Biological System of Analysis of Systems Math

H. R. MACMILLAN, M. J. M CCONNELL, and K. TAIDepts. of Mathematics, Pharmacology, and Chemistry and Biochemistry

University of California, San Diego9500 Gilman Dr., M/C 0365

La Jolla, CA 92032-0365USA

hughbob@mccammon.ucsd.edu http://mccammon.ucsd.edu/people/hmacmill/index.html

Abstract:- We present two examples of computational cellular biology that provide perspective on the com-plexity of creating predictive models of cell behaviors as they emerge from the interaction of molecular species.The first is a model of the diffusion and reaction of neurotransmitter in a neuromuscular junction using a con-tinuum based finite element formulation [1]. Whereas this formulation accounts for spatial variation of neuro-transmitter concentration, the single unknown, the second example is a system of ordinary differential equationsto describe the biochemical state of an embryonic mouse neuron in relation to the observed behaviors of death,division, or differentiation. Each model, born of a deterministic mathematical perspective, is early in its ownevolution as it grows to reflect true biology. The necessarily organic nature of these models, in simultaneouslyincorporating and stimulating understanding, is behind the perspective they provide on the future of computa-tional cellular biology. Whereas in written form, this material must be presented linearly, live presentation isdone in hypertext to more suitably portray the interconnected issues and methods on a multiplicity of scales.

Key-Words:- computational cellular biology, cell cycle, synapse, finite elements, dynamical systems

1 IntroductionTime offers a spectrum of scales within which com-plex systems are observed to emerge from the sim-ple dynamics of interacting, and sometimes unwitting,components. A cell, as a unit of life, is near the smallend of this spectrum. In the post-genomic era, anno-tation is underway of the specific roles gene productsplay within complex biochemical networks that giverise to discrete cellular behavior. This effort is cre-ating a burgeoning interface between computationalmath and cell biology that presents enormous chal-lenges if quantitative modelling is to be predictive andrelevant.

To sample and subsequently survey these chal-lenges, we present two early examples of computa-tional cell biology: the diffusion and reaction of neu-rotransmitter acetylcholine (ACh) in a neuromuscu-lar junction (NMJ) and the biochemical balance be-tween death, division, and differentiation of embry-onic mouse neurons. In exploring how the relevantbiology of each example is reflected in commonly dis-parate branches of mathematics, the need for organicyet rigorous mathematical formalisms is exposed. Inspite of being born of deterministic science, effectivemodels shall at once evolve with and further the holis-tic understanding of dynamical systems important to

biology. It is to this seeming paradox, the very exis-tence of a mathematical system of analysis of systemsbiology, that our title lightheartedly alludes.

2 Computational Cell Biology2.1 The Continuum ApproachGiven the complexity of cellular dynamics, due tothe relevance of multiple scales, nonlinear coupling,and stochastic effects, there is reluctance to believe inthe prospect of predictive models of cellular behavior.Afterall, the very term “behavior” has metaphysicalimplications, and the applied mathematician wouldperhaps prefer “physiology.” For example, relativequiet, within both applied mathematics and molecularcell biology communities, surrounds a leading exam-ple of continuum based efforts, the NIH Virtual CellProject [2]. Predating this approach is that of discretemodels such as MCell [5], which generate statisticsbased on the probabilistic trajectories of individualmolecules. Though the latter seems quite natural andis promising for small systems (such as a synapse!), itfaces tremendous hurdles regarding scalability, bothin terms of space and the number of distinct interact-ing molecular species . A compromise is the objectof “stochastic” continuum based approaches, which

acknowledge “finite-numbers” effects by incorporat-ing probablistic simulations as they quantify inherentnoise due to smaller scales.

Enthusiasm for finite difference methods, as cur-rently used in the Virtual Cell, is waning within thecomputational mathematics community due to theemergence of multilevel adaptive, and therefore scal-able, finite element techniques. Built-in adaptive con-trol of spatial approximation error in a variational set-ting yields a formulation imbued with a sense of op-timality. As such, finite element formulations appearmore amenable to a progressive incorporation withindynamical systems theory for coupled systems of par-tial differential equations — the study of the sensi-tivity of the solution relative to differences in priorstates, especially as a function of the parameters in theequations presumed to govern. This broad claim onthe wilderness of numerical analysis for solving cou-pled systems of partial differential equations is basedsolely on the flexible capacity of the variational set-ting to quantify relevant spatial differences betweenprior states, and it is merely a claim. However, regard-ing the stochastic nature of finitely many moleculesinteracting, the variational setting may again offer ad-vantage since, by design, it allows the scientist to in-corporate, directly into the approximation space usedfor the solution of the model, effects inferred fromparticle simulations such as Brownian Dynamics andMonte Carlo [5].

Although we subscribe to a finite element approachas the engine for resolving spatial effects in the firstexample here, we are fundamentally aligned with theVirtual Cell Project as it provides a substrate for thegrowth of effective continuum models. Only in theevent of significant spatial complexity does the dis-tinction between finite differences and finite elementsbecome a relevant. In many cases, such as in our sec-ond example, spatial resolution may be prohibitivelyexpensive to compute, impossible to verify by exper-iment, and/or of small consequence. Indeed there isirony in that an understanding of the extent to whicheach of these may be the case implicitly requires push-ing the envelope of realistic models.

2.2 Synaptic TransmissionMotivation for studying the diffusion and reaction ofACh at a NMJ is threefold. Immediately, a vari-ety of diseases are manifested at the neuromuscularjunction, either in the ultrastructure of an affectedNMJ or in the kinetic relationships of its molecularconstituents [1]. These aspects of a model can be

Figure 1: Artist’s rendition of synaptic transmissionat a neuromuscular junction.

varied to explore,in silica, the effect of potentiallytherapeutic treatments. Of more theoretical concern,given the wealth of data on NMJs, an efficient modelcan be used in taxonomic coevolutionary studies ofsynapse ultrastructure and kinetics as each relate tomuscular function. Lastly, and perhaps most impor-tantly, synaptic transmission serves as an instructiveparadigm for the maturation of mathematical modelsof cell biology.

Upon the arrival of a presynaptic actional poten-tial at the end of a nerve cell, a cloud of ACh is re-leased into the NMJ, pictured in Figure 1. The me-chanics of this release, which involve elastic changesto the nerve cell membrane, are the first reductionistcasualties in developing acomputationallytractablemodel. ACh chemically diffuses across the synapticcleft , where individual molecules encounter the en-zyme acetylcholinesterase (AChe) and embedded ionchannel acetylchonline receptors (AChR). When AChbinds to its receptor, the ion channel opens to permit aflux of ions into the muscle cell; this is the experimen-tally observable miniture endplate current (mEPC).Synchronization of this flux induces contraction of themuscle. An extremely fast enzyme, AChe providesthe efficient “switch” which allows for rapidly repeat-ing contraction of muscle.

2.2.1 Continuum ModelIn an attempt to capture the physical biology ofsynaptic transmission, the following partial differen-tial equation and boundary conditions for ACh con-centration,u(x, t), has been proposed [1]:

du

dt−∇ · a∇u = 0 in Ω, (1)

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n · a∇u = 0 onΓ/ΓAChE, (2)

n · a∇u = −κu on ΓAChE, (3)

with initial datau(x, 0) = u0, whereΩ is the NMJvolume with boundaryΓ, ΓAChE is the disconnectedsurface area attributed to the enzyme, andκ is the en-zyme’s linear kinetic reaction constant. Note that thisrepresents an outward flux due to consumption by theenzyme atΓAChE, and a zero flux elsewhere onΓ, in-cluding at the locations of AChR, since the receptorrereleases ACh back into NMJ. The latter conditionis another part of the model subject to improvementsince AChR binding of ACh may result in an “effec-tive,” albeit elusive, flux condition.

In [1], a discrete approximation to the solution of(1- 3) has been computed on several domains usinga method of lines to reduce the corresponding weakform to an elliptic PDE at each time step. Given thesolution at the previous time step, the elliptic PDE issolved by conjugate gradients using the finite elementpackage FEtk [3]. In Figure 2, the finite element dis-cretization of a realistic NMJ is reprinted from [1].This volume mesh was generated by K. Tai based ona realistic surface representation generously providedby Bartol et al. and the MCell project [3]. This en-ables benchmarking the finite element solution withMonte Carlo statistics via comparison of computedminiture endplate current. However, this benchmark-ing is subtle because relating the approximate solutionof (1 - 3) to mEPC requires accounting for the prob-abilistic nature of ACh binding to AChR. Currently,this is done by empirically determining a factor,α,which connects the mEPC,I(t), with a surface in-tegral of the concentration,u(x, t), weighted by thedensity of AChR,γAChR(x); that is,

α ≈ I(t)∫ΓAChR

γAChR(x)u(x, t)dx. (4)

Once selected, the constantα must be shown to bevalid for subsequent simulations, or else the assump-tion of a linear relationship, in time, between mEPC(i.e., AChR opening) and the weighted concentraionof ACh near AChR must be improved. Note thatthe second improvement may involve a constant timeshift between the quantities. This is an aspect inwhich the continuum model has not reached matura-tion, and yet one in which its use as feedback providesinsight on the actual dynamics at work. Particle meth-ods are well suited for estimating this relationship be-tween mEPC andu(x, t), yet less well suited for sim-ulations of increasing complexity. This is thereforea clear instance in which stochastic effects computed

Figure 2: Tetrahedral discretization of a realistic neu-romuscular junction. On top is the entire mesh, inthe center is a close up of the vesicle from within thenerve cell, and on the bottom the postjunctional foldsare pictured with the AChe clusters as interior bound-ary.

from particle methods can be incorporated into a con-tinuum based model.

2.2.2 Future WorkAnalysis is begun in [1] of the effect of NMJ archi-tecture on muscular function, as represented in theamplitude and duration of mEPC. Specifically, differ-ences due to muscle type (fast- or slow-twitch) as wellas due to affliction with muscular dystrophy are in-vestigated. Future studies will address physicochemi-cal manifestations of myasthenia gravis as well as thepresence of neurotoxins in NMJs. However, before

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Figure 3: A reduced network of the biochemical path-ways associated with the death, division, and differen-tiation behaviors of an embryonic neuron, as initiatedby DNA damage and its activation of repair, or sur-vival, mechanisms.

these simulations are trusted as insightful, the modelmust be further grounded in reality according to ex-perimental data. This is true not only in relation tothe definition in (4), but also in the representation ofstructural and kinetic differences. Note, however, thatthese are precisely the subtlties of synaptic transmis-sion that the model is poised to explore, mirrored byevolving experimental techniques.

2.3 Neuron Development2.3.1 Biochemical PathwaysOur second example of computational cell biology,which falls within biochemical systems theory [14],demands an increased mathematical complexity to de-scribe coupled molecular interactions. Studies of em-bryonic mouse neurons suggest that a balance be-tween DNA double-strand breaks and DNA damagesignaling pathways generates genetic diversity amongneurons [5-9] and influences the cellular decision todivide, differentiate, or die. In Figure 3, a networkof biochemical pathways is pictured that initially cap-tures key aspects of this dynamic. Note that thisnetwork is likely to evolve, to either expand or re-duce, with insight provided by successful mathemat-ical models and biomedical research. A given neu-ron with a minimal history of double-strand breakstends to remain stationary and divide, whereas sus-tained levels of DNA damage are observed to triggera cascade of events leading to cell death. However,during the transition between these states, neurons areobserved to emigrate away from a proliferative zoneand form the cerebral cortex, where they contribute tocognition. The hope is that a mathematical model of

this system can be developed to explore the relativeimportance of these pathways and shed light on thestructure of the network.

Motivation for the study of neuron developmentparallels that of our first example. First, imbalancein this dynamic is implicated as a potential root ofneurodegenerative disease [8]. More theoretically,mature neurons are quite special cells in that theylive throughout the organism’s lifetime without be-coming cancerous. Understanding the intricacies oftheir developmental cycle may provide insight onhow to bypass blockage of cell death pathways, andthereby avoid uncontrolled and cancerous prolifera-tion. Lastly, this example is representative of bio-chemical systems that will continue to confront com-putational mathematics in the post-genomic era.

There is a wealth of literature on various ap-proaches to examining the behavior of biochemicalnetworks, from graph theory to control theory to bi-furcation theory, as complexity of the underlying net-work decreases (see [10 - 12] and references therein).Predating much of the biological interest, past studiesof coupled chemical kinetics is its own mature fieldwith a great deal to contribute to biological under-standing [13]. Relative to many of these efforts, ourmodel is premature. Its presentation below, however,provides perspective on many concerns that must beaddressed for the consistent development of mathe-matical formulations to control the dynamics repre-sented by biochemical pathways. Preliminary numer-ical results, using Matlab, will be presented in Co-pacabana.

2.3.2 Mathematical ModelIn our previous example, complications arise from anirregular computational domain and the need to relatethe solution of the model to an experimentally observ-able quantity. The ACh reaction with other molecularspecies is limited to the boundary, and hence there isno internal coupling of unknowns. Because of the in-herent difficulty, alone, of a coupled system of differ-ential equations, and because of the near impossibletask of spatially resolving the governing physics, wefirst treat the unknowns as spatially averaged concen-trations. Note that we even do this to represent the ac-cumulation of double-strand breaks. As such, the bio-chemical pathways represented in Figure 3 may be in-terpretted mathematically as a system of coupled dif-ferential equations with time delays and general func-tional dependencies F and G, yet to be determined (!):

d

dt[DSB] = FDSB([Casp], [repair], [extD])

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d

dt[ATM ] = FATM([DSB], [Casp])

d

dt[Casp] = Gmito([BAX], [AKT ], t)

d

dt[repair] = GREPAIR([ATM ], t)

d

dt[Abl] = GAbl([ATM ], t)

d

dt[P53] = GP53([ATM ], t)

d

dt[AKT ] = FAKT ([Abl], [extS])

d

dt[BAX] = GBAX([P53], t)

d

dt[P21] = GP21([P53], t) .

The separate notationG is only to stress the relevanceof a time delay due to spatial effects; this is in lieuof introducing partial spatial derivatives to arrive at amuch more complicated system of coupled partial dif-ferential equations to capture the governing physics.Again, the nature of this time delay introduces furthercontrols on the model. As before, understanding howto quantify this delay will evolve alongside targetedexperiments.

We can begin to approximate the nature of couplingfunctionsF andG using classical biochemical kinet-ics. According to linear theory

F (c1, . . . , cn) = Kc ,

for a matrixK to be determined, and with appropri-ate time shifts inG. To incorporate saturation effectswhere necessary, Michaelis-Menten theory writes, inthe case of a single argument,

F (c) = Vc

c + k,

with k some constant. Toward further sophisticationin the representaion of F, a “power law” such as

F (c1, . . . cn) = kcα11 c

α22 . . . cαn

n

is espoused in [14]. For reactions involving multi-ple species, concatenated products of concentrationswith differing exponents quickly lead to intractablesystems of ordinary differential equations. However,the power law approach, as claimed in [14], does pro-vide a single framework which enables the discriptionof many systems. In the context of approximationtheory, each approach introduces parameters to rep-resent the coupling functions that must be estimated

and proven consistent with experimental understand-ing. However, in the spirit of constrained optimiza-tion problems, re-estimating the parameters must notcompromise the proof of this consistency.

This reveals that, in essence, the goal of biochem-ical systems theory is to learn how to pose an inverseproblem associated with a system of coupled differ-ential equations. Since the formulation of the math-ematical model is in fact the formulation of its solu-tion, we must try to learn to pose this inverse prob-lem well, rather than cavalierly “regularize” an “ill-posed” problem. This task is intimidating, but conso-lation can be found within the steady accumulation ofa priori information regarding the biochemical statesof the systems whose governing dynamics are in ques-tion. This information amounts to snapshots of thesolution for all concentrations, and should of coursesteer the formulation of its model.

2.3.3 This PerspectiveThe above suggests an iterative, or cyclical, perspec-tive in which the formulation offers a solution whoseanalysis informs the formulation. Instituting this per-spective requires wading through a confluence of sci-entific disciplines to provide two crucial links: the so-lution of the formulation and its subsequent analysisto improve the formulation. The former entails solv-ing differential equations numerically. The latter es-tablishes the dual faith that the system is not overlysensitive to small variations in the parameters that de-fine the nature of coupling, yet sensitive enough tocapture significant differences in behavior.

Potential sources of inaccuracy in the formulationare multifaceted. Most fundamentally, hypersensitiv-ity may be due to reducing a model to too few un-knowns. Also, simply assuming linear, Michaelis-Menten, or power law kinetic theory may be insuf-ficient. Another source may be associated with realiz-ing spatial effects as a time delay instead of solving apartial differential equation explicitly (on a geometrythat is itself approximate). Lastly, stochastic averag-ing to account for the interaction of discrete quantitiesis to be tuned. Therefore, understanding the relativeimportance of each potential source of trouble is thepath to a model that is “well-posed.”

Direction on this path can be obtained by sam-pling biochemical state space, both experimentallyand computationally and in parallel, to glean their rel-ative partitions into death, differentiation, and divi-sion. The capacity to make associations experimen-tally is improving through the use of high-throughput

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gene expression profiling, proteomics, and even videomicroscopy utilizing flourescent biosensors. In turn,the ability to draw parallels to mathematical mod-els hinges on finding tractable systems of differen-tial equations. Conserved quantities and special qual-ities will be learned from experimental observationsof state space, manifested according to geometry ofthis space, and modestly incorporated into a formula-tion asa priori information. This is a central conceptin a relatively new mathematical field, the geomet-ric integration of differential equations. For example,the qualitative behavior of many small systems of dif-ferential equations, such as nonlinear feedback loops,can be quantified as low dimensional attractors in statespace.

3 ConclusionThe nonreductionist perspective of systems biology[15] is at once convincing and slippery, which is em-blematic of the conundrum that life presents to sci-ence. As a movement within biology, it reflects alarger trend in science and social theory that is gain-ing momentum in concert with an increasing aware-ness of the need for sustainability on all scales of a“deep ecology” [16]: from cells to people to socioe-conomic development in lieu of conflict. As such, itis hoped that an improved systems understanding ofthe cell can, in turn, transcend scales and offer sup-port for, and perhaps even living proof of, the moralobligation of sustainably impacting one’s own “opensystem” environment [17]. The role of applied math-ematics (at the ”burgeoning interface”) is to permeateany systemic understanding of, for example, a cell,rather than reduce it and present it. Thus the goal isa biological system of analysis of systems math;, i.e.,in short, biology math and not math biology.

Acknowledgments :

Hugh R. MacMillan is grateful for financial supportfrom the Al P. Sloan/DOE postdoctoral fellowshipprogram in computational molecular biology, and toProf. J. Andrew McCammon, Dept. of Chemistry atUCSD, and Prof. Michael J. Holst, Dept. of Mathe-matics, for their postdoctoral sponsorship. Also, Drs.Steve Bond and Rich Henchman have provided fruit-ful discussions. Kaihsu Tai is a La Jolla Interfacesin Science graduate fellow, with partial support fromthe Burroughs Wellcome Fund. Mike J. McConnell issupported on a NIGMS training grant in Pharmacol-ogy from the National Institute of Health.

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