## A

Where Aij is the contact area of cells i, j, Ac is the surface area of the (larger) cell, nmoi the total number of adhesion molecules at the cell surface, Ws the adhesive energy of a single bond. Harmonic-like potential energy A simple model for the short range cell-cell interaction due to cell-cell adhesion and elastic deformations is to approximate two adhesively interacting cells by cuboidal objects with a non-deformable core, reversibly linked by linear springs 9, 20 2 dij (t) + y j _ yattr...

## B

FIGURE 1. (a) Streaming of Dictyostelium discoideum towards the aggregation center. Cells move chemotactically towards the aggregation center leading to formation of cell streams and finally mounds. (Reproduced from 19 with permission), (b) Example of a quasi-one-dimensional motion of Dictyostelium discoideum inside a stream (this picture is on much smaller scale in comparison with (a)). Cells are moving parallel to each other in the direction of chemical gradient (from left to right). Chemical...

## BiOlOgYand MeDiCiNe

Chaplain Katarzyna A. Rejniak Dr. Alexander R.A. Anderson Division of Mathematics University of Dundee 23 Perth Road Dundee DD1 4HN UK Dr. Katarzyna A. Rejniak Division of Mathematics University of Dundee 23 Perth Road Dundee DD1 4HN UK Prof. Mark A.J. Chaplain Division of Mathematics University of Dundee 23 Perth Road Dundee DD1 4HN UK Library of Congress Control Number 2007923086 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche...

## Contents

Hybrid Multiscale Models A Hybrid Multiscale Model of Tumour Invasion Evolution and the Microenvironment Lattice-gas Cellular Automaton Modeling of Developing Cell Systems 1.3 Mark Alber, Nan Chen, Tilmann Glimm, Pavel Lushnikov Two-dimensional Multiscale Model of Cell Motion in a Chemotactic Field II. The Cellular Potts Model and Its Variants 77 11.1 James A. Glazier, Ariel Balter, Nikodem J. Poplawski Magnetization to Morphogenesis A Brief History of the...

## Da T rl

x) jp j, Jem - XLt + -pc(x)j , D i -. (20) The conditions for applicability of Eq.(20) are given by (15), (16), (17) and (18). 2.4. Reduction to Keller-Segel model In this section we extend the model to include a time dependent chemical field c(x,t) (concentration of chemoattractant) by adding a diffusion equation with a source term determining cells' chemical secretion where Dc is a diffusion coefficient of a chemical, 7 is a decay rate of the chemical field and a is a cell production rate of...

## E AVoirva Vtra218

Where, v(a) is the volume in lattice sites of cell a, Vt its target volume, and Ay0i (t) the strength of the volume constraint. In Glazier and Graner's original papers, the value of Vt was constant for all biological cells and the volume of the generalized cell representing the surrounding medium was unconstrained (Ay0i (medium) 0) 17, 16 , If we add a constant to all Js, and also add the same constant to the interaction energy between like indices (which is 0 in Hcontact in Eq.(18) and...

## E E E Vsntrara I

Where ro is the center of mass of the cell. I must make absolutely clear that the use of the term internal energy is not of biological significance, but is merely a convenient theoretical quantity with which to gauge the accuracy of our computational procedures. In cell biology, when one thinks of energy, one usually considers cell metabolism and ATP production - here, we are simply tracking stored elastic energy in a coarse-grained model of the cell cytoskeleton. Typical late-time evolution of...

## Me r

Unless the partition function and the relevant expectation values are soluble analytically, which is rare, we must evaluate them numerically, which is effectively impossible because of the enormous number of configurations to enumerate (in the Potts model, qN, where N is the number of spins in the lattice). Computationally, Ashkin and Teller showed that we can neglect the vast majority of configurations which have high energies and thus very low probabilities, making...

## Saian2 6ata0n1 6atajn1

(with the sum running in cyclic order), and the local neighborhood contains more than two non-medium cells, changing the index at the site destroys the local connectivity. They then made cell fragmentation energetically costly by assigning a large energy penalty to updates that change local connectivity. No similar local rule has been developed in three dimensions (3D). Such an algorithm would be extremely useful. Inertia-tensor description of cell elongation can also be used to measure elastic...

## Fi Eexp1

Then each Fourier component evolves independently according to The one-step Boltzmann propagator (q) is defined as Tij(q) exp( l < q,a + exp(i < q,cp > )OjJ1,p . One can show that the eigenvalues of T are given as For the square lattice and -, the symmetries1 of ui can be used to obtain Ai(q) (cos(qi)+cos(g2)) + 4 (wi + W2) (sin2(gi)+sin2(q2)) - 4 (cos(gi) - cos(52))2 In two-dimensional systems, groups of unstable modes with identical absolute value of the wave number q simultaneously...

## Hybrid Multiscale Models

Many mathematical models of biological process that consider space explicitly, fall into one of two categories (i) continuum population models or (ii) discrete individual based models. Discrete, stochastic interactions between individual organisms cannot be captured by the continuum approach and likewise global population interactions cannot be captured by the discrete approach. In recent years a third category of models has emerged hybrid models which allow modellers to exploit the advantages...

## Ia iai5a jry iajJ5

Where a and denote directional indices. The inertia tensor translates any object into an equivalent ellipsoid. Let I > I2 > I3 be the eigenvalues of I. If cells are roughly ellipsoidal, the eigenvalues of the inertia tensor give the ratios of the lengths of their principal axes. Scaling these ratios by the volume of the cell gives us the lengths. We can now constrain cells to maintain a given length or shape. For instance, in 2D, we could impose a length constraint 20 Hi Ai (1(a) - Lt(a))2,...

## 4 The Glazier GranerHogeweg Model Extensions Future Directions and Opportunities for Further Study

Merks, Nikodem J. Poplawski, Maciej Swat and James A. Glazier Abstract. One of the reasons for the enormous success of the Model (GGH) model is that it is a framework for model building rather than a specific biological model. Thus new ideas constantly emerge for ways to extend it to describe new biological (and non-biological) phenomena. The GGH model automatically integrates extensions with the whole body of prior GGH work, a flexibility which makes it unusually...

## Offlattice Cell Models

Newman One of the more challenging biological characteristics to build into a cell model is cell shape, and, along with this, cell response to local mechanical forces. In this Chapter entitled Off-lattice Cell Models mathematical models of single cells are presented whereby each cell is modeled as a discrete entity with well-defined individual cellular characteristics such as intracellular reaction kinetics and biomechanical properties. In this way, cell shape...

## Info

P(r,L,i + e2At) l 2 - Ti(r - -ej, L + ee r, L,t) i i 2 -T2(r - ej, L - ee r, L, i) P(r, L, t) +Ti(r,L r + ej, L - ee,-, i)P(r + ej, L - ee,-, i) +T2(r,L r - ej, L - eej, i)P(r - ej, L - eej, i) +Ti(r,L r - ej, L + eej, i)P(r - ej, L + eej, i) +T2(r,L r + ej, L + eej, i)P(r + Ar, L + eej,i) , (33) where the terms Ti(r, L r', L') and T2(r, L r', L') are transition probabilities for transformations of a cells' length L' and position with center of mass x' to a cell of length L and position with...

## Ip

HDC tumour simulation results in a homogeneous MM (see Fig.3, top row) using the random mutation algorithm in combination with varying oxygen concentrations From t 0-40, oxygen is kept at a high concentration and then t 41-200 oxygen is switched to a very low concentration. Note the switch in morphology between high and low oxygen concentrations. A simulation movie showing the growth of the tumour as well as the other three variables (i.e. MDE, MM and oxygen) for this result can be...

## Viscoelastic Cell Models

Palsson cell elasticity is important sometimes cells need to have the ability to use other cells as stepping stones to move upwards, such as when Dictyostelium pre-spore cells crawl up the stalk cells to form a fruiting body. The force responsible for the upwards motion must be transmitted down to the ground and this can only be achieved if the cells have a solid elastic component In this chapter entitled Viscoelastic Cell Models, mathematical models are...

## Ii L

Velocity channels rest channel V(r) Dynamics in lattice-gas cellular automata. The dynamics of a LGCA arises from repetitive application of superpositions of local (probabilistic) interaction and propagation (migration) steps applied simultaneously at all lattice nodes at each discrete time step. The definitions of these steps have to satisfy the exclusion principle, i.e. two or more particles of a given species are not allowed to occupy the same channel. According to a model-specific...

## Jv

(b + Bfcs) u* +J2 Bfcc E Fi3 + ' (18) Note the analogy to equation (15). Here, Ui dRi dt, B s, B c and b are scalar friction coefficients with the substrate, the neighbor cell, and inner friction, respectively. A reasonable choice is B s 7yc, where b is the cell-internal friction. Fis a generalized force and may be defined by F dVij dRi. Stochastic radius fluctuations maybe neglected 13 . As shown in ref. 13 , this effective description yields a reasonable qualitative description of the growth...

## Mcs

Cell deformation as cell regulation, (a) Cell shape measurements of the polarity of sparse cultured cells ( o) and cells in confluent culture ( 1). (b) Scheme of the mitosis-regulated cell-cycle, used in Eq.(5), in which interphase, cell size, and cell shape are taken into account, (c) Evolution of a tumour, modelled as a 2D cross section (with substrate at the bottom). Mitosis distribution within the cell mass is given through a shade gradient from dark to light, with white...

## Medicine

Chaplain Katarzyna A. Rejniak Mathematics and Biosciences in Interaction Division of Theoretical Biology Botanical Institute University of Bonn Kirschallee 1 D-53115 Bonn e-mail wolf.alt uni-bonn.de Fred Adler (Dept. Mathematics, Salt Lake City) Mark Chaplain (Dept. Math. & Computer Sciences, Dundee) Andreas Deutsch (Div. Theoretical Biology, Bonn) Andreas Dress (Center for Interdisciplinary Research for Structure Formation (CIRSF), Bielefeld) David Krakauer...

## N

Here c represent the collagen fibers, b represents the fibrin fibers (the blood clot) and these two vectors taken together form u (in Eq.l), whereas f* represents the path of the fibroblasts and thus forms part of v* (in Eq.2). The parameters k, pc, dc and df are positive constants, the prime denotes differentiation with respect to time and Zc denotes the angle of the vector c. Eq.2 is given by where a is a positive constant and the function s is the speed of the cells which depends on the...

## N2ofc nk

Here r denote the position of the cell k. For the step sizes AL r - r' Ar eAr_, Ar G 0, aK) with < and the unit vector eAr is chosen due to an isotropic distribution (see section 3 for a description of how the step sizes are linked to the physical parameters). For the selection of the rotation angles we choose the method by Barker and Watts (see 49 ). This allows one to store the orientation of a cell as a vector. T is the characteristic time between two migration events....

## O o o o o

Example of adhesive interaction in the square lattice. Gray dots denote channels occupied by cells, while white dots denote empty channels. With probability W each node configuration on the right side (r)1) is a possible result of the interaction step applied to the middle gray node (-q) of the initial configuration. that rj1 is the outcome of an interaction at node r, when q is the pre-interaction state at r and G G(r) is the local density gradient, is...

## Ooo ooo ooooo oo ooo

-oo - - oo -o o - OO--OO --0*0 > oo -ooo -ooo -OOO -OOO -OO( Figure 3. Propagation in a two-dimensional square lattice with speed m 1 lattice configurations before (s) and after (sp) the propagation step gray dots denote the presence of a particle in the respective channel. (v*(r) (k)) Vi (f, k) (r, k) - rn(r, k) (6) rii r + ma,k + 1) - r i(r, k) Ci((r) (fc)) i 1 where the change in the occupation numbers due to interaction is given by 1 creation of a particle in channel (r, a) 0 no change in...

## P Q 2AAt

Because volume conservation is taking place on the level of the whole cell, pressure is constant throughout one cell. The main assumption for a volume conservation term at the cell level is that an infinitely fast redistribution of intracellular pressures occurs, i.e. pressure differences only occur between cells. This is a reasonable description, because the volume changes in biological cells are not due to compression of the cytoplasm (the fluid inside cells is effectively incompressible),...

## P2

The packing fraction p2 is the fraction of space occupied by close-packed spheres in a plane. By eliminating the ratio re rc from these two equations, we derive a self-consistent expression for the fraction of surface elements. Defining By equating a to unity, we can estimate the critical number of elements Nc below which all elements are essentially surface elements. We find Nc 54 2 32. For large N we can solve Eqn.(12) perturbatively, to find

## P3

Numerically, the leading term yields Ns 4.4 NFrom this, one finds the sobering fact, that even with N 256 elements, the fraction of surface elements is roughly 70 . Now, in the minimal energy configuration, each element-element interaction will contribute a gain of AV to the energy. Core elements, if close-packed, will have 12 neighbors, while surface elements, if close-packed, will have 9 neighbors. The minimum energy will then be buik surf yn 9a +12(1 (15) I have recorded the equilibrium...

## Pnrk HLArfcr[1 firk1i

For a given average number of cells per node p the non-linear Boltzmann equation has a spatially homogeneous stationary solution Question Can this solution be destabilized (i.e. is there pattern formation) In order to answer, we introduce fluctuations Taylor expansion yields the linearized Boltzmann equation Sfi(r + Ci,k +1) - 5fi(r,k) (10) where co 0 and the linearized Boltzmann operator is defined as

## Q1 Q2

Q1 Lx (1 + a)eAr, (3 + a)eAr, (5 + a)eA r, Q2 Ly (1 + (3)eA r, (3 + 3) eAr, (5 + 3) eAr, a 1 for - n, a 0 for - n + 1 2, n G N eAr eAr 3 1 for x- n, (3 0 for n + 1 2, n G N. (45) In what follows, we compare pcpm(r,t) for e 1 with p(r,t), a solution of the continuous equation (41) corresponding to the following choice of parameters Xx Ay 4 7,Lvt 1 ,Jcm 2,p 15, 0.1, Ar 1, At 1. The size of the CPM lattice is chosen to be Lrj pm Lycpm 100 and the simulation is typically run from to 0 to tend 200....

## R L

A3 Fij + 3-k R + Gn RFij + (3 7R)2) V2 Rrl + jr , and dij Ri + Rj 5 (i.e., 6 Si + Sj is the sum of the deformation along the axis between the centers of the closest spheres of the dumb-bells of cell i and cell j), and Ei, Ej are the elastic moduli, < 7j, aj the Poisson ratios of the cells i,j. This takes into account that a homogeneous, isotropic elastic body is completely characterized by two independent material constants, for example the Young modulus E and the Poisson ratio a. We have...

## T

4The existence of such a T depends on energies being similar for nearby configurations, which is true for Ising, Potts, CPM and GGH Hamiltonians. In this case, if we take one spin-copy attempt as our time unit, the average speed, from Si Si+1 is vel(Si Si+1) i VH (5i+1 - Si). (13) Thus, the average time evolution of the configuration obeys the Aristotelian or overdamped force-velocity relation where is an effective mobility. The movements of individual boundary elements of a domain may be quite...

## T T mm

Cell shape and simulation temperature, (a) Wulff construction diagram. The blossoms show the energy per surface length, e, as function of surface angle, 0. Lines shown how the final equilibrium shape (in dashed lines) is found geometrically. Diagrams shown are for nearest neighbour (nn) square lattice (al) and for nn hexagonal lattice (a2). (b) Actual and mean shape of a 2D cell, for different neighbourhoods and temperatures, (c) Actual and mean cell shapes for simulations on a square...

## Vr k 1 0 1 0 0VIr k 0 1 0 1

Example for reorientation of particles at two-dimensional square lattice node r gray dots denote the presence of a particle in the respective channel. No confusion should arise by the arrows indicating channel directions. In the deterministic propagation or streaming step (P), all particles are moved simultaneously to nodes in the direction of their velocity, i.e. a particle residing in channel (r, Cj) at time A is moved to another channel (r + mcj,cj) during one time step (Fig.3)....

## X

Probability densities of Monte Carlo simulations pcpm(i, t) (dotted line), p(x,t) of the Master Eq.(4) (solid line) and the Fokker-Planck Eq.(8) (dashed line) versus x for t tenj. (a) e 0.01 (b) e 0.1. The difference between solid and dashed curves is negligibly small in (a). Number of Monte Carlo simulations is N 2 X 10B. We used c(x) as given by Eq.(27). The initial conditions for each CPM run are chosen as follows. A random pixel in the interval 40, 60 is selected as a center of...

## 1

Typical results of cpm simulations. Time period is t 400. Number of Monte Carlo simulations is N 2 x 10B, c(x) cos(4-ra 100) and e 0.01. Notations for solid, dashed and dotted curves are as in Fig.3. The difference between solid and dashed curves is negligibly small. 3. Continuous limit of a two-dimensional CPM In this section we extend our results to a two dimensional case. We assume that each cell now has a rectangular shape and it moves or changes its shape by adding or removing a...

## II3 The Cellular Potts Model in Biomedicine

Merks Abstract. In this chapter we describe how the the Cellular Potts Model (CPM) has been applied to problems in the biomedical field. Examples are given in epidermal biology, cancer and vasculogenesis. They demonstrate the strength of the CPM and its rich set of extensions, in elucidating biomedically important phenomena. Keratinocytes are the principal cell type of interfollicular epidermis. They are shed at the skin surface and replaced by division in...

## III3 Modeling Multicellular Structures Using the Subcellular Element Model

This chapter describes a new method for simulating grid-free multicellular structures, in which the three-dimensional shape of each cell is dynamically adaptive to its local environment. This is achieved by constructing each cell from subcellular elements. I describe in detail the underlying mathematical equation of motion for the elements, and the additional algorithms which allow for cell growth and cell division. The model is illustrated with the simple example of a growing three...

## Twodimensional Multiscale Model of Cell Motion in a Chemotactic Field

Mark Alber, Nan Chen, Tilmann Glimm and Pavel Lushnikov Abstract. The Cellular Potts Model (CPM) has been used at a cellular scale for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angio-genesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. Continuous models in the form of partial differential, integral or integro-differential equations are used for studying biological...

## III2 Models with Latticefree Centerbased Cells Interacting with Continuum Environment Variables

In this chapter we describe a discrete continuum hybrid method applied to two biological systems. The cells are modeled as discrete objects which are free to move in space (lattice-free), the forces which act on the cells are applied to their center of mass (center-based), and the cells interact with something represented as a continuum variable. Dictyostelium discoideum is the first system modeled by the method. The cells move and communicate with each other through a diffusible...

## The Cellular Potts Model and Its Variants

In this chapter entitled The Cellular Potts Model and Its Variants mathematical lattice-based models are discussed that use stochastic changes at individual lattice sites to define configuration of all cells through the energy minimization process. The chapter by J. Glazier, A. Baiter and N. Poplawski, Magnetization to Morphogenesis a Brief History of the Glazier-Graner-Hogeweg Model, discusses development of the of the Glazier-Graner-Hogeweg (GGH) model starting with its ancestors the Ising...

## II2 The Cellular Potts Model and Biophysical Properties of Cells Tissues and Morphogenesis

Grieneisen and Paulien Hogeweg Abstract. In this chapter we examine the properties of the Cellular Potts Model (CPM) formalism which make it preeminently suitable for modelling biological cells. The most outstanding feature in which CPM differs from other modelling formalisms, is that a cell is modelled as a deformable object, and takes its shape from a combination of internal and external forces which act upon it. The energy minimisation based CPM formalism...

## Latticegas Cellular Automaton Modeling of Developing Cell Systems

Cellular automata can be viewed as spatially extended decentralized systems made up of a number of individual components and may serve as simple models of complex systems. Here, we show that a particular cellular automaton class, lattice-gas cellular automata (LGCA), is well suited for the modeling of developing cell systems characterized by motion and interaction of biological cells. As examples, we present LGCA models of adhesion and chemotaxis. We conclude with a detailed...

## D2r dl d2y dl dlx y35

Note the difference in the definition of the diffusion coefficient D2 here in comparison with the definition of the diffusion coefficient D in the ID case (see Eq.(20)). 3.2. Reduced model in the two-dimensional case where P oitz (r, L) is a Boltzmann distribution given by PBoltz(j, L) -i- exp( AElength) (37) AEiength E(r, L) Emin AXLX + XyLy + LxLy Emin is the minimal value of the Hamiltonian as a function of L for a given r, which is achieved at L limm) -(min) _ 0 2Ay(Jcm - XxLTx) + (Jem - y...

## Il Magnetization to Morphogenesis A Brief History of the Glazier GranerHogeweg Model

Glazier, Ariel Baiter and Nikodem J. Poplawski Abstract. This chapter discusses the history and development of what we propose to rename the Glazier-Graner-Hogeweg model (GGH model), starting with its ancestors, simple models of magnetism, and concluding with its current state as a powerful, cell-oriented method for simulating biological development and tissue physiology. We will discuss some of the choices and accidents of this development and some of the positive and negative...

## Hybrid Multiscale Model of Solid Tumour Growth and Invasion Evolution and the Micro environment

Cancer is a complex, multiscale process, in which genetic mutations occurring at a subcellular level manifest themselves as functional changes at the cellular and tissue scale. The importance of tumour cell microenvironment interactions is currently of great interest to both the biological and the modelling communities. In this chapter we present a hybrid discrete-continuum (HDC) mathematical model of tumour invasion that considers the tumour as a collection of many individual cancer...

## Il Centerbased Singlecell Models An Approach to Multicellular Organization Based on a Conceptual Analogy to Colloidal

In this chapter we present a model framework for multi-cellular simulations which is built on conceptual analogies to colloidal particles. Cells are approximated as homogeneous isotropic elastic sticky objects, capable of migrating, growing, dividing and changing orientation. A cell is parameterized by biomechanical, cell-kinetic and cell-biological parameters. Each model parameter can in principle be determined experimentally. We show some simulation results for in-vitro systems and...