Mathematical and Computational Challenges in the Biological Sciences

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Mathematical and Computational Challenges in the
Biological Sciences

• Gareth Witten
• Department of Mathematics and Applied
  Mathematics,
• University of Cape Town

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Contents

• 1. The Interface between Biology and
  Mathematics
• 2. Challenges Cellular and Molecular biology
• 3. Challenges Organismal biology
• 4. Challenges Ecology and Evolutionary biology

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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

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For Mathematics

• 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

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Explosion of biological data

• 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.

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Two further requirements

• 1. The need to integrate the information at
  different time and spatial scales.
• 2. The need for theoretical frameworks for
  approaching behaviour in
• spatially extended, hierarchical systems.

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Areas in biology devoid of mathematics

• 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.

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The Interface between Biology and 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
• Waddingtons (1957)
• idea of an epigenetic landscape
• Singularity theory
• Bifurcation of dyn. systems
• Alexander Woodcock A catastropheis 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

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Approaches have changed

• The ways in which whole fields of research are
  approached have changed.
• Examples Evolutionary genetics and
  evolutionary
• biology were fields historically concerned
• with inferring process from pattern.
• Problems of global change, biological
• diversity and sustainable development will
• require the integration of enormous data
  sets
• across disparate scales of space and time
  and
• organization
• (Lou Gross http//ecology.tiem.utk.edu/gros
  s/)

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Applications of mathematics

• 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, IBMs,
  networks, differential inclusions
• Some fundamental issues in biology appear to
  require new ways of thinking. For example,
  catastrophe theory etc

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Central Question in Science

• Classical mathematical approaches emphasised
• deterministic 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.

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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.

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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 interactionsgiven
  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.

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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
  functional
• hierarchy.               

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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?

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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.
•  

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Challenges Cellular and Molecular biology

• Structural analysis of macromolecules
• The 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
• structure
• Model 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.
•  
•  
•  
•  
•  
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•  

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Challenges Organismal Biology

• 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
  systems
• Dynamic 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.
•  

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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.
•  

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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)
•  

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Examples

• The proliferation of
• information from remote sensing etc introduces
  the need for GISs 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.

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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