Single-Cell-Based Models in Biology andMedicine ed.byA.R.A.Anderson,M.A.J.Chaplain,K.A.Rejniak Mathematicsand BiosciencesinInteraction, 79–106 c 2007 Birkh¨auser VerlagBasel/Switzerland
II.1 Magnetization to Morphogenesis: ABrief Historyofthe Glazier–Graner–Hogeweg Model
James A. Glazier,Ariel Balter andNikodem J. Pop lawski
Abstract. This chapter discusses the history and developmentofwhat we propose to rename the Glazier–Graner–Hogeweg model ( GGH model), start- ing with its ancestors, simple models of magnetism, and concluding with its current state as apowerful,cell-oriented method for simulating biological de- velopment and tissue physiology.Wewill discusssome of the choicesand accidentsofthis development and some of thepositiveand negativeconse- quencesofthe model’s pedigree.
1. Introduction Living cells, despite their greatinternal molecular complexity, do afew basic things.They stick to each other,moveactively up and down gradients in their external environment, change shapeand surface properties, exert forces on each other and theirenvironment, secrete and absorb materials, differentiate, grow, divide, and die. Afew maysend electrical signals or performother specialized functions. Manyapproaches to buildingphysicalmodels of tissues are possible. The GGH model uses aframework derivedfrom statistical mechanics to describe cellbehaviors, achoicewhichisnot at all obvious at first glance. The GGH model is a cell-oriented ,asopposed to a continuum or pointillistic model. Continuum models ignore cellsand treattissuesascontinuous materials withspecific mechanical properties, completely ignoring the division of tissues into cells. Pointillistic models treatbiological tissues as collectionsofpoint-like cells, ignoringmanycell characteristicsthat are importanttobiological behaviors, such as cellgeometry and the adhesiveinteractions between cells at theirmembranes. While both approaches areconvenientand have had many successes in explaining tissue developmentand physiology [33,and references therein], manybiological structureshave length scalesofafew celldiameters, e.g.,capillaries or pancreatic islets,and thus require explicit spatial descriptions of cells. The GGH model is actually aframework fordefining biological modelsrather than asingle model. GGH models define abiological structure consistingofthe configuration of aset of generalized cells ,eachrepresented on a cell lattice as a 80 J.A. Glazier,A.Balter and N.J Pop lawski
domainoflatticesites sharing the same cell index (generalized cellsmay represent all or partofareal cell or anynon-cellular material in thesimulation), aset of internal cell states for eachcell (whichmay be quite complex), and aset of auxiliary fields (whichmay includediffusingchemicals, extracellular matrix(ECM ), gravity, etc.).The heart of the GGH model is an effectiveenergy or Hamiltonian,which encapsulates almost all interactions between modelelements, and optionally,aset of partialdifferential equations(PDEs) and boundary conditionstodescribethe evolution of thefields,and adescriptionofthe evolutionofthe internalcell states. Terms in the effectiveenergyoften takethe form of potential energies andelastic constraints. We call theenergy effective because itsterms primarily describe cell responses,whichmay not resultfromexternal forces(e.g.,when acell uses its internalmotile apparatus to move up or down agradientofachemicaldiffusing in extracellular space ( chemotaxis ), or attached to asubstrate(haptotaxis)). 1 The GGH model also uses afew extra mechanisms,the most importantofwhichiscell division. The generalized-cell configuration evolves through stochastic changes at individual lattice sitestominimize the effectiveenergy.The classical GGHmodel uses a modifiedMetropolisalgorithm for this evolution. Since the GGH model uses the cell as its natural level of abstraction and treats subcellular behaviors phenomenologically,itreduces the interactions amongthe 105 –106 gene products within eachcell to aset of governing equations for the variation of the roughly ten phenomenologicalbehaviorswementioned above.Akey benefit of the GGH formalism is that we can include almostany biological mechanism or cellbehavior we like, simply by addingappropriate terms to the effectiveenergy. The GGH model then automatically handles the interactions between mechanisms (though we must be aware that itschoices maynot be the ones thatweexpect or want). Thus, the GGH model provides acompactand efficientway to describecomplex biological phenomena. 2 The global parameters of the effectiveenergy and those describing the prop- erties of cells maybestatic,orevolveaccordingtosimpleorcomplex descriptions of biological or non-biological processes. E.g.,the adhesionofcells might depend on amodel of cell signalling writteninthe form of reaction-kinetics ( RK )coupled ordinary differential equation ( ODEs), or the growth of cells might depend on a neural-network model of genetic regulatory pathways. The GGHmodel itselfis agnostic about the modelsrun inside eachcell or outside thecell lattice,using its Hamiltonian to translate the information those models provide into physical structure and physiological behaviors [18,19]. Glazier and Fran¸coisGraner derived their model as an extension of the large- qPotts model of statistical mechanics [17, 16], callingitthe ExtendedPotts model and later the Cellular Pottsmodel ( CPM). The namePottsassociates the CPM
1 Cells can alsohaptotax in response to gradients in substrate texture or rigidity. 2 The elements of the GGH model are separable. E.g.,wecould use aGGH Hamiltonian and evolveitinalattice-free way using aFinite-Element(FE)method(see chapter II.4, section 1.5), or we could use aforce-based formalisminstead of aHamiltonian formalism and evolveit on-lattice using the modified Metropolis algorithm. II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 81
with statistical models of equilibriumdomainformation,whichwas appropriate to Glazier and Graner’s simpleversion. Laterextensionstothe modeltodescribe cellbehaviorsmean that it nowhas little in common with itsPottsancestor. Key differences between thePottsand GGH models include: 1. Ashift from calculating staticequilibrium statistics to kinetics.While the modified Metropolis dynamics (whichthe GGHmodel usestoevolveasingle configurationquasi-deterministically) derives fromthe Metropolis algorithm traditionallyused with the Potts model, it does notobeythe detailed-balance conditions required to generate equilibrium ensembles. 2. Initial conditionsemulating aparticular biological configuration rather than random initial conditions. 3. Ashift fromphysically-motivated to biologically-motivateddomainproper- ties. E.g.,biological cells remain connected,while Potts cellsdonot, so GGH models often includemechanismstoenforce domain connectivity. Because theGGH model’s behaviors and goalsare almost totally different from thoseofthe Potts model, the analogy which the Potts name suggestsismisleading. Therefore, we propose to name it afterits originators, Glazier andGraner and the person who has donethe most to extend it and bring it to its currentprominence in biological modeling, Paulien Hogeweg. From itsancestors, the GGH model has inherited anumberofpeculiarities; we willdiscuss several of these and possible solutions to some in sections 6and 7. Becauseofits flexibility,extensibilityand ease of use, theGGH modelhas become the singlemost widelyused cell-level model of tissue development[18, 19, 20, 21, 31, 32, 50].
2. Historical Originsofthe Glazier–Graner–Hogeweg Model The GGH model began as an extension of thelarge-q Potts Model,itself an ex- tension of the IsingModel,asimple early model of ferromagnetismbased on the magnetic moments, or spins ( σ ), of individualatomsand their interaction energies ( J ). The interactionbetween asingle pair of neighboringspins is often called a link ,orabond.Spins interactvia an energy function called a Hamiltonian, H .His- torical usage explainsmanyofthe otherwiseobscure choices of terms and symbols in themathematical formalismofthe GGH model. We begin our historical survey withthe progenitor of the GGH model, the Ising model of magnetism.
2.1. Ferromagnetismand the IsingModel 2.1.1. Ferromagnetism. Ferromagneticmaterials developapermanentmagnetic fieldfrom anet orientation of thequantum-mechanical spins ( σ )oftheir compo- nentatoms. Themaingoal of early statistical-mechanical modelsofmagnetism wastoexplain the 2nd-order (continuous) phase transition whichoccursiniron at the Curietemperature ( T c ). Below this temperature, materials suchasiron are ferromagnetic.Inferromagnetic materials,stable domains (connected, spatially- extended areas withcoaligned spins) formwithoutanexternalfield, giving anet 82 J.A. Glazier,A.Balter and N.J Pop lawski
magneticpolarization. Fortemperatures above T c ,thermal energy disrupts domain formation, the material becomes paramagnetic and itsspontaneous magnetization drops suddenly to zero. 2.1.2. The Ising model. ErnstIsing constructed asimple model of magnetization by making four radical simplifications[24]: 1. His atomsreside at regularly spaced points i on alattice(throughout this chapter,the symbols i , j , k willdenote two-orthree-dimensional ( 2D or 3D) vectorsofpositive integers indexing latticesites, e.g., i =(k, l, m ): k, l, m ∈ N ). 2. Spinshave onlytwo allowed orientations, up ( σ =1)and down ( σ = − 1). 3. Eachatom only interacts with its nearestneighbors on the lattice. 4. The interactions are classical, rather thanquantum-mechanical, so the spins obey Boltzmann statistics. Accordingtoitem 4the relativeprobabilityofany configurationofspins { σ ( i ) } is its Boltzmannprobability,whichdependsonthe configuration energy,or(Hamil- tonian), H ( { σ ( i ) } ): − H ( { σ ( i ) } ) P ( { σ ( i ) } )=e kT , (1) where k is Boltzmann’s constantand T is the temperature in degrees Kelvin.3 Thus, the higher the energy of aconfiguration, theless probable it is. In the absence of anyexternalfield, the Ising Hamiltonianisthe sum of interactions J ( σ ( i ) ,σ( j )) between all pairsofspins ( i, j )that arenearest-neighbors ( | i − j | =1): 1 X H = J ( σ ( i ) σ ( j )). (2) Ising 2 ( i, j )neighbors 1 The factor of comes because the summationdouble counts the interactions. 2 In the Ising model J ( σ ( i ) ,σ( j )) favors co-alignedneighbor spins (energy − J )and penalizesanti-aligned neighbors (energy+J ), so we can write Eq.(2) as: J X H = − σ ( i ) σ ( j ) . (3) Ising 2 ( i, j )neighbors Lars Onsager analyticallysolved theIsing model in 2D and showedthat the expectedferromagnetic phase transition didoccur [38]. No analytical solutionis known in 3D; however, aferromagnetic phase transitionstilloccursfor T c > 0. In theIsing model, the transition between ferromagnetic and paramagnetic states occurs because eachunitofboundarybetween domains of opposite spin costs an energy 2 J ,soconfigurations withcontorteddomainboundaries andmany domainshave higher energies thanthosewithfewer, smootherdomains. On the other hand, thenumberofconfigurations composed of manycontorteddomains is much largerthanthe number composed of afew smoothdomains. If we pick aconfiguration at random from suchanensemble of configurationsdistributed
3 From nowon, we will set k =1,whichisequivalent to measuringtemperature in units of energy, and omit k from our equations. II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 83
according to the BoltzmannprobabilityinEq.(1), the domain structure we ex- pect to find dependsonthe productofthe multiplicityofthe structurewith the Boltzmann probability. If T 2 J ,the greater number of randomconfigurations winsout overtheirsmaller Boltzmann factors,sorandom configurations aremuch more probable. At lowtemperatures,the cost of domain boundaries is so high that thelowerBoltzmannprobabilities of randomconfigurations overcomes their large number, andlarge-domain configurations dominate.Inthe limit that T → 0, the most probable state has all the spins in the same direction (one infinitedomain). 2.1.3. Summary. The Ising model contains twokey ideas that carry forward to the GGH model: 1. Theenergy of mismatched links between neighboringspins on alatticerep- resents the energy perunit length of the boundaries between domains. 2. Atemperature or fluctuationamplitude determinesthe probabilityofacon- figuration. However, the Isingmodel is far frombeingamodel of biological cells because: 1. It lacks dynamics. 2. Many domains mayshare thesamespin, whilefor biological modelingwe needaunique label for each cell. 2.2.The Potts Model Renfrey B. Potts, in his PhD thesis,described asimple extensionofthe Ising model whichallowedmultiple degenerate values of the spin,(i.e. the energyofa link depends only on whetherthe neighboring spins are the same or differentand not on their particular values)[40, 41]. We can write the Potts version of Eq.(2) as: X H Potts = J (1 − δ ( σ ( i ) ,σ( j ))), (4) ( i, j )neighbors where δ ( x, y )=0if x = y and 1if x = y .Wedenote the number of possible spin values by q .The Potts model hasferromagnetic and other phase transitions [6, 71]. In the limit of large q ,the Potts model canhave manycoexistingdomains at low temperatures,but multiple domains can stillshare the same spin and it still lacks the concept of dynamics. 2.2.1.Summary. The Potts model containstwo keyidea for biologicalsimulations: 1. Individual domains can have individualspins (whichinCPM and GGH sim- ulations we refertoas cell indices.) 2. Domains have aboundaryenergy that can be used to model adhesivity. However, the Potts model still hasseveral shortcomings as abasicbiological sim- ulation: 1. It still lacks dynamics. 2. Manydomains can shareasingle spin. In biologicalsimulations we require thateachseparate domain representaunique object,suchasabiologicalcell or part of one. 84 J.A. Glazier,A.Balter and N.J Pop lawski
3. The Potts model specifies only asingle contact energy between all spin values. 4. The Potts model does nothave away to control domain size and shape. 2.3.From Statistics to Kinetics Accordingtostatisticalmechanics, the distribution of equilibrium configurations of aset of classical spins dependsonly on the Hamiltonianand the temperature. Mathematically, we encapsulate the statistics for all configurationsinthe partition function,whichsums theBoltzmannprobabilities of every configuration:
X − H ( { σ ( i ) } ) Z = e T . (5)
{ σ ( i ) } Thenthe expectation value for anyfunction f ( { σ ( i ) } )is:
P − H ( { σ ( i ) } ) f ( { σ ( i ) } ) e T { σ ( i ) } f = . (6) Z
2.3.1.Monte-Carlo methods. Unless thepartition function and therelevantex- pectation values are soluble analytically,whichisrare, we must evaluate them numerically, whichiseffectivelyimpossible because of the enormousnumberof configurationstoenumerate(in the Potts model, q N ,where N is the number of spins in the lattice). Computationally, Ashkin and Teller showedthat we can ne- glect the vast majorityofconfigurations which have high energies and thus very low probabilities,makingthe calculation tractable [5].Intheir Monte-Carlo method, we startwith anylatticeconfiguration and ’jump’randomlyfrom configuration to configuration with probabilities chosen so that thenumberoftimes we visit a configuration is proportionaltoits Boltzmann probability. If we thenaverage the values of f thatwecalculate forsuchasequence of configurations,the average converges to f .Ineffect,wehave replaced an integral overconfigurations witha time average. However, because theAshkin–Teller methodallows jumps between anytwo configurations, it still lacks the intrinsic time order that kinetic models require. The required probabilityfor atransition between configurations { σ ( i ) } and { σ ( i ) } is: p ( { σ ( i ) }→{σ ( i ) } ) e H ( { σ ( i ) } ) /T = . (7) p ( { σ ( i ) }→{σ ( i ) } ) e H ( { σ ( i ) } ) /T This conditioniscalled detailedbalance .Monte-Carlo methods do not obey de- tailedbalance at T =0. 2.3.2. The Metropolisalgorithm. The Metropolis algorithm [35] radicallymodifies the Ashkin–Teller methodbecause it is local ; i.e. ,instead of allowing transitions between anytwo configurations, it allowstransitions only between configurations differingintheirspin value at asingle latticesite.Wecan thinkofthe Metropolis algorithm as diffusion in configuration space. Thislocal behavior allowsanatural time orderingofconfigurations, because anyconfiguration retains amemory of past configurations, and limits allowedfutureconfigurations. The Metropolisalgorithm for aHamiltonian, H is: II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 85
1. Choose alatticesite at random.Wecall this the target site,whichwewil denote i target and itsspin, the targetspin,whichwewill denote σ target. 2. Pick anyvalueofspin at random.Wecall this spinthe trial spin and denote it σ trial . 3. Calculate thecurrentconfiguration energy, H initial,and the energy of the configuration if the targetspin were changed to thetrial spin value, H final. 4. Calculate thechange this substitution would cause in the total energy, i.e.
Δ H = H final −Hinitial, (8) 5. Accept this change ( i.e. really change thespin value at the lattice site)with probability: 1ifΔH < 0 , p ( σ ( i )=σ → σ ( i )=σ )= (9) target target target trial e − Δ H /T if Δ H > 0 . Steps1through 5together are calledaspin-copyattempt. 6. Go to 1. On alatticewith N sites, we defineone Monte-Carlo step ( MCS)as N spin-copy attempts.Wealso define the acceptancerate to be the averageratio of thenumber of spincopies accepted to thenumberofspin-copyattempts.Ifthe acceptance rate is small (as arule of thumb, the acceptancerate should be greaterthan 0 . 01), theMonte-Carlo methodisinefficientand we saythat the computationis stiff for thatHamiltonian. Clearly, the acceptancerate increases as T increases. For T>0 (because the transitionprobability,Eq.(9)obeys Eq.(7), the long-term distribution of configurations obeys Boltzmann statistics,Eq.(1). 2.3.3. The use of the Metropolisalgorithm for quasi-deterministickinetics. Two behaviors suggest the possibilityofusing the Metropolis algorithmfor kinetic simulations: 1. While the long-time behavior of the Metropolis algorithmispurely statistical, at lowtemperatures,overshort times,the transition probabilitytends to lowerthe configuration energy. 2. For T =0,the Metropolisalgorithm does not produceastatisticalequilib- rium. Insteaditdrives the configurations down energygradients to alocal energy minimum, where evolution stops. Consider asequence of configurations, eachdifferingbyone spin value, { S 1 , S 2 ,...} , withassociated energies, {H1 > H 2 >...} .Then the net rate of transition (the difference between the forward and backward transition probablilities): −Hi + H i +1 r ( S i → S i +1)=p ( S i → S i +1) − p ( S i +1 → S i )=1− e T . (10)
H i −Hi +1 4 If we choose T suchthat T is small, but not toosmall, for all i , then, −H + H i i +1 H i −Hi +1 −Hi + H i +1 2 1 − e T ∼ + O (( ) ) , (11) T T
4 The existenceofsuchaT depends on energies being similar for nearbyconfigurations, whichis true for Ising,Potts, CPM and GGH Hamiltonians. 86 J.A. Glazier,A.Balter and N.J Pop lawski
so the netrate: r ( S i → S i +1) ∼Hi −Hi +1. (12) In this case, if we takeone spin-copyattemptasour time unit, the average speed from S i → S i +1 is: 1 vel ( S → S )= H · ( S − S ) . (13) i i +1 T i +1 i Thus,the averagetime evolution of the configuration obeysthe Aristotelian or overdamped force–velocityrelation: H = F = µ ve l, (14) where µ is an effective mobility.The movements of individual boundary elements of adomainmay be quiterandom,but the average velocities of large domains will be deterministic when the argument of theexponential in Eq.(10)isnot toolarge. 5 When the argument is small enough, whichisthe caseinmost biological simula- tions,the deterministic velocityrelationship is indeedlinearand obeys Eq.(14), [70]. Thisresultisthe fundamental justification for using Metropolis-likedynamics in kinetic simulations.Changingthe dynamics, e.g.,from Metropolis to Glauber, maychange the results in complex andsometimes unpredictable ways[70]. Using Metropolisdynamics for kinetic simulations causes anumberofprob- lems. ThatEq.(13) requires that the argument of theexponential in Eq.(10) be small,makes our originaluse of the Boltzmann factor in Eq.(9) questionable.How- ever, no one has studied theeffects on GGHmodeling of switching to adifferent weighting factor in Eq.(9). The exact relation between Monte-Carlo spin-copyattempts andcontinuous time are stillthe subject of debate and are apersistentsource of criticismof kinetic applications of Metropolis-likealgorithms in GGH simulations. In addition, because only thetime-averaged movementobeys the deterministic kinetics, the time order of events occurring at differentlatticesites is ambiguous overshort times.Several more sophisticated approaches to dynamics are possible (see chapter II.4, section 3.1).
3. Kinetic Potts Simulations –From Metal Grains to Foams 3.1. From the Potts Model to Coarsening The use of Metropolismethods to revealthe quasi-deterministickineticsofcon- figurationsevolving under aHamiltonian, ledtoagreat expansion of the rangeof questionsthat Monte-Carlomethods couldaddress. One new area of interest in the early 1980swas the kineticsofmetallicgrain growth. Mostsimplemetalsare composed of microcrystals,or grains,eachofwhichhas aparticular crystalline lattice orientation.The atoms at the surfaces of these grains have ahigherenergy thanthose in the bulkbecause of theirmissing neighbors.Wecan characterize this
5 Likeaferromagnet, the CPM has acritical temperature analogoustothe Curie temperature (see section 6.1). Quasi-deterministic motion occurs only for temperatures well belowthis critical temperature. II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 87
excessenergyasaboundaryenergy .Atomsinconvexregions of agrain’s surface have ahigherenergythan those in concave regions,inparticular than those in theconcave face of an adjoining grain,because they have moremissing neighbors. Thus, an atomataconvexcurved boundarycan reduce itsenergy by “hopping” across thegrain boundary to theconcave side. The movementofatoms, which we can equivalently view as themovementofgrain boundaries, lowers the net con- figuration energy,but requires thermal activation because an atomhas ahigher energywhen it is in the spacebetween grains than whenitispart of one.Thus, while grains are stable at lowtemperatures, at high temperatures metallic mi- crostructure changes through annealing or coarsening,with the net sizeofgrains growingbecause of graindisappearance.
3.1.1. The Exxon model of graingrowth. In the early 1980s, agroup of researchers at Exxon Research, Michael P. Anderson, GaryS.Grest, ParadeepS.Sahni, and DavidJ.Srolovitz,noticed that thePottsHamiltonian is simply J timesthe total boundary length of the configuration[45,44, 46]. Theydrew an analogybetween grain growth and the Potts model, where they took the lattice sites to correspond to atoms,the specific spin values to differentcrystallineorientations, andlinks between differentspin domainstograinboundaries. Theyusually assumed that domains were initiallyconnected and compact, withadifferentspin assignedto eachgraintoavoid graincoalescence. 6 In grain growth heterogeneous nucleation does not occur, i.e. aspin of type σ willnot suddenly appear in themiddle of adomain of spin σ .Since the Metropolis algorithm allows heterogeneousnucleation, the Exxongroup modified the Metrop- olis algorithm to prevent it by selecting the trial spinfrom theneighborhood of the targetspin. We willcallthe lattice site of thetrial spin i source and its value σ source Though they did notrecognize it at the time, the concept of acopyoflatticevalue withasource and target impliedacopy direction,whichprovedcrucialfor later development of theGGH model.Forbidding heterogeneous nucleation means that evolution occurs onlyatdomainboundaries. It alsoviolates detailedbalance, a further move away from statisticalmechanics andtowards purely kinetic modeling. Forlow T ,using the Potts Hamiltonianand the modified Metropolis dynam- ics, individual domains evolveinamanner that resembles thegrowth of metallic grains during annealing at high temperatures.The simulated evolution of the dis- tributions of domain shapes, sizes and correlations agreed very well with experi- ments in metals[66].
3.1.2. Coarsening in foams. Glazier, working with the Exxongroup, later showed thatthe simulated evolution alsocloselymatched theexperimental evolution of bubbles in 2D liquid soap froths, where gas diffuses across soap films depending on theircurvatures [13,15]. In this case,alink between two different spinsrepresents
6 Allowing multiplegrains to have the same spin allows coalescence, whichoccurs in some metals. Since grain boundaries are simply links between lattice sites with differentspins, when two grains with the same spin come into contact, they immediately fuse into one large grain. 88 J.A. Glazier,A.Balter and N.J Pop lawski
Figure1. An evolvingsoap froth (left) and alarge-q Potts modelsimu- lation of grain growth (right) using the state of the soap froth at 2044 min as the initialcondition. Adapted from [15].
aphysical object,aunit of soap film, and thespins representthe gasinside the individual bubbles.The use of amismatched link to representaphysicalobject –asoap film–was an importantconceptual advance towards the GGH model. Glazier later extended these results to 3D [14].Fig.1shows theevolutionofasoap froth and aPottssimulationusing the configuration of the soap froth at 2044min as its initial condition[15]. The use of Potts-derived models to study coarsening remains an activefield of research[22, 30, 61, 60, 58]. Forareview of the earlier historyofthis work, see [66].
3.1.3. Lattice anisotropy. The Exxon group’s grain-growth simulationsrevealed aproblem with the Potts approach. The energyofaunit of grainboundaryis anisotropic; i.e. ,itdepends on the boundary’s orientationwith respect to the II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 89
lattice. Suchlatticeanisotropycan lead to alignment of grainboundariesalong preferred axes, and even to boundary pinningatlow temperatures.Holm et al.’s studies of these effects [23] led to the general adoption of longer interaction ranges for Potts simulations, i.e. when calculating theboundary energy,weconsider n th-nearest neighbors, where n is an integer. Larger values of n reduce lattice anisotropyeffects, but increasecomputation time, comparedtosimulations with smaller values of n .Hexagonal lattices greatlyreduce anisotropycompared to squarelattices withthe same neighborrange and are not difficult to implement computationally,but have not been much used in simulations. Theoptimum choice of lattice and theuse of adistance-dependent J factor to further reducelattice anisotropyremain to be examinedinasystematic fashion. 3.1.4. Summary. TheExxonsimulations introduced manykey ideas–the use of uniquely-labeledcompact domains to identifydifferent grains, thestudy of do- main kinetics under the influence of boundary energyand fluctuations, the use of mismatchedlinks to representmembranes, and the modificationofthe Metropolis algorithmtoprevent heterogeneousnucleation. They alsorevealed the problem of lattice anisotropy,whichstill afflicts GGH simulations. 3.2. From Grains to Relaxed Foams –ConstraintsinanExtended Potts Model One significantdifference between foams and metallic grains is thatthe growth rate and relaxationrateofboundary shape of ametallic grain are thesame, while in foams boundaryrelaxationismuchfaster than growth. The resultisthat grains can have irregular shapes, while foamboundarywalls(soapfilms) are near-perfect minimalsurfaces(circular arcsin2D). In abrilliantly-presented and careful study, Weaireand Kermode extended the Potts model using constraints to simulate coars- eningin2Dliquidfoams [67, 68]. 3.2.1. Constraints. The use of constraints to describeinteractions comes from clas- sical mechanics. E.g.,wecan describecircularmotion by imposing theconstraint thataparticle remains aconstantdistance from aspecified point. We use the cal- culus of variations to derive equationsofmotion under aconstraint, by minimizing an integral of aHamiltonian (or Lagrangian)with an added physical constraint condition,whichisthe productofaLagrange multiplier, λ (the generalized force needed to maintain the constraint), with afunctionwhich is minimalwhenthe constraintissatisfied. In the context of Monte-Carlo dynamics, we can write a constraint energy in ageneral elastic form: λ ( value − targetvalue) 2 .This constraintiszero if value = targetvalue and grows as value diverges from targetvalue.Wecall theconstraint elastic,because the exponent of 2occursinideal springs andelastic solids(we could, in principle, use anypositive even integer). Becausethe con- straintenergydecreases smoothly to aminimum when the constraint is satisfied, the modified Metropolis algorithmautomatically drives anyconfiguration towards one thatsatisfies the constraint. In the presenceofmultiple termsinaHamilton- ian,noconstraint is usually satisfied exactly,because no configurationwillsatisfy 90 J.A. Glazier,A.Balter and N.J Pop lawski allconstraintsand minimize all energies simultaneously (see chapter II.2, section 5).Whileincreasing the appropriate λ can force theconfiguration to satisfy any constrainttoany desired accuracy,increasing λ/T reduces the acceptance rate, whichslows the simulation timescaleand makes it computationally inefficient. If λ/T becomes toolarge, thesimulation will freeze andonly limited configuration evolution will occur. In the contextofnumerically solving differential equations, suchconstraintsare appropriately called stiff.Since we can make value depend on the configurationinany way we want,and alsomake targetvalue vary in any way we likeinspace or time, we can impose almost anybehavior using constraints (although itsexpression maybecumbersome). 3.2.2.The Weaire–Kermode model for soap froths. Because Weaireand Kermode wanted domain growth to be slowcompared to boundaryrelaxation andbecause they did notknowthatthe domains in the large-q Potts model already obeyed von Neumann’s law[65], they added an elastic constraintonthe volume of each domain and evolved thetargetvolumesvery slowly according to this law. Theythenused aPottsboundary-energy termintheir Hamiltonian to impose an effectivesurface tension on their domains, causing boundariestorelax towards foam-likeminimal- surfaceshapes [69]: X X 2 H = J (1 − δ ( σ ( i ) ,σ( j ))) + λ ( v ( σ ) − V t ( σ )) , (15) ( i, j )neighbors σ where λ is an inverse gas compressibility, v ( σ )isthe number of lattice sites in the domain withspin σ ,and V t ( σ )isthe target number of sites for that domain. One useful result from the constraintformalismisthat P ≡−2 λ ( v ( σ ) − V t ( σ )) is the pressure inside the domain.Adomainwith v
3.2.3.Summary. With Weaireand Kermode’s extension of the Potts model to include avolume constraint[69], all that amodel of biological cells still needed wasaboundaryenergy that depended on domain type,anidea that goes backto the Heisenberg model of magnetism [72].
4. TheOrigin of the Cellular Potts Model In this section we discussthe origin of the type-dependentboundaryenergies be- tween cellsused in the CPM and GGHmodel and write thefull Hamiltonian for the CPM.The inspiration for the CPM came from experimental and theoretical work by the biologistMalcolm S. Steinberg at Princeton University on biological cell-sorting experiments and from later experiments on regenerationinaggregates of hydra cells ( Hydravulgaris)bythe biophysicist Yasuji Sawada at Tohoku Uni- versity, Sendai, Japan [48, 25, 42]. II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 91
4.1.Cell Adhesion andCell Sorting Celladhesion is fundamental to multicellular organisms,and to manyunicellular organisms as well [10]. If cellscouldnot sticktoeachother andtoextracellular materials,buildingcomplex life would be impossible. Adhesion also providesa mechanismfor controllingstructures,aswell as holding them togetheronce they have formed. In the late 1950s, Steinberg, while tryingtounderstand howdifferencesin gene expression between cells couldtranslate into complexstructures in embryos, noticed that during embryonic development, the behavior of aggregates of cells resembled the behavior of viscous fluids. Forexample, arandom mixtureofem- bryonic cellsoftwo types, when formed into a3Daggregate,reorganized into a compact ballwith the more cohesivecell type surrounded by the less cohesivecell type in aphenomenon known as cell sorting [4, 3]. Differencesincohesion result- ingfrom differences in the numbers and typesofcell adhesion molecules on cell surfaces [11, 12, 9] could also explain the layered structure of the embryonic retina and the engulfmentofamorecohesivetissuebyalesscohesive tissue. Steinberg’s Differential Adhesion Hypothesis ( DAH )proposedthatthe final configurationof an initially arbitrary configuration of embryonic cells minimizedtheirtotal free energy,sotissues reallydid behave likeviscousfluids [54, 55, 56,57].7 The manyfamilies of adhesion molecules (CAMs, cadherins, etc.)provided amechanism for embryostocontrol therelativeadhesivities of their various cell typestoeachother andtothe noncellular ECM surrounding them, and thus to build complex structures. However, likethe Ising model, the DAHwas concerned onlywith equilibriumconfigurations, not kinetics. 4.2. The Cellular Potts Model In 1991, Glazier joined theSawada laboratoryatTohokuUniversityinSendai, Japan, which wasfamous for itsstudiesonthe regeneration of adult hydra from randomly mixedaggregatesoftheir dissociated cells [48, 47,49, 2, 53, 25, 42]. There,Glazier met Graner, who wasstudyingthe first phase of hydra regeneration, when endodermal cellssortedtothe center of the aggregate andectodermal cells to thesurface[59]. Graner wanted to see if theDAH explained hisresults. Glazier realized that he could extend his foam simulations with the Exxon group to ex- plore the kinetics and thermodynamics of biological cell sorting, e.g.,todetermine whetherthe cell sorting that Steinberg andArmstronghad observed experimen- tally required activecell motility, which would implythat the energy landscape of theconfiguration spacewas rough,with many local minima,orcouldoccurinthe absence of fluctuations, whichwouldimply that the energy landscapewas smooth, withasingleglobalminimum for the sorted state. Glazier alsorealized that the domainscouldrepresentmore than biological cells –inparticular he introduced the concept of adomainasageneralizedcell ,
7 In adultanimal tissues and in plants,cells usuallybind to eachother tightly via specialized junctional structures and do not move relativetoeachother. Exceptions includewoundhealing, immune response and cancer. 92 J.A. Glazier,A.Balter and N.J Pop lawski
Figure2. Atypical configurationofthe CPM or GGH model in 2D. The numeralsindicate cell index values. The levels of grayindicatecell types. A cell is acollection of lattice sites with the same index value.
which couldbeabiologicalcell, asubelement of acell,allowingcompartmental cellmodels,orpart of theextracellularmedium, afluid or asolid, dependingon the domains’ characteristics. Thisredefinition of everything in the simulation to be composed of generalizedcellsallows thesame model to treat manydifferent typesofobject,greatly simplifying model building. Glazierand Graner’s model discretized the continuous cellconfiguration onto asquarelattice. Acollection of lattice siteswith the same index represented agen- eralized cell, as showninFig.2, with aunique index for eachcell σ ( i ) ∈ [1,...,N] defined at each lattice site i ,and a cell type τ ( σ )for eachcell.Links between differentindices representedregionsofcontact between two cell membranes. From nowon, we willdrop the confusing term spin and refer to cell indices,since spin has no meaning in ourbiological context. Sincethe cellvolumeswere constantand uniform in cell-sorting experiments, Glazier and Graner usedanelastic volume constraintbased on Weaireand Ker- mode’s work on foams [69]tomaintainthe size of the biological cells. To represent variations in adhesion between cellsofdifferent types, theydefined aPotts-like boundary energy whichdepended on the celltypes at alink, J ( τ ( σ ) ,τ( σ )) (see Fig.2), so the boundary energy term in the Potts model (equation4)became: X H boundary = J ( τ ( σ ( i )),τ( σ ( j )))(1 − δ ( σ ( i ) ,σ( j ))), (16) ( i, j )neighbors
where the boundary energy coefficients are symmetric,
J ( τ,τ )=J ( τ ,τ) . (17)
Theyassumed that the boundary energieswere positive ( J>0), which provedto have unfortunate consequences (see section 6.2 below). The full Hamiltonian for II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 93
Glazier and Graner’sCPM is: X H CPM = J ( τ ( σ ( i )),τ( σ ( j )))(1 − δ ( σ ( i ) ,σ( j ))) ( i, j )neighbors X 2 + λ Vol ( τ )(v ( σ ) − V t ( τ ( σ ))) , (18) σ
where, v ( σ )isthe volume in lattice sites of cell σ , V t its targetvolume,and λ Vol ( τ ) the strength of the volume constraint.InGlazierand Graner’s original papers, the value of V t wasconstantfor all biological cells and thevolume of the generalized cellrepresenting the surrounding medium wasunconstrained ( λ Vol (medium) =0) [17,16]. If we add aconstanttoall J s, and also add the same constanttothe interac- tion energybetween likeindices (whichis0in H contact in Eq.(18) and requires us to change theform of theequation), the evolutionisunchanged. Since the kinetics in the modified Metropolis algorithm depends only on two things, the signofΔH and thevalueofΔH /T (see Eq.(9)),ifwemultiply all the termsinthe Hamil- tonian by apositive constantand multiply the temperature as well, the evolution of configurations remains unchanged.Therefore, we have two degrees of freedom, one additiveand one multiplicative, in setting the scale for the CPM parameters (see chapter II.2, section 4.1). Sincebiological cellsmove actively(in the case of vertebratecellsusually by extending and retracting theirmembrane using their cytoskeletons), sincethese membranefluctuations are very roughlyanalogous to thermal fluctuations(though they have anon-thermal origin)and since cells do not suddenly appear inside other cells, Glazier and Graner used the modified Metropolis algorithm of the Exxon Group for the dynamics of their model.
4.2.1.Smoothing (“annealing”). The CPM runswith T>0and the usual val- uesofofΔH /T are fairlysmall. Thus fluctuations are large, especiallyfor more cohesive cells(those with lower J values)and cell boundaries can becomehighly contorted. In this case,cells canbecome disconnected by spinning off small (usu- ally single lattice-site) blebs (wetakethe namefrom the small membrane-encased blobsofcytoplasmwhichmigrating cells sometimes leave behind). Neither blebs nor contorted boundaries are biologically realistic. Both can affect calculations of surface areas, neighbors and volumes.InGlazier andGraner’soriginal papers, before theycalculated statistics for aconfiguration, they eliminateddisconnected blebs and smoothedcell boundariesbyrunning their simulationwith T =0for five, or more, MCS, using the normal Hamiltonian andmodified Metropolis dy- namics. Rather confusingly,they calledthissmoothing annealing,eventhough it is not the same as norm al annealing in metals. Later studieshave generally not needed smoothingtoreproduce biologically-observed behaviors. 94 J.A. Glazier,A.Balter and N.J Pop lawski
5. Classical CPM Results Because of Glazierand Graner’s backgrounds in physics, their initial studies of cell sorting simulations resembled in manywaysthe Exxongroup’s statistical- mechanics studies of grain growth. Theyfirst validated the use of theCPM in a biological context by studying itsphase transitions and thebehaviorsofsimulated cellsofasingle type as afunction of individual model parameters ( e.g.,toestablish the optimum range for J/T and to checkfor excessive lattice-anisotropy effects). They then triedtosimulate Steinberg and Armstrong’s experiments on cell sorting, studying the behaviorofmixtures of two typesofcells, high-boundary-energy, low- adhesivitycells, and low-boundary-energy,high-adhesivitycells, surroundedbya single generalized cell representing fluid medium. Their single mostimportantresultwas to showthat cell sortingisanacti- vated processrequiring membranefluctuations [16]. In their CPM simulationsat lowtemperatures,cells clustered but clusters could not coalesce.This observation wasagood example of the way thatsimulations canclarify acomplex experimental situation.Experimentshad previouslyshown (and later experiments verified) that introducingdrugs whichblockedmembrane fluctuationsintothe fluid medium containing thecell aggregates inhibited cellsorting [4,36]. However, the drugs used interfered withthe cells’ cytoskeletons. Since the adhesion molecules, which determine cell-celladhesivities, bind to the cytoskeleton and change theiradhe- sivitywhen the cytoskeleton is disrupted, the experiments could not determine definitively whetherthe failure to observe sorting wasdue to lackofcell motility or to changes in cell-celladhesivity. As in this case,biological experimentsoften lackcleancontrol parameters. In their simulations,Glazierand Granerwere able firsttochange cellmotilitywhile keeping adhesion constantand thentochange cell adhesion while keeping cellmotility constant. By comparing theresults in these two cases, theywere abletoshowthat lossofcell motilityindeedprevented sorting. They then examined the variouspossible hierarchies of boundary energies ( J s) to characterize theclasses of typical patterns whichthey couldobtain. The varietyofoutcomes, even in this very simple situation, showedthe powerofvaria- tions in theexpression of cell adhesionmolecules to control embryonic morphology (see asimulationMovII.1.1form theaccompanying DVD).
6. Peculiaritiesofthe CPM Glazier and Graner were tryingtobuild amodel so simple that they could under- standthe physicsofall of its components. Thus the CPMhas certainpeculiarities, which, while not generally criticaltobiological simulations, maycause confusion and artifacts. If we are aware of these potentialproblems, we can usually take steps to makesurethattheydonot invalidate our results. II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 95
6.1.Temperature We mentioned in section 5that cellsorting (and indeed anyinteresting biological phenomenon we mightwanttoinvestigate) requirescell motilityboth experimen- tally and in simulations. Thus we must pickadynamicstogowith our Hamiltonian. From aphysicalpointofview, the modifiedMetropolisalgorithm is the simplest choice. Temperature can cause problems in anumberofcontextsinthe CPM (see chapter II.2, section 6). The CPM has severalphase transitionsintemperature, three of which,inparticular, are wholly non-biological and limitthe rangeof temperatures which simulations canuse.Asarough guide forpicking appropriate values of J to avoid ill effects, we want to accept asignificantfraction of index- copyattempts but notsomanythat the lattice patternmelts. Thus we need 0 . 2 ≤ Δ H /T ≤ 2. If thenumberofneighbors perlatticesite is n ,the typical fluctuationenergy perindex-copyis Jn/ 2, so we need to pickall of our J values suchthat 0 . 2 Ironically,given these issues, the use of T in GGH simulations has almost never been asignificantlimitation. Whereithas, simplefixes likeincluding an inertialconstraint(see chapter II.4, section 1.1)seem to have solvedthe problem. Nevertheless, improving the realismofGGH kinetics is worthyofattention,exper- imentally (to characterize actual kinetics),computationallyand theoretically (to understand the significance of differentkineticbehaviors to configuration evolution and scaling). 6.2.Diffusion, Energy and Parameter Choices Glazier and Graner’s originalmodel, following the physical realityinfoams and metallic grains, assumed that boundary energies were positive, so the boundary energyterm servedboth to minimize boundary areas andtodeterminethe optimal arrangementofcells. Anegative J results in the cellboundariesbreaking up to maximize their boundaries. When Glazier andcollaborators began studying cell diffusion rates, they found that more cohesivecells(smaller J>0) had more crumpledsurfaces, larger membrane fluctuations, and diffused further than less cohesivecells, exactlythe oppositeofthe expectedresult [42].8 Sincebiological cellsusually have adhesive interactions witheachother andthe ECM, the correct way to solvethis problemistouse negative J and constrain thesurface area separately (see section 7.1.1). Equivalently,Hogeweg further modified themodified Metropolis dynamics by shifting the BoltzmannprobabilityinEq.(9) to negative Δ E 0 ,whichgives the same effect: 1ifΔH < Δ E , p ( σ ( i )=σ → σ ( i )= 0 (19) target target target − (ΔH−Δ E 0 ) /T e if Δ H≥Δ E 0 . We can equivalently view this modified dynamics as introducing an extra dissipa- tion energyper index copy[31, 32]. Negative J sgivecorrect diffusion hierarchies [42].Since the configurations observed are independent of thesign of J (only the diffusion constantsare wrong), manyresearcherscontinuetouse simpler J>0 models, when they do not care about relativerates. Whilerelating GGH parameters to experimentally-meaningful material prop- erties has proveddifficult,the recentderivationofthe continuous limit of the CPM mayhelp to connect simulationand physical parameters [63]. 6.3. Intrinsic Dissipation and Viscosity To makeCPM simulationsmechanicallyrealistic, we would liketobeable to specify theresistancetomotion that cells experience when moving throughafluid due to dissipationand viscosity. Dissipation is theloss of energy duetomotion, i.e. resistance to all motion. Viscosity is dissipation whichresults from velocity gradients. The CPM is intrinsically dissipative,but not viscous. Consider achannel filledwith CPM cells,whichstrongly adhere to the walls of the channel(small J ). If we push the cells through the channel by applyingagradient, the cellswill experience plug flow, i.e. all cells move withthe same averagevelocity.The cells touching the walls do not move slowerthan thoseinthe centerofthe channel, as 8 In certain situations,strong binding between cells can lead to cytoskeletal changes whichincrease cell motility, in whichcase using positiveenergies maybemore appropriatebiologically. II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 97 we would expect in aviscous fluid. We can implementviscosityinthe CPM (see chapter II.4, section 1.2.1) to producethe correct velocityprofile. However, an objectmoving through afluid with CPM viscosityexperiences both viscousand intrinsic dissipation, while areal fluid has no intrinsic dissipation. 7. From the CPM to the GGH Model The basic CPM modelsonly the effects of differential adhesion, cell volumeand fluctuations.The GGH model addsmanyofthe other biological mechanismswe discussed in section 1and also addressessome of theissues in section 6. 7.1. Simple Extensions to the Hamiltonian 7.1.1. Surface area constraints and negativeboundaryenergies. Biological cells have adefined amountofcell membrane, whichwecan represent withasurface area constraint: X 2 H surface = λ S ( τ )(s ( σ ) − S t ( τ ( σ ))) , (20) σ where s ( σ )isthe surfacearea of cell σ and S t is its targetsurface area in lattice sites. Changing the ratio: √ 3 4 πS3 / 2 R = t , (21) V t changes therigidityofthe cell. Likeaslowly inflatedballoon(whichcorresponds to decreasing R ), for R>1the cell is floppy,while for R =1the cell is spherical and for R<1increasingly rigid.While Eq.(20)isthe simplest form of surface area constraint, otherforms are possible and maybepreferable, e.g. : X ` − 2 / 3 − 2 / 3 ´ 2 H surface = λ S ( τ ) s ( σ ) v ( σ ) − S t V t , (22) σ keeps the R of acell constantasitgrows. Constraining the cell surfacearealeavesusfree to define cells’ boundary energies J to be eitherpositive or negative, dependingontheir real biological values,eliminatingthe diffusion-hierarchy problem we notedinsection 6.2. 7.2. Non-Hamiltonian Extensions 7.2.1. Cell growth and proliferation. The simplest way to simulate the growth of cellsistoallow V t and S t to increase gradually with time fromagiven initial value, V t0,todouble that value, 2 V t0 [7],atarate proportional to theconcentration of anutrientorgrowth factor C ([39], see alsoasimulation MovII.1.2 fromthe accompanying DVD): dV t ∝ C . (23) dt growth factor Forlarge C ,areal cell’s growth rate saturates, which we can include by replacing C C α by a Michaelis–Menten or Hill form β + C α ,where α and β are constants depending on celltype. Cells can alsogrowwhen they are stretched [43,18]. 98 J.A. Glazier,A.Balter and N.J Pop lawski 7.2.2. Cell division. To model cell division ( mitosis )when acell σ reaches agiven doubling volume, we assign anew index σ to half of the existing cell’s lattice sites, dividing the cell eitheralong the cell’s shortest axis [18, 19], or randomly [37,39]. In both cases,the new cells have targetvolumes V t / 2[7]. The new target volumes ensure thatthe pressure inside the cells (chapter II.2, section 5) does not change during mitosis. 7.3. Fields, Forces andDiffusion Akey early GGH extension of the CPM wasthe inclusion of multiple, additional lattices to trackthe concentrations of molecules(or external drivingpotentials) thataffect the cell behaviors. We refertothe lattice of cell indices as the cell field and theadditional lattices as external or auxiliaryfields .The CPM automatically handlescontactforces between cells. However, we would liketodescribeother influences on cell movementaswell,for instance,externalphysical forces, or indi- rect effects likechemotaxis.Inthe context of the GGH model, we implement any influence on cell movementeither through ageneralized potential energyadded to the Hamiltonian or by directional biasing of index-copyacceptance probabilities. The latter has the form of a generalizedforce (chapter II.2, section 3). The primaryuse of fields is to recordthe concentration of signaling chemicals or other biomolecules, whichwewill usually denote C ( i ), thatmay diffuseand react (section 7.3.2) andinfluence cell behavior via chemotaxis (section 7.3.1) or in other ways. Usually simplefieldsare thoughtofasoccupyingthe same spaceasthe cell lattice ( i.e. cells do not excludefields),although we can use repulsive haptotaxis to keep cellsand fields spatiallydistinct. Fieldsmay alsobeattached to cellsor subcells rather than to the lattice (see chapter II.4, section 1.2.2) in whichcase we denote the field C ( σ )and itsvalueatalattice site is: C ( i )=C ( σ ( i ))v − 1 ( σ ( i )), (24) i.e. the amount of chemical at aparticular lattice site equalsthe amountinthe cell divided by the cell volume. In some simple cases, we can representthe fieldasananalyticfunction of position. E.g. ,agraviationalfield in directionˆn producesagravitational potential energy: X H gravity = − µ gravity ( τ ( σ ( i )))(i · nˆ ) , (25) i where µ gravity ( τ ( σ ( i ))) is the density of cell σ [28]. Yi Jiang in her PhD thesis[27] applied shear forces directly to cells by in- cluding aterm in the Hamiltonian of form: X H = f ( i, σ, t ) . (26) i The force candepend on position, celltypeand time. II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 99 7.3.1.Chemotaxis and haptotaxis. If cell adhesion is themost importantbiologi- calmechanism permitting multicellularity (allowing cells to sticktogether to form larger groups), chemotaxis is thesecond most important (seechapter II.2, sec- tion 7.3). Adhesion by itselfcan onlyproduce layers andblobs,while adhesion in conjunctionwithchemotaxis canproduceaplausible facsimile of amulticellular organismlikeaslime mold [32]. Cells canrespond to diffusiblechemicals which maybepresent in the ECM(chemotaxis )aswell as to chemicals bound to sub- strates ( haptotaxis,whichalso includes cell movementinresponse to changes in substraterigidity, texture or strain). Chemotaxis is crucial forlong-range signaling during morphogenesis, allowing cells at one end of atissuetocontrol themotion of cells at theother end.Haptotaxis is crucial for shapestabilization. Forexam- ple, pre-cartilage mesenchymal cellswilltraveltoregions where the substrate has more non-diffusing fibronectin, where they will differentiate into cartilage and then bone. The abilitytosimulate chemotaxis in the GGHmodel dependsoncells re- spondinginasimplefashion to external chemicalgradients. Thesimplestprototype for such aresponse is if, v ∝ µ cell C, (27) where C ( x )isachemical field and µ cell is the cell’s chemotactic response param- eter.Then C hasthe formofachemicalpotential and µ cellC has the formofa potential energy, so adding theterm, H chemotaxis = µ cellC ( x ) , (28) to the main Hamiltonian causesthe celltomove up ( µ<0) or down ( µ>0) the chemicalgradient.9 Clearlythis functional formisanoversimplification. Cellscan- not respond to infinitelysmall gradients and cellsdonot go infinitelyfast in large gradients. Asigmoidal response is more appropriate.Additionally,the value of µ is not static for asinglecell.Cells adapt to external concentrationsofchemicals, becominglesssensitive when chemicalconcentrationsare high and more sensitive when theyare low[52]. The timescalefor adaptation varies greatly.Inextreme cases cells respond only to abrupt temporal changes in chemical concentration rather than to spatial gradients. Exactly howwetranslate thesimple ideainEq.(28) into an effectiveenergy is somewhatcomplicated and depends on the ideaofanindex copy direction (see section3.1.1), which correspondsphysically to the direction of motion of aunit of cell membrane. Thedifferentimplementations of chemotaxis/haptotaxisinthe GGH model depend on whether we we considerthe chemical field at thetarget lattice site only,atboth thetargetand source lattice sites, or at the source lattice site only,and whetherweuse thechemotacticresponseofthe sourcecell,target 9 Mostbacteria respond to chemical gradients in atotally differentway,through chemokinesis. In thiscase, the bacterium moves(runs )inaroughly straightline for aperiodoftime, then turns and randomizes its direction and movesoffagain in astraight line. If the external chemical concentration at the bacterium is increasing, the duration of the runs increases. If the concen- tration is decreasing, the duration of the runs decreases. The net result favors movementupthe gradientand the net movementofthe bacterium follows Eq.(28). 100 J.A. Glazier,A.Balter and N.J Pop lawski cell, or acombination.Depending on the particular biological situation, different choices maybeappropriate. E.g. ,when acell forms aleading edge, it typically responds to chemical concentration changes at theleading edge only and not at the trailing edge, in whichcase we willwanttoconsider thechemotacticresponse of thesourcecell only. The simplest formofchemotaxiscomes from[27], Δ H chemotaxis =(µ ( σ target) − µ ( σ source ))C ( i target) . (29) This equationhas theadvantage that it has the potential energyform of Eq.(28). However, it only acts at boundariesbetween chemotactingdifferentlycells. Thus, abig block of identically chemotacting cellswillrespondvery weakly because they onlysense the chemicalfield at their edges. Savill and Hogeweg [50],intheir modelsof Dictyostelium discoideum chemo- taxis, usedanenergy change proportional to thedifference between the local chem- icalconcentrations at the destination and sourcesites: Δ H chemotaxis = µ ( σ source )(C ( i source ) − C ( i target)). (30) An advantage of this formisthat acell responds to achemical field in the sameway,regardlessofwhat typesofneighbors it has.Merks, in his model of vasculogenesis, modeledchemotaxis with contact inhibition, i.e. chemotaxisonly at cell-mediuminterfaces, not at cell-cell interfaces [34]: Δ H chemotaxis =(µ ( σ target) − µ ( σ source ))(C ( i target) − C ( i source )). (31) CompuCell3D(see chapter II.4, section 5.1.1) [26] uses yetanother form: Δ H chemotaxis = µ ( σ source ) C ( i target) , (32) whichcan describeleading-edge-led cell movement. See also asimulation MovII.1.3 fromthe accompanying DVD. Finally,weknowthat for large C or C ,the cell’s response saturates. We can include this effect using aMichaelis–Menten or Hill form: C α H = µ . (33) chemotaxis β + C α Since all of these options maybeappropriate in differentbiologicalcircumstances, we propose to encapsulate all of them in ageneral term: α ( cC( i source ) − dC( i target)) Δ H chemotaxis =(aµ( σ source )+bµ( σ target)) , (34) α β +(cC( i source ) − dC( i target)) where a , b , c , d , α and β are constants.Inparticular, α is the Hillcoefficient,which determines howsteeply thechemotactic responserises at itsthreshold value C = β . To model haptotaxis [62, 73], we can use achemotaxis formwith anon-diffusing C or makethe surface energies J ( τ ( σ ( i )),τ( σ ( j ))) dependonthe concentration C .The simplest form is C ( i )+C ( j ) J ( τ ( σ ( i )),τ( σ ( j ))) = J ( τ ( σ ( i )),τ( σ ( j ))) − β , (35) 2 where β is apositive constant.The lineardecrease of the values of J ,asthe concentration of the chemical correspondingtocell adhesion molecules grows, leads to observed density-dependentpatterns in mesenchymal condensation in II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 101 vitro where the averagesizeofclusters is smaller at higher concentrationsofthese molecules, [73]. 7.3.2. Diffusion on external-field lattices. Cells respond to diffusible signalsfrom other cells or external sources. Typically,weimplementdiffusionusingaseparate solver whichacts on the external fields, whichwecall acertain number of times perMCS. The simplest method,the forward-Euler method,evolves the diffusion equation by redistributing concentration between neighborlatticesites fromthose with higher concentrations to those with lowerconcentrations. The proportion of concentration redistributed at eachstep relates to the diffusion constantfor that substance.The Euler methodisunstablewhen the ratio D Δ t/Δ x 2 is bigger than about 1 / 2 d ,where d is the dimension of the space,but we can maintainstability by calling the diffusionsolver multiple times perMCS and using asmaller Δ t each time. An advantage of forward-Euler and other finite-differenceschemes(for ex- ample,Crank–Nicholson) is that regions of the external field lattice corresponding to differentcell typescan have differentsecretion rates, decayratesand diffusion constants, including no diffusion and anisotropic diffusion. Since these properties correspond to individual cells (see section 7.4), whichcan move,wegain some of thebenefits of advection–diffusion,atvery little computationalexpense.For example, one iteration of a2Ddiffusionequationwith localdecay rate d ( i, j ) would be: C ( i, j, t +Δt )=C ( i, j, t ) + D ( i +1,j) C ( i +1,j,t)+D ( i, j +1) C ( i, j +1,t) + D ( i − 1 ,j) C ( i − 1 ,j,t)+D ( i, j − 1)C ( i, j − 1 ,t) − 4 D ( i, j ) C ( i, j, t ) − d ( i, j ) C ( i, j )+f ( i, j )Δt, (36) where f ( i, j )describes secretion, absorption and reaction of the chemical. This formula multiplies the concentration by the diffusion,secretion anddecayrates site-by-site overthe whole lattice priortothe iteration, whichwesometimes call applyingamask.Asthe cell latticeevolves,the diffusion, secretionand decayrates update automatically. 7.4. Internal CellStates As researchers have attemptedmore realistic biological simulations, they have devised methodstoimpart specific biological behaviors to individualcells. As a result, GGHsimulations have focused more on the properties and interactions of generalized cells and less on theproperties of individual lattice sites, though, of course, the actualmovement of cellsstill occurs at the lattice-sitelevel. This change in focus has inspiredmethods to describeand generate increasingly complex internal cell states and to describegeneralized-cellinteractions. We callthis general class of approaches, off-lattice extensions of theGGH. In Hogeweg’s model of genetic evolution[18, 19, 21, 29], cells have simple models of agenome andintergenomicpathwaysthat determine cell–cell adhesion, cell division and death. Between cellgenerations, thegenetic code of eachcell 102 J.A. Glazier,A.Balter and N.J Pop lawski evolvesvia gene mutations. The evolving regulatorypathwayscreate cellcolonies withunique morphogenetic tendencies, including manyexperimentally observed morphogenetic mechanisms and morphologies. Alternativemethods forcell differ- entiation use preprogrammed type changes of thecells [51]. 8. Outlook The great advantages of the GGH model are its simplicity and extensibility, which have made it themost widely-usedapproachtocell-level modelingbiology.GGH cellsmove according to effective-energy gradients v ∝ H ,whichmeansthat F ∝ v ,asinbiological experiments. As in experiments, the positionand movement of membranes determine celldynamics. Adding new biologicalmechanisms is as simple as adding new potential energies or constraints to the Hamiltonian. While the lattice discretization and modified Metropolis dynamics of the GGH model can causecertain artifacts,these rarely cause serious difficulties.Recentextensions of the GGH usingsubcells to model thebehavior of fluids [8] andelastic media have addressedmanyofthese issues (see chapter II.4,section1). We discuss additional extensions to the GGH,which use generalized cells to provide manyoff-lattice enhancementstothe GGH without abandoning the convenienceofthe GGH’s underlying fixedlattice, in chapter II.4. As our understandingofthe GGH model improves, we expect to be abletofurther improve both its accuracy andthe range of biologicalproblemsitcan address (see chaptersII.2 and II.3) and to see it even more widely adopted. This chapter has focused on theorigin and developmentofthe GGH model without discussing the computer-engineeringaspects of its implementation. One of themost importantdevelopments in GGH modeling in the pastfew years has been thecreation and release of open-sourcemodeling packages like the Tissue Simulation Toolkit ( TST)[33],10 or CompuCell3D [26], 11 whichprovide standard platforms for model development (seethe accompanying DVDfor simulations: MovII.1.1, MovII,1,2 and MovII,1,3). The use of one of these standardpackages allowsusers to reproduce published resultsand share new algorithms relatively painlessly,and opens the fieldofGGH modeling to amuchbroaderaudience. Acknowledgments We gratefullyacknowledge support from the National Institutes of Health, Na- tional Institute of GeneralMedical Sciences, grant1R01 GM076692-01, an IBM Innovation Institute award, NASA Glenn ResearchCenter,grantNAG 2-1619, the National ScienceFoundation,Division of IntegrativeBiology, grant IBN-0083653, andthe Office of the Vice President for Research,the CollegeofArtsand Sciences, 10http://sourceforge.net/projects/tst. 11http://dustbunny.physics.indiana.edu/∼ mswat/wwwCompuCell/index.html, https://simtk.org/home/compucell3d, http://www.nd.edu/ lcls/compucell/linux.htm. II.1 Magnetization to Morphogenesis: AHistoryofGGH Model 103 the PervasiveTechnologies Laboratories, the AVIDD and ODIN computerclusters and the BiocomplexityInstituteatIndiana University. References [1] B. Alberts,A.Johnson, J. Lewis,M.Raff, K. Roberts, and P. Walter. Molecular biology of the cell.Garland,New York, 2002. [2] H. Ando, Y. Sawada, H. Shimizu, and T. Sugiyama. 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Pop lawski BiocomplexityInstitute and Department of Physics, Indiana University Swain Hall West 117,727 East ThirdStreet, Bloomington, Indiana 47405-7105, USA e-mail: [email protected]