Transcript Mathematical and Computational Challenges in the

Mathematical and Computational
Challenges in the Biological Sciences
Gareth Witten
Department of Mathematics and Applied Mathematics,
University of Cape Town
Contents
1. The Interface between Biology and
Mathematics
Challenges: Cellular and Molecular biology
3. Challenges: Organismal biology
4. Challenges: Ecology and Evolutionary biology
2.
The Interface between Biology and Mathematics
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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
For Mathematics…
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“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
Explosion of biological data…
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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.
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.
Areas in biology devoid of mathematics
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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
differentiate and organize itself into one of the
myriad species of living things?
How can the individual characteristics of
neurons and the neural network give rise
to thought and consciousness.
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
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…”
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/~gross/)
Biological diversity during the Cambrian explosion (530 million years ago)
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, IBM’s,
networks, differential inclusions…
Some fundamental issues in biology appear to require new ways of
thinking. For example, catastrophe theory etc…
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.”
A Challenge
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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.
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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.
2.
3.
There is the sheer complexity of the system and the number of its
components.
The components operate over radically different time scales and
spatial scales.
The processes are occurring in a system that is spatially extended
and organized within a structural and functional
hierarchy.
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
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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.
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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)
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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.
Challenges: Organismal Biology
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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.
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Some mathematical models have illuminated problems in this area:
Example: In the biomechanics of feeding aqueous organisms where solving the
NavierStokes equation for flow through bristled appendages have shown how the geometry permits
the appendages to function either as a paddle or a rake.
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.
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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
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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
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The proliferation of
information 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…
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These challenges: aggregation of components to
elucidate the behaviour of ensembles, integration
across scales, and inverse problems are basic to all
sciences.
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The uniqueness of biological systems, shaped by
evolutionary forces, will pose new difficulties, mandate
new perspectives, and lead to the development of new
mathematics