighted with Y(i
j), calculation of the ensemble average of a physical quantity is
obtained by computing the average of / Y.
Division of by Y effectively corrects for the bias introduced in
the sampling probability distribution.
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Probability-Weighted DMCProbability-weighted DMC incorporates
weighted sampling into DMC.
Steps (iii) and (iv) of the DMC algorithm are replaced by
(Step iii) Using the current component/species distribution,
prepare a probability table of all the events Ei that can take
place,
(Step iv) define the weight factor scale and compute the inverse
probability weight tableResat et al., J.Phys.Chem. B 105, 11026
(2001)for all events.Note that the stochastic simulations mentioned
here use discrete numbers of molecules, i.e. the species are
produced and consumed as whole integer units.
Therefore, the weight table w(E ) must contain only integer
values.
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Probability-Weighted DMC(Step v) Prepare the weighted
probability tableResat et al., J.Phys.Chem. B 105, 11026
(2001)(Step vi) Compute the total probability by summing the
weighted probabilities of all individual events(Step vii) Pick two
random numbers r1,r2 [0...1].
Determine which event E occurs next as before using r1.
(Step viii) Propagate the time as before using r2.
The speed-up achieved by the PW-DMC algorithm stems from the
fact that the reactions with large probabilities are allowed to
occur in bundles.
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Comparison of DMC and PW-DMCDMC is essentially a method to solve
the master equation that rules how the probabilities of the
configurations are related to each otherResat et al., J.Phys.Chem.
B 105, 11026 (2001)W : transition probability of going from
configuration to P : probability of configuration .
Using the master equation, the statistical average X of the rate
of change of the property X can be expressed as:In PW-DMC, this
relation is rearranged using the weight factor w asPW-DMC leaves
the ensemble averages unchanged.However, the fluctuations increase
with w.
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Integrated Model of Epidermal Growth Factor Receptor Trafficking
and Signal TransductionThe EGF receptor can be activated by the
binding of any one of a number of different ligands.Each ligand
stimulates a somewhat different spectrum of biological
responses.
The effect of different ligands on EGFR activity is quite
similar at a biochemical level the mechanisms responsible for their
differential effect on cellular responses are unkown.
After binding of any of its ligands, EGFR is rapidly
internalized by endocytosis.Resat et al. Biophys Journal 85, 730
(2003)
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Computational modelling of EGF receptor system(1)trafficking and
ligand-induced endocytosis(2)signaling through Ras or MAP
kinases
This work combines both aspects into a single model.
Most approaches to building computational kinetic models have
severe drawbacks when representing spatially heterogenous processes
on a cellular scale.
Review: In the traditional approach, we- formulate set of
coupled ODEs (reaction rate equations) for the time-dependent
concentration of chemical species- use integrator to propagate the
concentrations as a function of time given the rate constants and a
set of initial concentrations.Resat et al. Biophys Journal 85, 730
(2003)
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Multiple time scale problemIn Dynamic Monte Carlo, reactions are
considered events that occur with certain probabilities over set
intervals of time.
The event probabilities depend on the rate constant of the
reaction and on the number of molecules participating in the
reaction.
In many interesting natural problems, the time scales of the
events are spread over a large spectrum.Therefore it is very
inefficient to treat all processes at the time scale of the fastest
individual reaction.
In the EGFR signaling network, - receptor phosphorylation after
ligand binding occurs almost instantaneously- vesicle formation or
sorting to lysosomes requires many minutes.Resat et al. Biophys
Journal 85, 730 (2003)
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Solution to multiple time scale problemComputing millions and
billions non-correlated random numbers can become a time-consuming
process.
Resat et al. (2001) introduced Probability-Weighted DMC to
speed-up the simulation by factor 20 100.
Different processes are only tested at variant times depending
on their probabilities = very unlikely processes compute MC
decision very infrequently.Resat et al. Biophys Journal 85, 730
(2003)
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Signal transduction model of EGF receptor signaling pathwayResat
et al. Biophys Journal 85, 730 (2003)
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Species in the EGF receptor signaling modelResat et al. Biophys
Journal 85, 730 (2003)
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Receptor and ligand group definitionsResat et al. Biophys
Journal 85, 730 (2003)
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Early endosome inclusion coefficientsResat et al. Biophys
Journal 85, 730 (2003)These are adjusted to yield the
experimentally determined rates ofligand-free and ligand-bound
receptor internalization.
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Time course of phosphorylated EGF receptors(a) Total number of
phosphorylated EGF receptors in the cell. Curves represent the
number of activated receptors when the cell is stimulated with
different ligand doses at the beginning. The y axis represents the
number of receptors in thousands.
(b ) Ratio of the number of phosphorylated receptors that are
internalized to that of the phosphorylated surface receptors.
(c) Ratio of the number of internalized receptors to the number
of surface receptors. Curves are colored as: [L] = 0.2 (magenta), 1
(blue), 2 (green), and 20 (red) nM.Resat et al. Biophys Journal 85,
730 (2003)
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Distribution of the receptors among cellular compartmentsResat
et al. Biophys Journal 85, 730 (2003)
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Stimulation of EGFR signaling pathway by different
ligandsComparison of the results when the EGFR signaling pathway is
stimulated with its ligands EGF (red) and TGF- (green). (a ) Total
number of receptors in the cell as a function of time after 20 nM
ligand is added to the system. Red diamond (EGF) and green square
(TGF-) points show the experimental results.
(b) Distribution of the receptors between intravesicular
compartments and the cell membrane.
(c) Distribution of the phosphorylated receptors between
intravesicular compartments and the cell membrane. In the figures,
y axes represent the number of receptors in thousands.Resat et al.
Biophys Journal 85, 730 (2003)
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Ratio of internal/surface receptorsThe ratio of the In/Sur
ratios when the EGFR signaling pathway is stimulated with its
ligands EGF and TGF- at 20 nM ligand concentration.
Comparison of computational (solid lines) and experimental
(points) results. Ratio of the ratios for the phosphorylated (i.e.,
activated) (blue), and total (phosphorylated + unphosphorylated)
number (magenta) of receptors.Resat et al. Biophys Journal 85, 730
(2003)
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SummaryLarge-scale simulations of the kinetics of biological
signaling networks are becoming feasible.
Here, the model consisted of hundreds of distinct compartments
and ca. 13.000 reactions/events that occur on a wide
spatial-temporal range.
The exact Dynamic Monte Carlo algorithm of Gillespie (1976/1977)
was a breakthrough for simulations of stochastic systems.
Problem: simulations can become very time-consuming. In
particular if the processes occur on different time scales.
Methods like the probability-weighted DMC are promising tools
for studying complex cellular systems using molecular quanta.
Many other variants of DMC have and are being development.
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Bacterial Photosynthesis 101PhotonsLight Harvesting
Complexeslight energyelectronic excitationReaction
CentereH+pairsATPasechemical energycytochrome bc1 complexH+
gradient; transmembrane potentialubiquinoncytochrome c2electron
carriersoutsideinside
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Modelling as metabolic networkChemical reactions involved:
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Photosynthesis cycle viewlight energyelectronic
excitationeH+pairschemical energyH+ gradient,transmembrane
voltageoutsideinsideThe conversion chain: stoichiometries must
match turnovers!
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LH1 / LH2 / RC a la textbookCollecting photonsHu et al, 1998
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The Cytochrome bc1 complexthe "proton pump"
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The FoF1-ATP synthase Iat the end of the chain: producing ATP
from the H+ gradient
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The electron carriersCytochrome c: electrons from bc1 to RC heme
in a hydrophilic protein shell 3.3 nm diameterUbiquinon UQ10:
carries electronproton pairs from RC to bc1 hydrophobic tail long
(2.4 nm) isoprenoid tailtaken from Stryer
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Tubular membranes photosynthetic vesicleswhere are the bc1
complexes and the ATPase?Jungas et al., 1999
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Chromatophore vesicle: typical form in Rh. sphaeroidesLipid
vesicles3060 nm diameterH+ and cyt c insideVesicles are really
small!average chromatophore vesicle, 45 nm :surface 6300 nm
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Photon capture rate of LHCs+ Bchl extinction coeff.normalization
(Bchl = 2.3 2)relative absorption spectrumof LH1/RC and LH2sun's
spectrum at ground(total: 1 kW/m)multiplytypical growth condition:
18 W/mLH1: 16 * 3 Bchl 14 /sLH2: 10 * 3 Bchl 10 /sCogdell etal,
2003Feniouk et al, 2002Franke, Amesz, 1995Gerthsen, 1985
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LH1 / LH2 / RC nativeSiebert etal, 2004electron micrographand
density map125 * 195 , = 106Chromatophore vesicle, 45 nm :surface
6300 nm
Area per:per vesicle (45 nm)LH1 monomer(hexagonal)146 nmLH1
dimer234 nmLH2 monomer37 nmLH12 + 6 LH2456 nm11
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Photon processing rate at the RC Which process limits the RCs
turnover?Unbinding of the quinol 25 msMilano et al. 2003
+ binding, charge transfer 50 ms per quinol (estimate)
with 2e- H+ pairs per quinol 4050 /s per RC 22 QH2/s1 RC can
serve 1 LH1+ 3 LH2= 44 /sLH12 + 6 LH2 456 nm 11 LH1 dimers
including 22 RCs on one vesicle 480 Q/s can be loaded @ 18 W/m per
vesicle
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The F1F0-ATP synthase"mushroom like structures observed in AFM
images" ATPase is "visible"per turn: 1014 H+ per 3 ATP
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How much bc1?measured enzymatic activity per bc1 dimer: 84 cyt c
are reduced / s400 H+ / s per ATPase 1 dimer can process 42 QH / s
1 dimer "pumps" 168 H+ / s 11 bc1 dimers per vesiclecan be loaded
by RCs proton generation by 2.4 bc1 dimers per vesicleenough to
drive 1 ATPasex5 !!!inoutsafety first!number of bc1 complexesshould
be limited by howmany protons can bepumped out by ATPaseXiao et al.
2000
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bc1 Placement Diffusional limits?Roundtrip timesmaximal capacity
of the carriers:T = TRC + Tbc1 + TDff Cytochrome c:TRC 1 msTbc1 12
msTDiff 3 sTround-trip = 13 ms 3 cyt c per vesiclesufficient to
carry e-savailable: 22 cyt c per vesicle Quinol:TRC 50 msTbc1 23
msTDiff 1 msTround-trip = 75 ms 7 Q per vesicle sufficient to carry
e-s.available: 100 Q per vesicle poses no constraints for the
position of bc1
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Proposed setup of a chromatophore vesicleblue: small LH2 rings
(blue)
blue/red: Z-shaped LH1/RC dimers form a linear array around the
equator of the vesicle, determining the vesicles diameter by their
intrinsic curvature. At the polesgreen/red: the ATPase light blue:
the bc1 complexes
Increased proton density close to the ATPase.favors close
placing ATPase and bc1 complexes.yellow arrows: diffusion of the
protons out of the vesicle via the ATPase and to the RCs and
bc1s.Geyer, Helms, Biophys. J. (2006) 91, 921
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reconstituted LH1 dimers in planar lipid membranesUpper panel:
drawn after the AFM images of Scheuring et al (4) of LH1 dimers
reconstituted into planar lipid membranes. The Z-shaped dimers are
seen from their cytoplasmic side. The average height of the
membrane is indicated in the upper panel by the grey level of the
background. Scheuring et al measured the distances between the
minima of the AFM height scan to be 38.5 nm and the distance be
tween the lowest and the highest parts to be 3.8 nm.Lower panel:
interpretation how alternatingly oriented LH1 dimers may explain
the observed height scan indicated by the dashed lines. These
values are nicely reproduced by the proposed arrangement of the LH1
dimers, when one assumes that they are stiff enough to retain the
bending angle of 26 that they would have on a spherical vesicle of
45 nm diameter and taking into account the length of a single LH1
dimer of about 19.5 nm.
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Photosynthesis: textbook view
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Viewing the photosynthetic apparatus as a conversion chainThick
arrows : path through which the photon energy is converted into
chemical energy stored in ATP via the intermediate stages (rounded
rectangles).
Each conversion step takes place in parallely working proteins.
Their number N times the conversion rate of a single protein R
determines the total throughput of this step.
: incoming photons collected in the LHCsE : excitons in the LHCs
and in the RCeH+ electronproton pairs stored on the quinolse for
the electrons on the cytochrome c2pH : transmembrane proton
gradientH+ : protons outside of the vesicle (broken outine of the
respective reservoir).
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Modelling of internal processes at reaction centerAll individual
reactions with their individual rates k together determine the
overall conversion rate RRC of a single RC. Thick arrows : flow of
the energy from the excitons through the cyclic charge state
changes of the special pair Bchl (P) of the RC. Rounded rectangles
: reservoirs
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Parameters
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Stochastic dynamics simulations II Production runsfixed set of
parameters
integrate rate equations with:
- Gillespie algorithm (associations)
- Timer algorithm (reactions); 1 random number determines when
reaction occurs
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example: binding and e- transfer at reaction center
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Stochastic simulations of a complete vesicleModel vesicle:12
LH1/RC-monomers1-6 bc1 complexes1 ATPase
120 quinones20 cytochrome c2
integration time step: 10 s
simulating 1 minute real time requires 1.5 minute on one opteron
2.4 GHz proc
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simulate increase of light intensityduring 1 minute,light
intensity was slowly increased from 0 to 10 W/m2(quasi steady
state)
there are two regimes- one limited by available light- one
limited by bc1 throughputlow light intensity:linear increase of ATP
production with light intensityhigh light intensity:saturation is
reached the later the higher the number of bc1 complexesGeyer,
Lauck, Helms, J. Biotech (to be accepted)
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oxidation state of cytochrome c2 poollow light intensity:all 20
cytochrome c2are reduced by bc1high light intensityRCs are faster
than bc1,c2s wait for electrons
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oxidation state of cytochrome c2 poolmore bc1 complexescan load
more cytochrome c2s
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total number of produced ATPlow light intensity: any
interruption stops ATP production
high light intensity: interruptions are buffered up to 0.3 s
durationblue line:illumination
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c2 pool acts as bufferAt high light intensity, c2 pool is mainly
oxidized.
If light is turned off, bc1 can continue to work (load c2s, pump
protons, let ATPase produce ATP) until c2 pool is fully
reduced.
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SummaryChromatophore vesicles appear an ideal model system to
interface molecular simulations and ODE models.
Even though some parameters are still not known experimentally,
the available knowledge allows to construct detailed kinetic and
spatial models.
Such systems allow to test how much knowledge is required about
a particular system.
Predictions need to be tested by new experiments!Such tests are
essential before one can call a model correct.
T_{c_2} = 2\, \frac{(2R_i)^2}{6\, D_0}T_Q = 2\,
\frac{(\pi\,R_m)^2}{4\, D_Q}