Mathematical and Computational Challenges in the Biological Sciences Gareth Witten Department of Mathematics and Applied Mathematics, University of Cape.

Mathematical and Computational Challenges in the

Biological SciencesGareth Witten

Department of Mathematics and Applied Mathematics,

University of Cape Town

Contents

1. The Interface between Biology and Mathematics

2. Challenges: Cellular and Molecular biology

3. Challenges: Organismal biology

4. Challenges: Ecology and Evolutionary biology

The Interface between Biology and

Mathematics The interface between mathematics and biology presents

challenges and opportunities for both mathematicians and biologists.

For biology, the possibilities range from the level of the cell and molecule to the biosphere

“Traditional” areas: mathematical statistics, dynamical systems etc. “Non-traditional” areas: knot theory (De Wit Sumners, 1995),

interval graphs and algorithms for DDP (double digest problem) (Waterman, 1999), and differential inclusions (Aubin, 1991).

How can one deduce enzyme mechanisms from observed changes in DNA geometry

and topology

Biologists can identify intervals between sites but not the order of these intervals

For Mathematics…

Opportunities have surfaced within the last three decades because of the enormous increase in the quantity and quality of biological data due to advances in technology and the availability of powerful computing power (hardware and software) that can potentially organize the plethora of biological data.

Explosion of biological data…

Two further requirements

1. 1. The need to integrate the information at different time and spatial scales.

2. 2. The need for theoretical frameworks for approaching behaviour in

spatially extended, hierarchical systems.

There exist areas in biology that are virtually devoid of mathematical theory, and some must remain so for years to come. In these anecdotal information accumulates, awaiting the integration and insights that come from mathematical abstraction.

How is it that a homogeneous ball of cells can How can the individual characteristics of

differentiate and organize itself into one of the neurons and the neural network give rise

myriad species of living things? to thought and consciousness.

Areas in biology devoid of mathematics

In other areas, theoretical developments have run far ahead of the capability of empiricists to test ideas, developments that may capture few biological truths.

Morphogenesis Catastrophe theory

Waddington’s (1957)

idea of an epigenetic landscape

Singularity theory

Bifurcation of dyn. systems

Alexander Woodcock: “A catastrophe…is any discontinuous transition that occurs when a system can have more than one stable state, or can follow more than one stable pathway of change…”

The Interface between Biology and Mathematics

The ways in which whole fields of research are approached have changed. Examples: Evolutionary genetics and evolutionary

biology were fields historically concernedwith inferring process from pattern.

Problems of global change, biologicaldiversity and sustainable development willrequire the integration of enormous data setsacross disparate scales of space and time

andorganization(Lou Gross: http://ecology.tiem.utk.edu/~gross/ )

Biological diversity during the Cambrian explosion (530 million years ago) 

Approaches have changed

Most applications of mathematics to biology will have little effect on core areas of mathematics

Routine application of existing mathematical techniques to biological problems, for example, Lotka Volterra, Navier-Stokes, etc.

Existing mathematical techniques are inadequate and new mathematics must be developed, within conventional frameworks, for example, IBM’s, networks, differential inclusions…

Some fundamental issues in biology appear to require new ways of thinking. For example, catastrophe theory etc…

Applications of mathematics

Central Question in Science

Classical mathematical approaches emphasiseddeterministic systems of low dimensionality, and thereby swept as much stochasticity and heterogeneity as possible under the rug. New techniques and the advances in technology and advances in algorithm development has led to the development of highly detailed models in which a wide variety of components and mechanisms can be incorporated, for example, IBM models.

A central question in science:“What detail at the level of individual units is essential to understand more macroscopic regularities.”

A Challenge

Part of the problem is the use of mathematical models to represent model structures and processes are modelled as different types of mathematical objects; for example, the muscle fibre orientation is modelled by a tensor, while action potential in a cell can be modelled by solutions of differential equations.

The answers lies in the principles of dynamic organisation that are still far from clear, but that involve emergent properties that resolve the extreme complexity of gene and cellular activities into robust patterns of coherent order.

What is needed?

The reductionist approach (for eg. HGP) ignores the fact that an organism is not a thing composed of parts, but a system of interacting processes.

What is needed is a means of reconstructing the behaviour of a system from a detailed knowledge of its components and their interactions…given the baroque complexity of living systems any such reconstruction must be constructive and

computational.

Example, Organismal biology deals with all aspects of the biology of individual plants and animals, including physiology, morphology, development, and behaviour. It interfaces cellular and molecular biology at one end, and ecology at the other.

Further considerations

However, there are several problems in understanding the behaviour of a biological system even when a detailed and accurate description of its components is available:

               1. There is the sheer complexity of the system and the number of its components. 2. The components operate over radically different time scales and

spatial scales.3. The processes are occurring in a system that is spatially extended

and organized within a structural and functionalhierarchy.               

Summary

A number of fundamental mathematical issues cut across all of these challenges:

1. How can we incorporate variation among individual units in nonlinear systems?

2. How do we treat the interaction among phenomena that occur on a wide range of scales or space, time and organisational complexity?

3. What is the relation between pattern and process?

Challenges: Cellular and Molecular biology

The grand challenges at the interface between mathematics and computational and cellular and molecular biology relate to two main themes:

 1.               Genomics:

critical for sequencing human and other genomes

2.               Structural biology: structural analysis, molecular dynamic simulation, and drug design.

Challenges: Cellular and Molecular biology

Structural analysis of macromoleculesThe area of molecular geometry with visualisation has been under-represented and significant advanced are being pursued.

  How do proteins fold?     Relatively short polypeptides can have significant secondary

structureModel structures with predicted motifs are synthesised by chemical means.

Structural analysis of cells  A major goal of cell biology is to understand the cascade of events that controls the response of cells to external ligands. (eg

hormones)

  Molecular Dynamics Simulation

3-D structures as determined by x-ray crystallography and NMR are static since these techniques derive a single average structure. In nature, molecules are in continual motion.

Organismal biology deals with all aspects of the biology of individual plants and animals; including physiology, morphology, development, and behaviour. It interfaces cellular and molecular biology and ecology.

The study of complex hierarchical biological systemsDynamic aspects of structure-function relations.

Some mathematical models have illuminated problems in this area:Example: In the biomechanics of feeding aqueous organisms where solving the Navier-Stokes equation for flow through bristled appendages have shown how the geometry permits the appendages to function either as a paddle or a rake.

Challenges: Organismal Biology

Examples

Organ physiology:Solving the appropriate equations of fluid mechanics and elasticity can help

us understand the relationships between the structure of the heart and its function of providing appropriate blood flow in response to changing environmental conditions. 

  Organ morphogenesis:Includes finite element analysis of mechanical stress fields in the cellular

continuum of growing tissue; optimisation models to understand the functional significance of morphologies, and hydrodynamic models for nutrient transport in plants in plants and animals

  Demographic models to predict cell cycle duration, age distribution, and

family trees of cells in developing tissue. 

Challenges: Ecology and Evolutionary

biology

Two grand challenges:1. Global change:

Includes the relation to biodiversity and sustainable development of the biosphere as well as global changes in the carbon cycle, climate and the distribution of greenhouse gases.

2. Molecular evolution:Builds bridges between population biology and the problems of cellular and molecular biology(application of population genetic theory to molecular evolution)

Examples

The proliferation ofinformation from remote sensing etc introduces the need for GIS’s that provide a framework for classifying information, spatial statistics for analysing patterns, and dynamic simulation models that allow the integration of info across multiple scales.

Further challenges…

These challenges: aggregation of components to elucidate the behaviour of ensembles, integration across scales, and inverse problems are basic to all sciences.

The uniqueness of biological systems, shaped by evolutionary forces, will pose new difficulties, mandate new perspectives, and lead to the development of new mathematics