Abstract

Unlike physical sciences, mathematics has been less intrusive in the biological sciences because of the largely descriptive nature of the later. One of the major reasons is the apparent non-determinism and lack of invariance principles inherent in every biological system. However, with the advent of quantum theory, theory of chaos, sophisticated statistical tools and non-linear mathematical models of physical systems, an explosive synergy between the two fields with immense potential was readily discernable. Mathematics has become pervasive across all the branches of biological sciences in the recent decades. The interface between mathematics and biology has initiated and fostered newer approaches in both the fields.

Key Words: Mathematics; Biology; Explosive Synergy, Immense Potential.

Introduction

To most of our ordinary experiences the world of our conscious perceptions that surrounds us is a crooked conundrum of randomness. The world contains the seemingly non-deterministic ‘free will’ of organisms, the puzzling purple haze of consciousness, complex emotions as love, the unpredictable tsunamis uprooting lives, and the infinite alter-egos in parallel universes. The infinite nature of our mysterious universe filled with galaxies, black holes tend to force a limit to the extent things can be known. But, Man has accepted the challenge with the discovery of mathematics, the precision language in which Mother Nature speaks. Galileo once wrote: “The book of nature is written in the language of mathematics”. Platonics may even argue that the world of mathematics exists independent of our naming it. There is remarkable depth, subtlety and mathematical fruitfulness in the concepts that lie latent within any physical system. Mathematics strongly prizes rigor and precision. Mathematical fact is immutable, and successful mathematical theories have lifetimes of hundreds or thousands of years.

By contrast, most of our knowledge of biological systems is comparatively recent, and most biological theories evolve rapidly. Historically, it has been excluded from the quantitative mathematical culture.One of the major reasons is the apparent non-determinism in every biological system which superficially seems to fall out from precise mathematical linearity. However, with the advent of quantum theory, theory of chaos, and non-linear mathematical models of physical systems, an explosive synergy between the two fields with immense potential was readily discernable. The interface between mathematics and biology has initiated and fostered newer approaches in both the fields.

The Impact of Biology on Mathematics

Accomplishments of the Past

The application of mathematics to biology is not new; neither is evidence of impacts on mathematics. Robert Brown, a botanist, discovered what is now called Brownian motion while watching pollen grains in water. Today, the mathematical description of such motion is central to probability theory. The theories of dynamical systems and partial differential equations represent areas of mathematics in which numerous fruitful lines of inquiry were prompted by biological questions, and in which such influences continue to be felt. In theoretical fluid mechanics, the dominant classical stream of development was toward understanding of high Reynolds number (almost inviscid) flow and of compressible flows; biology has motivated a great many new developments in viscosity dominated flows (Purcell 1977). More recently, molecular biology has stimulated advances in analysis and low-dimensional topology and geometry.

Statistics and Stochastic Processes

Statistics is perhaps the most widely used mathematical science. It has achieved its present position as a consequence of an intellectual development begun during the 19th century. Stigler (1986) noted that from the doctrine of chances to the calculus of probabilities, from least squares to regression analysis, the advances in scientific logic that took place in statistics before 1900 were to be every bit as influential as those associated with the names of Newton and Darwin. The quantitative study of biological inheritance and evolution provided an outstanding context for statistical thinking, and quantitative genetics remains the best example of an area of science whose very theory is built out of the concepts of statistics. The great stimulus for modern statistics came from Galton's invention of the method of correlation, which he first conceived not as an abstract technique of numerical analysis, but as a statistical law of heredity (Porter 1986). The profound problems raised by Darwin's insight have led to new fields of mathematical science. Likewise, problems in eugenics and plant breeding were the motivation for Fisher's statistical work (Fisher 1930). The analysis of variance and the theory of experimental design were developed to interpret and plan plant breeding experiments at the Experimental Station at Rothamsted, an institution that continues to be a major influence on statistical theory and practice. The benefits to mankind of these and later biometrical developments have been enormous. The "Green revolution" in agriculture would have been quite impossible without these tools.

The influence of biology on probability theory and statistics has been equally strong in later years of this century. Neyman, Park and Scott (1956) developed stochastic models in order to interpret experiments of Park on flour beetles. In these experiments, two competing species of beetles were pitted in competition. To Park's surprise, the outcome of a given experiment could not be predicted; but in a long series of experiments, the statistical distribution of outcomes was predictable.

Population Models

The study of simple population models provides a classic example of stimulation of mathematics by biology with resulting benefits to both. For example, iterations of a single nonlinear function, described via a population model of a simple kind, capture the dynamics of an isolated population with discrete generations, subject to influences that regulate the population numbers exclusively through the population size. More explicitly, the population size at generation (n+1) is assumed to be a given nonlinear function of the population size at generation (n). Models of this type were introduced in population studies a long time ago. However, it was only in the 1970's that a widespread appreciation for the depth and beauty of the mathematical phenomena involved in these mathematical problems emerged. The motivation from population biology was an important part of the chain of historical events that led to very significant scientific and mathematical discoveries.

Nonlinear Partial Differential and Functional Equations

Nonlinear partial differential and functional equations traditionally have been applied in the physical sciences. But several examples highlight the seminal impact of biological ideas on mathematical research in this area. The theory of diffusion, which describes the behavior of a population of randomly moving particles or molecules, exemplifies an area traditionally viewed within the context of chemistry or physics. However, the mathematics of nonlinear diffusion equations has received much of its impetus from biology. Fisher's (1937) interest in the problem of the spread of advantageous genes in a population stimulated his consideration of an equation that incorporates diffusion augmented by a simple ("logistic") nonlinear growth term. It was treated simultaneously by Kolmogorov et al. (1937), who proved the existence of a stable traveling wave of fixed velocity representing a wave of advance of the advantageous gene.

These problems will continue to be a fertile area of mathematical research since current mathematical and numerical approaches are only partially adequate for addressing these issues.

The Impact of Mathematics on Biology

Impact on Cellular and Molecular Biology

The application of mathematics to cellular and molecular biology is so pervasive that it often goes unnoticed. Biological complexity derives from the fact that biological systems are multifactorial and dynamic. Quantitative research in these fields is based upon a wide variety of laboratory techniques, with gel electrophoresis and enzyme-based assays among the most common. Measurements include activity, molecular weight, diameters, and size in bases and with all these an understanding of the accuracy, precision, sources of variation, calibration, etc. In short, the quality of the measurement process is of central significance.

The determination of the dynamic properties of cells and enzymes, expressed in the form of enzyme kinetic measurements or receptor-ligand binding are based on mathematical concepts that form the core of quantitative biochemistry. Molecular biology itself can trace its origins to the infusion of physical scientists into biology with the inevitable infusion of mathematical tools. The utility of the core tools of molecular biology was validated through mathematical analysis. Examples include the quantitative estimates of viral titers, measurement of recombination and mutation rates, the statistical validation of radioactive decay measurements, and the quantitative measurement of genome size and informational content based on DNA (i.e., base sequence) complexity.

While the experimentalist strives to isolate single variables in order to make statistically significant measurements, many systems are not amenable to such single factor examination. Therefore, mathematically based computational models are essential to meaningful analyses.

DNA Structure

Differential geometry is the branch of mathematics that applies the methods of differential calculus to study the differential invariants of manifolds. Topology is the mathematical study of shape. It defines and quantizes properties of space that remain invariant under deformation. These two fields have been used extensively to characterize many of the basic physical and chemical properties of DNA. Specific examples of particular note follow.

The recent review of Dickerson (1989) summarizes how geometric concepts of tilt, roll, shear, propeller twist, etc. have been used to describe the secondary structure of DNA (i.e., the actual helical stacking of the bases that forms a linear segment of DNA). In addition, these concepts can be used to describe the interaction of DNA with ligands such as intercalating drugs (Wang et al. 1983).

Macromolecular Sequences

DNA sequences are collected in the GenBank database, and protein sequences are collected in the Protein Identification Resource (PIR). When a new DNA sequence is determined, GenBank is searched for approximate similarities with the new sequence. Translations of the DNA sequence into the corresponding amino acid sequence are used to search the protein database. Sensitive search methods require time and space proportional to the product of the sequences being compared. Searching GenBank (now more than 40 x 106 bases) with a 5000 bp sequence requires time proportional to 2 x 1011 with traditional search techniques. Lipman and Pearson (1985) have developed techniques that greatly reduce the time needed. Using their techniques, one can screen the databases routinely with new sequences on IBM PCs, for example. These methods rapidly locate diagonals where possible similarities might lie and then perform more sensitive alignments. This family of programs, FASTA, FASTN, etc., are the most widely used sequence analysis programs and have accounted for many important discoveries. An example of the impact of such analysis is the unexpected homology between an oncogene and a growth factor. This discovery became the basis of the molecular theory of carcinogenesis.

Apart from screening databases, a lot of questions remain to be answered on DNA sequences. Are sequences descended from a common ancestral sequence? Do they serve similar functions? One problem has been to calculate the probability of a long matching region between two DNA sequences, where some level of dependence occurs as a result of overlapping regions. Strong limit laws have been established that give rates for the longest matching sequences between different sequences (with a given proportion of mismatches) as the length of the sequences increases. Detailed distributional behavior has been obtained using the Chen-Stein method of approximation by a Poisson random variable. These new distributional results are now used as a basis for statistical tests. Arratia et al. (1990) contains a snapshot of current mathematical work on these questions. Relevant statistical questions include the calculation of Markov-type probabilities and likelihoods over directed graphs; maximum likelihood estimation for multinomials with highly non-regular parameter spaces involving large numbers of nuisance parameters; model selection from among large numbers of hypotheses of the same dimension and selection among small numbers of non-nested hypotheses of different dimension.

Model Based Approach for Biological Sequence and Networks

Mathematical model based approach to unravel the mysteries of complex biological sequences and cellular networks have revolutionized our understanding. The recent probabilistic graphical models have immense potential for understanding the interaction and regulation of biological system. The computational procedure for reasoning in graphical models is derived from the basic principles of probability theory. One major application of such models is taking place in genetic clusters and expression analysis. The key advantage of such models is the precision with which the actual regulation-interaction within the biological system can be captured allowing for sufficient flexibility to fit in heterogeneities (Friedman, 2004).

Genetic Mapping

Genetic mapping deals with the inheritance of certain "genetic markers" within the pedigree of families. These markers might be genes, sequences associated with genetic disease, or arbitrary probes determined to be of significance (e.g., Restriction Fragment Length Polymorphism [RFLP] probes). The sequence of such markers and probabilistic distance (measured in centiMorgans) along the genome can often be determined by hybridizing each family member's genome against the predetermined probes. In essence, the genetic map most likely to produce the observed data is constructed. The availability of complete genome sequences facilitates the development of high throughput assays that can probe cells at a genome wide scale. Such assays can measure molecular networks and their components at multiple levels.

Cell Motility

Cells can move, monitor changes in their environment, and respond by migrating towards more favorable regions. It is a remarkable fact that a bacterial flagellum is driven at its base by a reversible rotary motor powered by a transmembrane proton flux, and analysis of models for this device has been prolific. The study of bacterial chemotaxis (the migration of bacteria in chemical gradients) has been particularly rewarding, in part because organisms such as Escherichia coli are readily amenable to genetic and biochemical manipulation, and in part because their behavior is closely tied to the constraints imposed by motion at low Reynolds number and by diffusion (of both the cell and the chemoattractant). Mathematics has helped us learn how a cell moves (Dembo 1989), how it counts molecules in its environment (Berg and Purcell 1977), and how it uses this information (Berg 1988). It also has made it possible to relate the macroscopic behavior of cell populations to the microscopic behavior of individual cells (Rivero et al. 1989).

Structural Biology

Mathematics has made perhaps its most important contribution to cellular and molecular biology in the area of structural biology. This area is at the interface of three disciplines biology, mathematics and physics. Success in this field has involved the use of sophisticated physical methods to determine the structures of biologically important macromolecules, their assembly into specialized particles and organelles, and even at higher levels of organization more recently. A wide array of methods has been employed. X-ray crystallography and Nuclear Magnetic Resonance Spectroscopy (NMR) have allowed us to study each individual molecule of interest. With remarkable accuracy NMR can differentiate between malignant and inflammatory areas of brain and also can quantitatively tell about each hypothesized molecules (above a critical molecular weight) in a particular disease. This has opened up new vistas in the field of medicine.

The beneficial aspects of Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) scans have already been felt. Moreover, these methods have become the preferred tool for studying brain structure in various diseases.

Mathematics plays three roles. First, computational methods lie at the heart of these techniques because a large amount of information about local areas or short distances is encrypted in the raw data, and it is a major computational task to deduce a structure. Second, new mathematical methods of analysis are continually being developed to improve ways of determining the structure. Third, increasingly sophisticated computer graphics have been developed in response to the need to display and interpret such structure.

Molecular Dynamics Simulation

Three-dimensional 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; it is this motion that allows them to function (a static molecule is as functional as a static automobile). Mathematical and computational methods have been able to complement experimental structural biology by adding the motion to molecular structure. These techniques have been able to bring molecules to life in a most realistic manner, reproducing experimental data of a wide range of structural, energetic and kinetic properties. At present, some of the most extensive molecular dynamics simulations have been used to study proteins and segments of DNA in solution.

Drug Design

One of the most interesting and potentially useful molecular interactions concerns drugs that bind with very high affinities to protein and nucleic acid macromolecules and either block the normal function of the macromolecule or mimic other ligands for such structures as receptors and induce a normal physiological response. Inhibition can be advantageous if the protein is made in excess, or if normal cellular control of the protein's activity has been lost. Because drug binding involves spatial complementarity, and because the aim is to design a molecule that binds with the highest affinity possible, it should be possible to use the three-dimensional structure to aid design. Current work in this area has followed several directions. The most direct approach is to crystallize the protein together with the drug. Study of the structure of the complex can suggest modifications to the drug expected to enhance its affinity for the receptor or enzyme active site. For this method to work, one needs an initial drug known to bind to the protein. Other methods aim to circumvent this requirement by deducing the structure of the drug directly from the structure of the protein. While these methods are able to suggest completely new drug molecules, they involve a search for structures that fit a binding site. The theoretical underpinnings of such searches require further theoretical development. More specifically, they would benefit from application of better methods in global optimization and graph theory.

The Impact of Mathematics on Organismal Biology

Organismal (sometimes called organismic) biology deals with all aspects of the biology of individual animals and plants, including physiology, morphology, development, and behavior. Mathematical theorists have made signal contributions to organismal biology. Examples range from technological advances to theories of biological structure and function, and rely on a wide range of mathematical techniques.

Mathematics and Cardiovascular Functions

One of the most exciting areas of applications of mathematics has been to cardiac function. The outstanding milestone in early history of biological quantization was the work of William Harvey in the early 17th century. Harvey’s demonstration that the blood circulates was the pivotal founding event of the modern interaction between mathematics and biology (Cohen, 2004).

A major cause of death from malfunction of the heart is the phenomenon called ventricular fibrillation, wherein properly coordinated heart action is replaced by purposeless local oscillations of the ventricles. Mathematical modeling has revealed why this phenomenon occurs. Major experimental efforts have been suggested by the modeling.

In related work, powerful numerical algorithms and state-of-the-art computing have been applied by Peskin and McQueen (1989) to study blood flow in the heart. A computational method has been introduced to solve the coupled equations of motion of the muscular heart walls, the elastic heart valve leaflets, and the viscous incompressible blood that flows in the cardiac chambers. Variants of this method have been applied to other problems in bio-fluid dynamics, including platelet aggregation during blood clotting, aquatic animal locomotion, and wave propagation along the basilar membrane of the inner ear.

Mathematics and Skeleto-muscular Functions

Another major contribution of mathematics to physiology is the theory of cross-bridge dynamics in striated muscle. Introduced by Huxley (1957) and further developed by Podolsky and others, this theory not only has provided a satisfying explanation of the mechanical behavior of muscle, but it also has served to provide organizing principles for biochemical research on the fundamental energetic and control mechanisms of muscle contraction.

Mathematics and Morphogenesis of Organs

Mathematical methods for the quantitative description of morphogenesis of organs composed of nonmigrating cells (including plants, animal bone and skin, and shells) were suggested by Richards and Kavanagh (1943) and by Erickson and Sax (1956). These methods, which involve evaluation of velocity gradients from empirical data, have provided the phenomenological basis for understanding the physiology of growth.

Mathematics and Immunology

Modeling the immune system requires the same type of hierarchical approach as does neurobiological modeling. New ideas and mathematical representations are required to handle systems with large numbers of constantly changing components. Some promising approaches involve the formulation of models in terms of a potentially infinite dimensional "shape space," wherein emphasis is placed on determining interactions among molecules based on their shapes. In computer models binary strings have been used to represent molecular shape, with the obvious advantage of fast algorithms to determine complementarity and the ability to represent 4 x 109 different molecular shapes with 32 bits (Farmer et al., 1986). To handle the perpetual novelty that the elimination of old components and the generation of new components introduces into the immune system, models can be formulated using "metadynamical" rules, wherein an algorithm is used to update the dynamical equations of the model depending upon the components present in the system at the time of update (Bagley et al. 1989).

Mathematics and Neuroscience

The Hodgkin and Huxley Model

A famous contribution in this area is the theoretical model made by Hodgkin and Huxley (1952) of the electrical signals in the squid axon. This Nobel prize-winning work incorporated the findings of a series of brilliant experiments concerning the ion permeability of the axonal membrane into a set of mathematical equations that predicted the shape and speed of the "action potential" wave that moves down the axon. Patch clamp recordings now permit investigators to relate the Hodgkin-Huxley membrane models to the opening and closing of the molecular channels that span the membrane and are responsible for their ionic conductance. Hodgkin and Huxley's inferences from macroscopic current measurements have been confirmed in basic form, but greatly expanded with respect to their descriptions of configurations and transition mechanisms. In recent years the work of Hodgkin and Huxley has found unexpected application in non-neural systems in which electrophysiology plays a surprising regulatory role. One example of this is the control of insulin secretion by the electrically active beta cells of the pancreas.

Functional Neuroimaging and Electrophysiology: A Window to Study Brain Function

Evolving functional brain-imaging techniques nowadays represent the most powerful tools for characterizing in vivo human anatomy, neurophysiology and neurochemistry at modest temporal and spatial resolution. Positron Emission Tomography (PET), Single Photon Emission Computed Tomography (SPECT), functional MRI (fMRI) scans and Quantitative EEG (QEEG) have caused a quantum leap in the understanding of cognitive processes.

The whole neural network, according to the current understanding, behaves as a complex system (having many degrees of freedom) and probably follows non-linear chaotic dynamics (Nunez, 1995). Complex linear and non-linear mathematical models are increasingly applied in cognitive neuroscience research for understanding the brain-behavior relationship.

Mind, Brain and Mathematics

A Digression in Philosophy

Cartesian substance dualism pictures the world as constituting of two independent domains, the mental and the material, each with its distinct defining properties (consciousness and spatial extendedness respectively). There is causal interaction between the two but they are ontologically independent of each other and it is metaphysically possible for one domain to exist in the total absence of the other (Kim, 2000). However, from scientific point of view, to think of a dualistic ‘mind’ that is external to the body and influencing our choices may be arguably unreasonable. It can be argued that if ‘will’ could somehow influence Nature’s choice of alternatives then why an experimenter cannot influence the result of a quantum experiment via his ‘will power’ (Penrose, 1994). Over time the dualistic viewpoint has been replaced by the more familiar multilayered system model that views the world to be hierarchically organized into various ‘level’s. This is essentially a grand synthesis of diverse elements into an integrated hierarchical system which allows us to formulate testable hypothesis of a particular ‘function’ at one level by another one. However, the system of organization, as limited by our current understanding, is fundamentally a bottom-up perspective.

If it is possible to explain the behavior of ions with the help of physics, there is no reason that the same principle will not be effective to explain as complex phenomenon as human consciousness (Given that both are part of the same complex system, rejecting dualism).

Consciousness: Does Classical Physics Has Got the Answer?

By now we know that computational algorithms play a vital role in modeling biological system. But, when it comes to the riddle of consciousness it surprisingly transcends the deterministic computational algorithms of classical physics (Penrose, 1994). Apparently it seems appropriate that nerve signals themselves are things that can be treated in a classical way. As far back in 1949, Donal Hebb suggested a simple rule of this kind. Modern connectionist models have modified the Hebbian procedures and an operational algorithm was made for neuronal networks. Scientists like Edelman, drawing upon Darwinian principles, further advanced the understanding.

If the synaptic connections and their strengths are kept fixed, then the way in which a neuron’s firing affects the next one can be treated classically. The action of brain consisting of such neuronal network would then be effectively simulated by computational neuronal network algorithms. The much debated ‘Artificial Intelligence’ can be created then. But it seems extremely improbable (and possibly impossible) that such a scheme ever can model human consciousness. The barriers are inherent randomness of the information flow across brain, the time-to- time variability of ‘synaptic talk’, and the phenomenon of synaptic plasticity which changes the strength of synapses. Therefore, with the added non-computability, consciousness could probably never be understood from the point of view of classical physics.

Quantum Theory and Consciousness

Scientists today believe that quantum processes could underlie the phenomenon of consciousness. There would be no exaggeration to say that this is the most complicated and most challenging mysteries of Nature that mathematics is trying to unravel. In Quantum Physics, matter is ultimately not a solid substance and there can be complex number weighted simultaneous coexistence of infinite states of a same particle (Penrose, 1994). The problem of large-scale quantum coherence in as ‘hot’ a structure as brain was resolved when Bose-Einstein condensation was shown to be likely even at body temperatures in living matter. Thus our brain "instantiates" not one but two systems: a classical one and a quantum one.

It is known that microtubules inside the neurons control the function of synapses. The membrane of microtubules has been postulated to effectively protect the quantum processes from getting lost to the environmental randomness (the wave-function collapse). Thus, consciousness is a manifestation of the quantum cytoskeletal state. Consciousness may arise from the "excitation" of such a Bose-Einstein condensate. Whenever the condensate is excited by an electrical field, conscious experience occurs.

Life and consciousness are ultimately due to the mathematical properties of the quantum wave function. The wave aspect of nature yields the mental; the particle aspect of nature yields the material.

The Way Forward...

Biological Challenges that Could Stimulate Innovations in Mathematics

• Many current and future challenges for statistics and probability that are motivated by questions in molecular biology, genetics, and molecular evolution will require new techniques and theories. One such set of challenges involves the use of DNA sequence data to reconstruct phylogenetic trees, analyze genetically complex traits precisely, and study other problems. This is particularly important after the completion of the Genomic Project.

• The complex networks of gene interactions, proteins, and signaling between cells and the abiotic environment is probably incomprehensible without innovative mathematical structure.

• Understanding Brain-behavior relationship is a complex system problem which might require modification in the current mathematical approaches.

• Monitor living systems to detect large deviations such as natural or induced epidemics or physiological or ecological pathologies (Weinstein et al., 1997).

Mathematical Challenges that Would Contribute to Progress in Biology

• Appropriate statistical tools need to be developed to model multi-level biological data. It is clear now that no single model will meet all the analysis needs. To deal with the complexity of biological processes, computational biology must develop methodology to explore different models with varying details and rapidly apply them to diverse data sets. The language of graphical models is well suited for composing different sub-models in a principled and understandable fashion.

• Understand probability, risk, and uncertainty:Despite three centuries of great progress, we are still at the very beginning of a true understanding. There is a need to synthesize existing theories or invent newer innovative approaches.

• There is a need to set standards for clarity, performance, publication and permanence of software and computational results.

• Probably the ultimate understanding of biological systems (as well as Nature) depends upon the successful unification of Quantum theory with classical physics or modification of both.

Conclusion

Unlike physical sciences, mathematics has been less intrusive in the biological sciences because of the largely descriptive nature of the later, lacking the invariance principles and fundamental natural constants of physics. However, in the recent decades mathematics has become pervasive across all the branches of biological sciences, allowing for healthy interaction between the two. There is immense scope for understanding the fundamental rules of Nature when mathematics marries biology. The coming decades will see enormous growth of mathematical application in biological sciences, providing biologists the ‘extra-sense’ which Darwin once longed for.

References