VII.2 Mathematical Biology Michael C. Reed
VII.2.Mathematical Biologyuses all available information; it is related to the problem of reconstructing a 3D structure from a 2D projection. The operation has been fully described and is nowavailable in Mathematica.The generalized inverse also enables one to handleredundant axes in quasicrystals, but usually the interesting problems are nonlinear. Other inverse problemsinclude the following.(i) Finding the arrangement of atoms that gives riseto the observed scattering patterns of X-rays orelectrons from a crystal.(ii) Reconstructing a 3D image from 2D projections inmicroscopy or X-ray tomography.(iii) Reconstructing the geometry of a molecule givenprobable interatomic distances (and perhaps bondangles and torsion angles).(iv) Finding the way in which a protein molecule foldsto give an active site, given the sequence of constituent amino acids.(v) Finding the pathway to producing a molecule synthetically, given that it occurs in nature.(vi) Finding the sequence of rules that generate a membrane or a plant or another biological object, giventhat it takes a certain shape.Some questions of this type do not have unique answers. For example, the classic question as to whetherthe shape of a drumhead can be determined from itsvibration spectrum (can you hear the shape of a drum?)has been answered in the negative: two vibrating membranes with different shapes may have the same spectrum. It was thought that this ambiguity might also bethe case for crystal structures. Linus Pauling suggestedthat there might be two different crystal structures thatwere homometric (that is, giving the same diffractionpattern), but no definite example has been found.5ConclusionAs the examples in this article show, mathematics andchemistry have a symbiotic relationship, with developments in one often stimulating advances in the other.Many interesting problems, including several that wehave mentioned here, are still waiting to be solved.Further ReadingCotton, F. A. 1990. Chemical Applications of Group Theory.New York: Wiley Interscience.Hollas, J. M. 2003. Modern Spectroscopy. New York: JohnWiley.837Hyde, S., S. Andersson, K. Larsson, Z. Blum, T. Landh, S.Lidin, and B. W. Ninham. 1997. The Language of Shape.The Role of Curvature in Condensed Matter: Physics,Chemistry and Biology. Amsterdam: Elsevier.Parr, R. G., and W. Yang. 1989. Density-Functional Theory ofAtoms and Molecules. Oxford: Oxford University Press.Thomas, J. M. 2003. Poetic suggestion in chemical science.Nova Acta Leopoldina NF 88:109–39.Wales, D. J. 2004. Energy Landscapes. Cambridge: Cambridge University Press.Wells, A. F. 1984. Structural Inorganic Chemistry. Oxford:Oxford University Press.Wolfram, S. 2002. A New Kind of Science. Champaign, IL:Wolfram Media.VII.2 Mathematical BiologyMichael C. Reed1IntroductionMathematical biology is an extremely large and diversefield. It studies objects ranging from molecules to global ecosystems and the mathematical methods comefrom many of the subdisciplines of the mathematicalsciences: ordinary and partial differential equations,probability theory, numerical analysis, control theory,graph theory, combinatorics, geometry, computer science, and statistics. The most that one short article cando is to illustrate by selected examples this diversityand the range of new mathematical questions that arisenaturally in the biological sciences.2How Do Cells Work?From the simplest point of view, cells are large biochemical factories that take inputs and manufacturelots of intermediate products and outputs. For example, when a cell divides, its DNA must be copiedand that requires the biochemical synthesis of largenumbers of adenine, cytosine, guanine, and thyminemolecules. Biochemical reactions are usually catalyzedby enzymes, proteins that facilitate a reaction but arenot used up by it. Consider, for example, a reaction inwhich chemical A is converted to chemical B with thehelp of an enzyme E. If a(t) and b(t) are the respective concentrations of A and B at time t, then one typically writes down a differential equation for b(t), whichtakes the formb (t) f (a, b, E) · · · · · · .Here, f is the rate of production, which typicallydepends on a, b, and E. Of course B may be produced
838by other reactions (which would lead to additional positive terms · · · ) and may be used as a substrate itselfin still other reactions (which would lead to additionalnegative terms · · · ). So, given a particular cell function or biochemical pathway, we can just write downthe appropriate set of nonlinear coupled ordinary differential equations for the chemical concentrations andsolve it by hand or by machine computation. However,this straightforward approach is often unsuccessful.First of all, there are a lot of parameters (and variables)in these equations and measuring them in the contextof real living cells is difficult. Second, different cellsbehave differently and may have different functions, sowe would expect the parameters to be different. Third,cells are alive and change what they are doing, so theparameters may themselves be functions of time. Butthe greatest difficulty is that the particular pathwayunder study is not really isolated. Rather, it is embedded in a much larger system. How do we know that ourmodel system will continue to behave in the same waywhen embedded in this larger context? We need newtheorems in dynamical systems that answer questionssuch as this, not for general “complex systems” but forthe particular kinds of complex systems that arise inimportant biological problems.Cells continue to accomplish many basic tasks eventhough their environments (i.e., their inputs) are constantly changing. A brief example of this phenomenon,which is known as homeostasis, will illustrate the problem of “context.” Let us suppose that the chemical reaction above is one step in the pathway for making thethymines necessary for cell division. If the cell is a cancer cell, we would like to turn off this pathway, and areasonable way to try to do this would be to put into thecell a compound X that binds to E, thereby reducing theamount of free enzyme available to make the reactionrun. Two homeostatic mechanisms immediately comeinto play. First, a typical reaction is inhibited by itsproduct: that is, f decreases as b increases. This makesbiological sense because it ensures that B is not overproduced. So, when the amount of free E is reduced andthe rate f declines, the resulting decrease in b drivesthe rate up again. Second, if the rate f is lower thanusual, the concentration a typically rises since A is notbeing used up as quickly, which also drives the ratef up again since f increases as a increases. Given thenetwork in which A and B are embedded, one can imagine calculating how much f will drop if we put a certain amount of X into the cell. In fact, f may drop evenless than we calculate because of another homeostaticVII. The Influence of Mathematicsmechanism that is not even in our network. The enzymeE is a protein produced by the cell via instructions froma gene. It turns out that sometimes the concentrationof free E inhibits the messenger RNA that codes forthe production of E itself. Then, if we introduce X andreduce free E, the inhibition is removed and the cellautomatically increases its rate of production of E, thusraising the amount of free E and with it raising thereaction rate f .This illustrates a fundamental difficulty in studying cell biochemistry, indeed a difficulty in studyingmany biological systems. These systems are very largeand very complex. To gain understanding, it is naturalto concentrate on particular relatively simple subsystems. But one always has to be aware that the subsystems exist in a larger context that may contain variables (excluded by the simplification) that are crucialfor understanding the behavior and biological functionof the subsystem itself.Although cells exhibit remarkable homeostasis, theyalso undergo spectacular changes. For example, celldivision requires unzipping of the DNA, synthesis oftwo new complementary strands, the movement apartof the two new DNAs, and the pinching off of themother cell to produce two daughters. How does a celldo all this? In the case of yeast cells, which are comparatively simple, the actions of the biochemical pathwaysare quite well understood, partly because of the mathematical work of John Tyson. But as our brief discussionmakes clear, biochemistry is not all there is to cell division; an important additional feature is motion. Materials are being transported all the time throughout cellsfrom one specific place to another (so their motion isnot just diffusion), and indeed, cells themselves move.How does this happen? The answer is that materialsare transported by special molecules called molecularmotors that turn the energy of chemical bonds intomechanical force. Since bonds are formed and brokenstochastically (that is, some randomness is involved),the study of molecular motors leads naturally to newquestions in stochastic ordinary and partial differential equations [IV.24]. A good introduction tothe mathematics of cell biology is Fall et al. (2002).3GenomicsTo understand the mathematics that was involved insequencing the human genome it is useful to start withthe following simple question. Suppose that we cut up aline segment into smaller segments and are presented
VII.2.Mathematical Biologywith the pieces. If we are told the order in which thepieces came in the original segment, then we can putthem back together and reconstruct the segment. Ingeneral, since there are many possible orders, we cannot reconstruct the segment without extra information of this kind. Now suppose that we have cut upthe segment in two different ways. Think of the linesegment as an interval I of real numbers, and let thepieces be A1 , A2 , . . . , Ar when you cut it up the firstway, and B1 , B2 , . . . , Bs when you cut it up the otherway. That is, the sets Ai form a partition of the interval I into subintervals, and the sets Bj form anotherpartition. For simplicity, assume that no Ai shares anendpoint with any Bj , except for the two endpoints ofI itself.Suppose that we know nothing about the order inwhich the pieces Ai and Bj come in I. In fact, supposethat all we know about them is which Ai overlap withwhich Bj : that is, which of the intersections Ai Bj arenonempty. Can we use this information to work out theoriginal order of the pieces Ai and thereby reconstructthe interval I (or its reflection)? The answer will sometimes be yes and sometimes no. If it is yes, then wewould like to find an efficient algorithm for doing thereconstruction, and if it is no, then we would like toknow how many different reconstructions are consistent with the given information. This so-called restriction mapping problem is really a problem in graphtheory [III.34]: the vertices of the graph correspondto the sets Ai or Bj , and there is an edge between Aiand Bj if Ai Bj .A second problem is whether we can find the originalorder of the Ai (or the Bj ) if what we are told is thelength of each set Ai and each set Bj , and the set ofall the lengths of the intersections Ai Bj . The catch isthat we are not told which length corresponds to whichintersection. This is called the double digest problem.Again one would like to be able to tell when there is onlyone solution, or to place an upper bound on the numberof possible reconstructions if there is more than one.Human DNA is, for our purposes here, a word oflength approximately 3 109 over a four-letter alphabet A, G, C, T. That is, it is a sequence of length 3 109in which each entry is A, G, C, or T. In the cell, theword is bound letter by letter to the “complementary”word, which is determined by the rule that A can only bebound to T, and C can only be bound to G. (For example,if the word is ATTGATCCTG, then the complementaryword is TAACTAGGAC.) In this brief discussion we willignore the complementary word.839Since DNA is so long (it would be approximately twometers if one stretched it out into a straight line) it isvery hard to handle experimentally, but the sequenceof letters in short segments of approximately five hundred letters can be determined by a process calledgel chromatography. There are enzymes that cut DNAwherever specific very short sequences occur. So ifwe digest a DNA molecule with one of these enzymesand digest another copy with a different enzyme, wecan hope to determine which fragments from the firstdigestion overlap fragments from the second digestionand then use techniques from the restriction mappingproblem to reconstruct the original DNA molecule. Theinterval I corresponds to the whole DNA word, and thesets Ai to the fragments. This involves sequencing andcomparing the fragments, which has its own difficulties. However, lengths of fragments are not so hardto determine, so another possibility is to digest withthe first enzyme and measure lengths, digest with thesecond and measure lengths, and finally digest withboth and measure lengths. If one does this, then theproblem one obtains is essentially the double digestproblem.To completely reconstruct the DNA word one takesmany copies of the word, digests with enzymes, andselects at random enough fragments that together theyhave a high probability of covering the word. Eachof the fragments is cloned, in order to get enoughmass, and then sequenced by gel chromatography. Bothprocesses can introduce errors, so one is left with avery large number of sequenced fragments with knownerror rates for the letters. These need to be comparedto see if they overlap: that is, to see if the sequencenear the end of one fragment is the same as (or verysimilar to) the sequence at the beginning of another.This alignment problem is itself difficult because of thelarge number of possibilities involved. So, in the end wehave a very large restriction mapping problem exceptthat we can only say that given fragments overlap withprobabilities that are themselves hard to estimate. Afurther difficulty is that DNA tends to have large blocksthat repeat in different parts of the word. As a result ofthese complications, the problem is much harder thanthe restriction mapping problem described earlier. Itis clear that graph theory, combinatorics, probabilitytheory, statistics, and the design of algorithms all playcentral roles in sequencing a genome.Sequence alignment is important in other problemsas well. In phylogenetics (see below) one would likea way of saying how similar two genes or genomes
840VII. The Influence of Mathematicsare. When studying proteins, one can sometimes predict protein three-dimensional structure by searchingdatabases for known proteins with the most similaramino acid sequence. To illustrate how complex theseproblems are, consider a sequence {ai }1000i 1 of onethousand letters from our four-letter alphabet. We wishto say how similar it is to another sequence {bi }1000i 1 .Naively, one could just compare ai with bi and define a metric [III.56] like d({ai }, {bi }) δ(ai , bi ). However, DNA sequences have evolved typically by insertions and deletions as well as by substitutions. Thusif the sequence ACACAC · · · lost its first C to becomeAACAC · · · , the two sequences would be very far apartin this metric even though they are very similar andrelated in a simple way. The way around this difficultyis to allow sequences to include a fifth symbol, –, whichstands for the place of a deletion or a place opposite aninsertion. Thus, given two sequences (of perhaps different lengths), we wish to find how they can be augmented with dashes to give the minimum possible distance between them. A little thought will convince thereader that it is not feasible to use a brute-force searchfor a problem like this, even for the fastest computers—there are so many potential augmentations that thesearch would take far too long. Serious and thoughtful algorithm development is required. Two excellentintroductions to the material discussed in this sectionare Waterman (1995) and Pevzner (2000).4Correlation and CausalityThe central dogma of molecular biology is DNA RNA proteins. That is, information is stored in DNA,it is transferred out of the nucleus by RNA, and the RNAis then used in the cell to make proteins that carry outthe work of the cell through the metabolic processesdiscussed in section 2. Thus DNA directs the life of thecell. Like most things in biology, the true situation ismuch more complicated. Genes, which are segments ofDNA that code for the manufacture of particular proteins, are sometimes turned on and sometimes turnedoff. Usually, they are partially turned on; that is, theprotein they code for is manufactured at some intermediate rate. This rate is controlled by the binding (orlack of binding) of small molecules or specific proteinsto the gene, or to the RNA that the gene codes for. Thusgenes can produce proteins that inhibit (or excite) othergenes; this called a gene network.In a way, this was obvious all along. If cells canrespond to their environments by changing what theydo, they must be able to sense the environment andsignal the DNA to change the protein content of thecell. Thus, while sequencing DNA and understandingspecific biochemical reactions are important first stepsin understanding cells, the hard and interesting workto come is to understand networks of genes and biochemical reactions. It is these networks, in which proteins control genes and genes control proteins, thatcarry out and control specific cellular functions. Themathematics will be ordinary differential equations forchemical concentrations and variables that indicate towhat extent a gene is turned on. Since transport intoand out of the nucleus occurs, partial differential equations will be involved. And, finally, since some of themolecular species occur in very small numbers, concentration (molecules per unit volume) may not be auseful approximation for computations about chemical binding and dissociation: they are probabilisticevents.Two kinds of statistical data can give hints aboutthe components of these gene networks. First, thereare large numbers of population studies that correlate specific genotypes to specific phenotypes (such asheight, enzyme concentration, cancer incidence). Second, tools known as microarrays allow us to measurethe relative amounts of a large number of different messenger RNAs in a group of cells. The amount of RNAtells us how much a particular gene is turned on. Thus,microarrays allow us to find correlations that may indicate that certain genes are turned on at the same timeor perhaps in a sequence. Of course, correlation is notcausality and a consistent sequential relationship isnot necessarily causal either (sure, football causes winter, a sociologist once said). Real biological progressrequires understanding the gene networks discussedabove; they are the mechanisms by which the genotypesplay out in the life of the cell.A nice discussion of the relationship between population correlations and mechanisms occurs in Nijhout(2002), from which we take the following simple example. Most phenotypic traits depend on many genes; suppose that we consider a trait that depends on only twogenes. Figure 1 depicts a surface that shows how thetrait in an individual depends on how much each ofthe genes is turned on. All three variables are scaledfrom 0 to 1. Suppose that we study a population whosemembers have a genetic makeup that puts the individuals near the point X on the graph. If we do a statisticalanalysis of the population, we will find that gene B ishighly statistically correlated to the trait, but gene A is
VII.2.Mathematical 0.6Gene A0.4Gene B0.20.2Figure 1 A phenotypic surface.not. On the other hand, if the individuals in the population all live near the point Y on the surface, wewill discover in our population study that gene A ishighly statistically correlated to the trait, but gene B isnot. More detailed examples with specific
Mathematical biology is an extremely large and diverse field. It studies objects ranging from molecules to glob-al ecosystems and the mathematical methods come from many of the subdisciplines of the mathematical
animation, biology articles, biology ask your doubts, biology at a glance, biology basics, biology books, biology books for pmt, biology botany, biology branches, biology by campbell, biology class 11th, biology coaching, biology coaching in delhi, biology concepts, biology diagrams, biology
Biology, Mathematics, and a Mathematical Biology Laboratory 1.1 The Natural Linkage Between Mathematics and Biology Mathematics and biology have a synergistic relationship. Biology produces interest-ing problems, mathematics provides models to understand them, and biology
DAT Study Tips* Biology Materials: DAT Destroyer, Feralis Biology Notes, Cliff's AP Bio 3rd Edition, DAT Bootcamp (Both Cliff’s AP Bio and Feralis Notes are free online) Biology is one of the most time consuming sections to study for, given that the scope of the material covered in DAT biology is so randomly big. Cliff's AP Bio 3rdFile Size: 527KBPage Count: 9Explore furtherDAT Bootcamp Biology Flashcards Quizletquizlet.comHow to Study for the DAT Biology Section the Right Way .datbootcamp.comFeralis Biology Notes DAT Study Tips Free Downloadferalisnotes.comFeralis Biology Notes? Student Doctor Network Communitiesforums.studentdoctor.netBiology Cumulative Exam Flashcards Quizletquizlet.comRecommended to you b
Mathematical biology is an interdisciplinary field in which mathematical methods are developed and applied to gain understanding of biological phenomena. In ex-ploring any topic in mathematical biology, the first step is to develop a good under-standing of the biology a
J.D. Murray Mathematical Biology (3rd edition) (see schedules) - the classic text on mathematical biology, covering a range of applications D. Neal Introduction to Population Biology - much overlap with this course in mathematical detail, but explores the biological princip
This century is the century of mathematical biology, which is to translate Charles Darwin’s (1809-1882) theory into mathematics 3/24. Mathematical Modeling and Biology Bo Deng Introduction Examples of Models Consistency Model Test Mathematical Biology Conclusion Model as approxim
IB Biology 9780198307747 IB Biology Course Book (Print Online) 134.95 IB Biology 9781927173930 Biozone IB Biology Student Workbook 49.95 IB Biology 9781927173947 Biozone IB Biology Model Answers 12.95 IB Biology 9780198393511 Biology for the IB Diploma - IB Study Guide 63.95
Double Concept Modal Modal Concept Examples Shall (1) Educated expression Offer Excuse me, I shall go now Shall I clean it? Shall (2) Contractual obligation The company shall pay on January 1st Could (1) Unreal Ability I could go if I had time Could (2) Past Ability She could play the piano(but she can’t anymore) Can (1) Present Ability We can speak English Can (2) Permission Can I have a candy?
