A Brief History of the Glazier–Graner–Hogeweg Model
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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. Poplawski 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 Poplawski 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 Poplawski 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