Multiscale modelling of tumour growth induced by circadian rhythm disruption in epithelial tissue 1

Transcription

1 Multscale modellng of tumour growth nduced by crcadan rhythm dsrupton n epthelal tssue 1 D. A. Bratsun D. V. Merkurev A. P. Zakharov L. M. Psmen Abstract We propose a multscale chemo-mechancal model of cancer tumour development n an epthelal tssue. The model s based on transformaton of normal cells nto the cancerous state trggered by a local falure of spatal synchronsaton of the crcadan rhythm. The model ncludes mechancal nteractons and chemcal sgnal exchange between neghbourng cells, as well as dvson of cells and ntercalaton, and allows for modfcaton of the respectve parameters followng transformaton nto the cancerous state. The numercal smulatons reproduce dfferent dephasng patterns spral waves and quasstatonary clusterng, wth the latter beng conducve to cancer formaton. Modfcaton of mechancal propertes reproduces dstnct behavour of nvasve and localsed carcnoma. Keywords Cancer modellng Sgnallng Crcadan rhythms Tmedelay Systems bology 1 The research has been supported by the Mnstry of Educaton of Perm Regon (grant C-26/244) and the Russan Fund for Basc Research (grant r_ural_a). A. P. Zakharov L. M. Psmen Department of Chemcal Engneerng, Technon - Israel Insttute of Technology, 32, Hafa, Israel Tel.: Fax: E-mal: psmen@technon.ac.l D. A. Bratsun Theoretcal Physcs Department, Perm State Humantaran Pedagogcal Unversty, 61499, Perm, Russa D. V. Merkurev Department of Hosptal Pedatrcs, Perm State Medcal Academy, 61499, Perm, Russa

2 1 Introducton Cancer modellng has, over past decades, become really one of the challengng problems for mathematcans, physcsts and other researchers workng n the bologcal scences [1]. One of the man problems of mathematcal modellng of cancer (as, ndeed, of any bologcal system) s the multscale nature of ths phenomenon [2, 3, 4, 5]. One can dentfy at least three characterstc spatal scales ncludng the mcroscopc (the sze of cell s nucleus, less than 1 m ), mesoscopc (cell s sze, 1-1 m ) and macroscopc scale (the sze of the tumour or the organsm as a whole). At the mcroscopc scale, a system of ordnary dfferental equatons can be used to smulate the system wthn the populaton dynamcs, where each varable corresponds to a some bologcal property characterstc to all cells of the same populaton. The paper [6] seems to be the frst to apply of populaton dynamcs models focused on cancer. Ths drecton has been further developed to account for more fne effects (see, for example, the survey [7]). The advantage of ths approach s that the models are easly manpulable, and enable a relatvely fast determnaton of the governng parameters. It omts, however, spatal effects and some other mportant aspects. Another type of modellng, supplementng the populaton dynamcs by certan varables determnng the average structure of a populaton of cancer cells (for example, the age of cells), can be attrbuted to the class of sem-phenomenologcal models wth nternal structure [8]. A large body of lterature has been dedcated to models lnkng the mcroscopc scale to the tumour scale [2]. Ths approach allows one to understand how changes n cell-cell communcatons nfluence macroscopc propertes of the malgnant growth. Most commonly, the tumour s consdered as a contnuous medum. It s descrbed by a system of partal dfferental equatons ncludng the mass balance equaton for the cellular medum and reacton-dffuson equatons descrbng the feld of sgnallng between cells. The tumour may be vewed as a sold porous matrx [9] whch nteracts wth a fllng cellular lqud of healthy cells. In ths way, one can descrbe some spatal features of a real tumour (for example, ts fractal structure). In a more recent work of ths knd [1], the tumour was represented as a multphase medum smulated n a three-dmensonal geometry by a fnte elements method. A dsadvantage of such models s ther nherently phenomenologcal character. An alternatve approach s cell-based dscrete modellng. In contrast wth contnuous medum models, dscrete models enable to trace the functonng of ndvdual cells. Thanks to the progress n botechnologes, there s an ncreasng amount of expermental data avalable at a sngle cell level whch should be taken nto account n mathematcal models. Smpler models of ths knd are based on cellular automata [11] or random walk of cells [12]. The latter approach allows one to consder stochastc processes of tumour formaton but cannot account for ther mechancal propertes. A large group of models combne nteractons at both sub-cellular and cellular level [2, 4]. Ths approach s motvated by the fact that the bologcal system as a whole s drven by genetc mutatons: for example, too strong expresson of

3 certan genes can lead to malfuncton of the cells,.e. transton to the cancerous state [13]. Stochastc descrpton of gene regulaton processes can be naturally appled here [14]. Snce cancer s a multscale phenomenon, the most realstc approach to modellng requres nvolvng processes at all levels of descrpton. Due to the dffcultes of ths approach, there are not so many works of ths knd. One of the frst attempts to mplement a hybrd approach ncluded a model of cellular automata whose state s determned by a contnuous dstrbuton of oxygen around a blood vessel near the orgn of the tumour [15]. In another remarkable work [16], the authors examned a sphercally growng tumour wthn the framework of a lattce model of mechancally nteractng dscrete cells, n whch the gene regulaton processes have been smulated separately for each cell n the populaton. A good revew of recent developments n multscale cancer modellng can be found n Ref. [4]. We would also lke to menton the book [5] coverng state-of-the-art methods of multscale cancer modellng. We conclude that any realstc model of tumour growth should nvolve a dynamcal model of nteracton of a large number of cells [17]. Such a model must take nto account the physcal propertes of ndvdual cells n the ensemble, e.g. to descrbe dynamcally changng volumes and surfaces of cells, ther elastc response to external mechancal mpacts, the ablty to move, dvde, etc. These processes must be nfluenced by the exchange between the cells of the varous sgnals generated at ether subcellular (transcrpton/translatons) or macroscopc level (collectve behavour). The model should combne therefore a dscrete descrpton of ndvdual cell dynamcs and a contnuous descrpton of chemcal felds that are common to the whole ensemble. The development of such models s not easy but the progress n ths area over the last decade and the smultaneous breakthroughs n the computer technology are brngng the researchers closer to the realstc smulaton of the lvng tssue functonng. Ths communcaton combnes modellng of sgnallng regulaton of cancer formaton wth cell-based mechancal modellng of deformatons and rearrangement of epthelal tssue. It has been recognsed n recent years that core crcadan genes are mportant n tssue homeostass and tumourgeness. In recent years, a sgnfcant number of works have shown that dsrupton of the crcadan mechansm s mplcated n gene deregulaton leadng to the development of cancer and other dseases [18, 19, 2]. The man dea of ths work s that the tumour occurance s nduced by a local dsrupton of the crcadan rhythm n the epthelal tssue when the control of the perpheral crcadan oscllators by the suprachasmatc nucle weakens or vanshes completely. We have supplemented the mechancal epthelum model whch was developed earler [21] by two smple models of crcadan rhythms suggested n our prevous papers [22, 23, 24], where the crucal element of the mechansm of oscllatons s a delay of proten synthess durng the process of gene regulaton. These consttutve parts are supplemented by a smple phenomenologcal model of the cell transformaton nto the cancerous state leadng to modfcaton of ts physcal propertes.

4 2 Descrpton of the model 2.1 General prncples Epthelum can be defned as a relatvely avascular aggregaton of cells whch are n apposton over a large part of ther surfaces, and are specalsed for absorptve, secretory, protectve, or sensory actvtes [25]. Epthela may occur as sheets of cells, charactersed as a coverng and lnng epthelum, as n the lnng of the ntestne, or as sold aggregatons of cells, as n glandular organs. We focus our attenton on the frst type, and construct a two-dmensonal model of a sngle layer of epthelal cells. Epthelal cells can be shaped n dverse ways. The three basc cell shapes n coverng and lnng epthela, dstngushed by ther appearance n mcroscopc sectons, are squamous, cubodal, and columnar but the dstnctons are often blurred. The cells adhere to each other by formng specalsed attachment structures (adherens junctons or desmosomes) that ensure coherence and strength of tssue. We wll further concentrate on modellng of carcnomas. Carcnoma s a type of cancer that develops from epthelal cells. It may affect the skn, colon, prostate, breast, and lung, and s among the most wdespread forms of cancer n adults, caused when DNA s altered or damaged to such an extent that the cells start to exhbt abnormal malgnant propertes. The followng propertes of cancer cells wll be taken nto account n our model: a normal cell can turn nto a cancer cell ([1], p.7); cancer cells do not undergo reverse transformaton ([1]; p.7); cancer cells exhbt a number of alteratons of ther physcal propertes n comparson to normal cells ([1], p.285); cancer cells show uncontrolled mtotc dvsons causng dsorgansed growth ([1], p.45; [26, 27]); cancer cells are far less adhesve than the normal cells, and therefore tend to wander through tssues causng cancerous growth n dfferent parts of the body ([1], p.215); cancer cells exhbt a number of alteratons n ther genes: for example, ther crcadan rhythm s dsrupted, and they cease to exchange sgnals wth normal cells ([1], p.39; [26, 27, 28]). In order to arrve at a realstc descrpton enablng to obtan stress feld and velocty dstrbuton n the epthelum, we have developed a dscrete chemo-mechancal model. The man components of our model are mechancal, nvolvng elastc nteractons between cells, ther prolferaton and dvson, and genetc, nvolvng the mechansm of crcadan rhythms and ntercellular sgnallng. The mechansm of the cell transformaton s the thrd mportant element of the model. Ther relatonshps are schematcally shown n Fg.1.

5 Fgure 1: Prncpal components of the model and ther relatonshps The mechancal submodel presents the epthelum as an elastc twodmensonal array of cells, approxmated by polygons. The evoluton of the system starts from a hexagonal cells consttutng the plane. In the course of prolferaton, dvson and transformaton, the ntal lattce deforms ncorporatng also rregular polygons. The deformaton of the epthelal layers composed of cells that tghtly adhere to one another, s a crucal process n tssues snce t plays a central role n gastrulaton, early phase n the embryonc development that ntates the formaton of three-dmensonal structures of tssues and organs [29]. The physcal propertes of cancer cells sgnfcantly dffer from those of the normal cells [1, 26, 27]. Thus, the transformaton of the cell nto cancerous state changes the mechancs of cell behavor. Ths assumpton establshes the mportant lnk between the elements of the model shown n Fg.1. The key feature of our genetc model s the nfluence of perodc nputs. Bologcal rhythms are perodcal fluctuatons, whch are qute characterstc for all forms of lfe on earth. Rhythms are closely connected adaptve phenomena to perodc changes n envronmental factors. We concentrate here on crcadan rhythms that synchronse wth daly changes of the envronment. A remarkable feature of these rhythms s that they are produced by autonomous ntracellular mechansm [3]. To date, t s wdely accepted that the genetc mechansm s responsble for crcadan oscllatons [31]. It s now realsed that crcadan rhythms manfest themselves even on the subcellular scale n the form of RNA and proten fluctuatons. As soon as transport protens pass through the cell membranes and trgger ntracellular nteractons, crcadan oscllatons nevtably develop at the mesoscopc and macroscopc scales. At the organsm scale, the sgnals from ndvdual cells are synchronsed, thus formng generalzed rhythm for the organsm as a whole. Now t s establshed that the man crcadan rhythm n mammals s produced by the suprachasmatc nucle (SCN), the group of cells located n the hypothalamus. One of the prncpal dscoveres was the fndng of ultra-fast synchronzaton of oscllatons wthn 2-3 perods due to strong couplng between SCN neurons va a unque mechansm of neuropeptdergc sgnallng [32]. As a consequence, any pattern formaton cannot occur n such a medum: SCN mantan full synchrony as a populaton [33]. It may mply that the crcadan clock of hgher organsms can behave dfferently [34, 35]. For example, the authors of [35] have proposed to use phase response curve analyss to show that the coupled perod wthn the SCN stays near the populaton mean nstead of a Hll-type regulaton wdely assumed n prevous mathematcal models.

6 Though the SCN clock s the crcadan pacemaker for the whole organsm, the perpheral crcadan oscllators n mammals are not just a passve medum. It was shown that perpheral clocks are cell-autonomous and can produce rhythm rrespectve of SCN [36]. Thus, the master clock rather synchronzes perpheral clocks whch work locally and autonomously, than sends the drect commands to control the local crcadan physology. The problem of the spatal synchronsaton of a large number of nteractng oscllators s actvely dscussed n the physcal lterature [37]. Bologcal applcatons were hndered, untl recently, by the lack of expermental data but the stuaton has begun to change due to the appearance of sophstcated expermental technques. Some examples are gene expresson data analysed from postmortem brans [38], and attempts at spatal analyss of fluorescent genes [39]. But stll most studes largely concentrate on the temporal rather than the spatal organsaton of rhythms. The prncpal ssue for pattern formaton s a local couplng between clocks n the perpheral cells. Although peptdergc sgnallng s absent n the perpheral tssues, other mechansms should be present. For example, t was reported n [4] that the perpheral tssues of SCN-lesoned mouse stll demonstrate some degree of synchronzaton n the local crcadan clocks. Of course, the couplng between perpheral oscllators s much more weaker that n SCN, but t stll exsts [36]. It means that when the SCN control dsappears, the weak couplng of local oscllators leads to greater freedom of each of them n a whole group, and hence to the possble formaton of collectve patterns. Snce there s extensve expermental evdence that gene deregulaton nfluenced by crcadan clock s nvolved n the development of cancer [18, 19, 2], the molecular mechansm of the crcadan rhythm dsrupton and ts propagaton due to cell-to-cell sgnallng should become an essental factor n transformaton of cells. In mammals, t s known now that f the connecton between the perpheral clocks and the pacemaker n SCN s destroyed the organsm becomes at hgh rsk of cancer. For example, t was demonstrated n [41] that when the perpheral tssues n mce deduced from the SCN control were artfcally synchronzed by meal tmng, the cancer growth was reduced by 4%. Analyzng all the above, one can conclude that wthout the SCN control the local crcadan oscllators are left to themselves, and can weakly nteract wth each other, formng poorly synchronzed patterns whch nduce locally the cell transformaton. Ths assumpton establshes the prncpal lnk between these two elements of the model (Fg.1). Ths lnk operates n both drectons: on the one hand, the crcadan patterns cause the cell transformaton ([28, 42, 43], see dscusson below); on the other hand, the crcadan rhythms n cancer cells are ether swtched off or serously damaged ([26, 27, 28]). 2.2 Mechancal cellular model of epthelal tssue We adopt a two-dmensonal mechancal model of the epthelal layer along the lnes of an earler study of wound healng [21]. The cells always reman attached to each other formng a contnuous two-dmensonal

7 epthelal surface (Fg. 2). The curvature of the layer (presumed small compared to the cell sze) and thckness nhomogenetes are neglected. The followng features of the model descrbed n detal below make t sutable for realstc smulatons of the epthelum: allowng for changng the sze and shape of cells; allowng for tssue spreadng va the mechansm of cell dvson (Fg. 2a); allowng for moton of cells by the mechansm of ntercalaton (Fg. 2b); ncludng the dynamcs of sgnallng speces takng part n the regulaton of ntracellular processes; ncludng the exchange of chemcal sgnals between adjacent cells through ther shared borders (Fg. 2c). (a) (b) (c) Fgure 2: Elements of chemomechancal model of epthelum: (a) - cell dvson; (b) - cell ntercalaton; (c) - cell-to-cell communcatons The mechancal model s based on the elastc potental energy U of the tssue [44]: U = P ( A A ), 2 (1) cells where P and A stand for the permeter and area of cell respectvely. Here the coeffcent characterses the effect of contractle forces wthn the permeter of the cell, characterses the elastc resstance to stretchng or compressng the cell wth respect to the reference cell area A. Vertces of the polygons representng the cells form a lattce. Evoluton of the tssue s modeled by movng the lattce nodes (Fg. 2). We defne the mechancal force actng on any j th node n a standard way:

8 where R denote the poston of the node. j F j U =, (2) R We consder the nternal movement of cells n the tssue as strongly overdamped process, so the equaton of moton for th node can be wrtten as dr V = = KF H F F, (3) dt where V s the velocty, K s the moblty coeffcent, H s the Heavsde functon. The threshold force F has been ntroduced n (3) to take nto account the stuaton when the node remans mmoble even f force s not zero. An mportant feature of the model s the ablty of cells to dvde. Ths allows the tssue to carry out nternal movement through the redstrbuton of nternal stresses n the envronment. It s assumed that the dvson occurs when the longest edge of the polygon and ts correspondng opposte edge are dvded n half (Fg. 2a). In order to mnmze the growng dsorder n the dstrbuton of nodes n the process of cell dvson, the probablty to dvde was connected to the number of vertces of the polygon n accordng to the followng formula (see [21] for more detal): n6 p = p q, (4) dv where q s a dstrbuton constant and p s a scalng factor. One can see that for q > 1, the polygons wth a number of sdes over 6 experence the dvson more frequently. Thus, the most lkely shape of the cell accordng to (4) s a hexagon. For cancer cells, the dependence of the number of nodes s suspended and the dvson rate s set at consderably hgher level. In fact, the real epthelal tssues can demonstrate the effect of extruson of some cells leadng to ther death. There s also ncreasng evdence that clones of cells exhbtng dfferental prolferaton rates, such as those nduced through oncogenc transformaton, lead to cell competton effects [45]. For sake of smplcty, we here neglect all these phenomena. Another mechansm whch ncreases the lqudty of the tssue and excludes the severe deformaton of some cells s ntercalaton [21, 46]. We ntroduce here a specal parameter l whch determnes the moment when the ntercalaton can occur. Then the probablty of the event can be wrtten n the smple form: 1, l < l pnt =. (5), l l One can see from (5) that f the length of the border separatng two cells becomes less than l, t s substtuted by a lnk of a slghtly larger length n the normal drecton (Fg. 2b). The ntercalaton s known to be mportant n many tssue reshapng processes. Snce cancer cells are far less adhesve than normal cells, the respectve l s set at a hgher level, thereby makng the ntercalaton of cancer cells more probable. Altogether, Eqs. (1-5) defne the mechancal dynamcs of tssue on the cellular level. j j

9 2.3 Genetc models of crcadan rhythms As mentoned above, we ntend to nvestgate the effect of pattern formaton of perpheral crcadan rhythms on the occurrence of cancer. For ths reason, the mathematcal models of rapd synchronzaton of the entre communty of oscllators developed for SCN cells smlar those n [33, 34, 35], are not very sutable for our purposes. In ths paper we use two smple models of crcadan oscllatons. The frst s a sngle-gene auto-repressor model wth dmerzaton where the negatve feedback loop s delayed n tme suggested n [23]. The second s a two-varable model wth both negatve and postve feedback loops [22], whch s a smplfed verson of tme-delayed models proposed earler [47, 48] for Neurospora. Both models allow not only a full synchronzaton of spatally dstrbuted oscllators, but also demonstrate the pattern formaton at a small enough couplng between oscllators. (a) (b) Fgure 3: Network archtecture of the molecular components of the crcadan rhythm: (a) - Sngle-gene auto-repressor model wth tme-delay where only x gene defnes the oscllatory actvty. stands for the concentraton of proten produced by x gene; (b) - Two-genes model wth postve and negatve feedback loops where there s a par of the prncpal genes, x and y, whch are responsble for mantanng the rhythm n cell. See text for detals, symbol represents steps wth a delay, and stand for correspondng proten concentratons Crcadan model I: Sngle-gene auto-repressor wth tmedelay The classcal Goodwn model s consdered to be a mnmal oscllator based on a negatve feedback loop [49]. It ncludes three-step chan of reactons for the clock gene mrna, the clock proten and the transcrptonal repressor whch nhbts the producton of the frst component. The problem s that no lmt-cycle oscllatons can occur n ths model untl the degree of a Hll term becomes greater than 8 [5]. However, a generalzaton of the

10 Goodwn model to the case of tme-delayed feedback s much more practcal because the oscllatory behavour can be obtaned for the smaller value of a Hll coeffcent [33]. Tme-delay seems to be the most common cause of oscllatons n genetc systems, snce gene regulaton processes are typcally very slow and comprse multstage bochemcal reactons engagng the sequental assembly of long molecules, and therefore are lkely to generate tme delays. In fact, a three-stage scheme of the Goodwn model can be further smplfed, f we take nto account that the delay may also nclude other ntermedate steps of the process. Let us consder a sngle-gene proten synthess wth negatve auto-regulaton Fg. 3a. Ths s a popular motf n genetc regulatory crcuts, and ts temporal dynamcs has been analyzed wthn both determnstc and stochastc framework [51]. The generalzed verson of ths system takng nto account that the producton of the autorepressor proten takes a fnte amount of the delay tme has been studed n [23, 24]. Suppose that the proten whch s responsble for the crcadan rhythms can exst both n the form of solated monomers and dmers D. Both forms are actvely nteract wth each other va the reactons dmerzaton and undmerzaton: k d D, D, (6) where k d, kd stand for reacton rates. We denote the unoccuped and occuped state of the promoter ste of the gene as S and S 1 respectvely. Let us postulate that the chemcal state of the operator stes S { S, S } 1 determnes the producton of correspondng proten at tme t. If the operator at tme t s unoccuped (S ) then the proten may be produced at tme t. Otherwse, f the operator s occuped (S 1 ), the producton at tme t s blocked. The transtons between operator states for each proten occur wth rates k 1, k 1 when some dmer bnds to the promoter or unbnds from t respectvely: k 1 S DS1, S1 S D. An mportant role n ths model comes from the delay n the synthess: AI t S S k d k 1 t t where A I s the reacton rate of a tme-delayed producton. Fnally, the system should be supplemented by the effect of proten degradaton wth rate B :. (9) Altogether, Eqs. (6-9) defne the gene regulaton at a sngle cell level. The dynamcs of the system (6-9) s governed by the followng set of delay-dfferental equatons (DDEs): d ( t) 2 = 2k d ( t) 2kd D( t) AI S ( t ) B ( t), dt B, (7) (8)

11 dd( t) 2 = 2kd ( t) 2kd D( t) k 1S ( t) D( t) k 1S1( t), dt ds ( t) = k 1D( t) S ( t) k 1S1( t), (1) dt ds1( t) = k 1D( t) S ( t) k 1S1( t), dt where the average number of monomers and dmers at tme t are denoted as (t) and D (t) respectvely. The contnuous varables S ( t ) and S ( ) 1 t stand for the average number of unoccuped and occuped operator stes of clock gene at tme t respectvely. Then we assume that S S 1 = 1, (11) where the copy number whch ndcates how many copes of monomers are produced n a sngle act of transcrpton/ translaton s assumed to be 1. The man approxmaton we make here s an assumpton that the reactons of dmerzaton (6) and bndng/unbndng (7) are fast n comparson wth producton/degradaton (8-9),.e. k >> AI, B. Thus, we can suppose that dynamcs of operator-ste and dmers quckly enters nto a local equlbrum, where the concentratons of reagents become 2 2 D =, S = S, (12) 1 where kd / kd, k 1 / k 1. We ntroduce the total number of the folded monomers n all forms as 2 2 T = 2D 2S1 = 2 2, (13) 2 1 where t was taken nto account the Eqs. (11-12). In fact, the last term n (13) s suffcently small compared to the other terms f << 1 and << 1. By takng nto account (11-13), the system (1) can be reduced to the followng sngle equaton for the total number of monomers: T d ( t) AI = B ( t). (14) 2 dt 1 ( t ) In fact, ths equaton s a further smplfcaton of the Goodwn model because ntermedate steps of mult-stage reacton wth the formaton of mrna and transcrptonal repressor have been hdden here n the delay term. A typcal tme seres of the number of the monomers obtaned by solvng Eqs. (14) for a sngle cell s shown n Fg. 4a. For the parameter values gven n the fgure capton the perod of oscllatons s about 27 h. Ths s about three tmes greater than the tme delay. In order to descrbe the effect of ntercellular sgnallng, we suggest for smplcty that the monomer can penetrate to other cell va the mechansm of the dffusve transport. Thus, we can generalze Eq. (14) for the case of a mult-cellular system wth the crcadan sgnallng n the followng way: d ( t) 1 AI = B ( ) 2 t dt 1 4 ( t) 1 ( t ) P ( ( t) ( t)), (15) j jadj( ) j

12 where the subscrpt refers to the -cell. The prefactor has appeared n Eq. (15) due to formula (13) where the number of dmers bndng to the promoter has been neglected. The last term n Eq. (15) descrbes the transport of the sgnallng speces between neghbourng cells and j ( adj () stands for adjacent to -cell ), wth the flux beng proportonal to the boundary length P and tuned by the coeffcent. Ths mples that the transport rate s j lmted by the transfer through cell membranes. After a cell dvdes, the daughter cells nhert the phase of the crcadan rhythm of the parent cell. We assume for the smplcty that the same proten transports the crcadan sgnal outsde the cell. (a) (b) Fgure 4: The typcal tme seres obtaned durng a sngle-cell smulaton: (a) - Dynamcs of the proten wthn the Crcadan Model I defned by Eq. (14) wth parameters = 8 h, A = 5 nm h 1, B = 5 h 1, =.1 nm 1, I =.2 nm 1 ; (b) - Dynamcs of the (sold lne) and (dashed lne) protens wthn the Crcadan Model II defned by Eqs. (26,27) wth parameters = 6 h, A = 8 nm h 1, A = 4 nm h 1, B =.3 h 1, B =.4 h 1, = 5 nm 1, = 5 nm 1 At a sngle-cell level wthout the ntercellular sgnallng ( = ) the equaton (14) has only a postve steady-state soluton: 2 * = B B 1. (16) 2 2 2A 4 2 I A I By lnearzng Eq. (14) near the fxed pont (16), and lookng for a soluton of t the form ( t) : e, where =, we obtan 1 * B = Re( W ( 2A I x e )) B, (17) 1 * B = Im( W ( 2A I x e )), (18) W ( z) where W (z) stands for the Lambert functon defned as W( z) e = z. The condton for the vanshng of the functon (17) gves the analytcal expresson for the Hopf bfurcaton curve, and the frequency of neutral oscllatons s defned by (18). The bg advantage of the models wth delay s the weak dependence of the oscllaton perod on the reacton rates of genetc reactons. Ths s especally mportant when descrbng the crcadan rhythms, as t

13 sgnfcantly ncreases the robustness of the model. The perod of oscllatons n the models wth a strong delay heavly depends on the delay tme. Mathematcally, t can be explaned by the asymptotc behavour of the Lambert functon n (18) Crcadan model II: Two-genes oscllator wth postve and negatve feedback loops Fgure 3b shows a graphcal scheme of the general network archtecture. The genes are labeled as x and y, and ther protens, as and, respectvely. They consttute a system havng both actvatng and repressng elements [22]. One can see from Fg. 3b that the proten dmerses and then works as a postve regulator of by ntatng ts transcrpton. The operates symmetrcally wth respect to the. Durng the crcadan cycle the protens are degraded by turns whch enables the cycle to restart agan and agan. The scheme also takes nto account that the can be also removed by assocatng wth to form a heterodmerc / complex. The synthess of both and occurs wth a delay of several hours (Fg. 3b). The molecular mechansm descrbed above can be presented as a set of bochemcal reactons: d k d D D, (19) D S S 1 d k, d, D, k1 k k k1 (2) D S S D S, (21) k1 A 1 S S S 1 D 1, 1 k1, S1 S B B,, (24) k D A t t, S1 S1,, (22) (23). (25) Here, the reactons of dmersaton and dedmersaton are gven by Eqs. (19) and (2); the dynamcs of operator-stes durng bndng/unbndng of dmers are descrbed by Eqs. (21) and (22); the tme-delayed synthess of protens s gven by Eq. (23); the lnear and non-lnear degradaton are descrbed by Eqs. (24) and (25), respectvely. Assumng that reactons of dmersaton and bndng/unbndng are fast compared to the processes of producton and degradaton of protens, so that they quckly reach a local equlbrum, we obtan two delay dfferental equatons (n fact, the procedure of dervaton s smlar to those for the Crcadan Model I): T d ( t) A = A B ( t) k ( t) ( t), 2 (26) dt 1 ( t ) T d ( t) = A dt 1 A B t 2 ( t ) ( t) k ( t) ( ), (27)

14 where kd / kd, k / 1 k 1, kd / kd, k / 1 k 1 are equlbrum constants of the fast stages. The average number of the and T monomers at tme t are denoted as (t) and (t) respectvely. The and T are the total number of the folded and monomers n all forms defned n the same way as t was done n (13). For smplcty, the delay tme s assumed to be same for both genes. In fact, the delay n the model (26,27) ncludes all possble mult-stage events between the moment of the transcrpton start and the moment when the proten becomes able for regulaton. For example, the delay n Eq. (26) ncludes complexaton and nuclear entry of the complex, translaton and multple post-transcrptonal phosphorylatons. The delay n Eq. (27) s the tme taken for proten to actvate y -gene and the producton of. It ncludes nuclear entry, translaton and possble multple phosphorylaton of. Thus, the phosphorylaton reactons of and / complexes are not taken nto consderaton explctly and are lumped n the delays because the number of phosphorylatons are not defntely known, but only appears to be hgh. Ths s the dfference wth 23-varable model suggested n [47]. The same way was chosen n the paper [48] (see the dscusson n reference). A typcal tme seres of the total number of the and monomers obtaned by solvng Eqs. (26,27) for a sngle cell s shown n Fg. 4b. One can see that the proten domnates n the day tme", reachng ts maxmum at noon when the concentraton of the proten s close to zero. Durng the nght", the stuaton s opposte:, and s maxmal. As before, we can generalze Eqs. (26,27) for the mult-cellular case wth the crcadan sgnallng between cells n the followng way: d 1 t) dt = 1 4 A ( t) 1 ( 2 2 ( t ) B ( ) ( ) ( ) t k t t (28) ( t ) P ( ( t) ( t)), j jadj( ) 2 1 A ( ) t d ( ) = ( ) ( ) ( ) t dt, ( ) B t k t t t 1 ( ) (29) t where the subscrpt refers to the -cell and and are defned as specfc quanttes (per square unt) and are set pontwse n each cell. We assume that the proten transports the crcadan sgnal outsde the cell, whle ts partner s confned wthn the cell volume. Smlar dynamc equatons were studed earler n the framework of a contnuous spatallyextended reacton-dffuson model [22]. In fact, the ntracellular crcadan clock can be generally affected by a cell sze change. The robustness of the oscllatons n ths case was studed recently n [52] wthn the stochastc descrpton. Snce we work here n terms of the concentraton, rather than the number of molecules as t s commonly used n the stochastc models, ths possble dependence s not taken nto account. j

15 2.4 Submodel of transformaton of cells As mentoned above, the man crcadan genes appear to greatly nfluence the tumourgeness. As s known, the rhythmcty demonstrated n the expresson of clock-controlled genes regulates varous functons of cells, such as ther dvson and prolferaton (see, for example, theoretcal papers [53, 54] and expermental observatons [55]). Desynchronsaton of ths rhythmcty can be nvolved n some pathologes, ncludng the development of tumours. There s ncreasng evdence lnkng malfuncton of the boclockwork and pathogeness of cancer [18, 19, 2, 27, 28, 56, 57]. In the revew [28], t was gven the large number of examples of the connecton between crcadan genes, crcadan perodcty, agng-related phenotypes, and cancer. For nstance, t was shown expermentally that mce homozygous for dsrupton of crcadan clock genes that result n loss of rhythmcty have many phenotypc anomales. Among these phenotypes are ncreases n basal cancer occurrence or n the occurrence of cancer followng genotoxc stress or n genetcally cancer-prone models [56]. In another revew [57], t was dscussed the emergng role of crcadan genes n hematologcal and hormone-related malgnances and even was declared that manpulatng crcadan bology s a possble way to fght cancer. But what s the exact mechansm lnkng the crcadan rhythmcty and the occurrence of cancer? Some detals of ths connecton began to appear n the lterature. For example, t was shown recently that transcrptonal regulator at the core of the crcadan clock mechansm hper2 drectly affects hp53 levels whch plays a key role as human tumour suppressor [42]. The mportant result obtaned expermentally was reported n [58]. The authors have checked whether the crcadan rhythm was connected to the cell-cycle oscllaton n mmortalzed rat 1 fbroblasts by observng cell-cycle gene promoter-drven lucferase actvty. It was revealed that there was no drect phase relatonshp between the crcadan and cell rhythms. These data mply that the crcadan system does not govern the cell-mtoss rhythm n rat 1 fbroblasts. These expermental results rather dffer from numerous studes n whch assume a pror that cell mtoss s regulated by crcadan system n mammalan tssues n vvo. Thus, t was suggested that there s no drect couplng between the crcadan rhythm and cell cycle but the tmng of cell mtoss s synchronzed wth the rhythmc host envronment [58]. Another study mplyng that the couplng among cell populaton can plays the role n preventng the carcnoma s [43] where authors theoretcally show that the couplng between neghborng cells sgnfcantly mproves the sustanablty of hp53 pulses. Snce the exact mechansm lnkng crcadan oscllatons wth cancer s stll unknown, we propose a smple phenomenologcal model. Based on the papers cted above, the man dea of the alteraton mechansm s a synchronsaton falure of a local oscllaton phase n the common feld of spatally synchronsed crcadan rhythms n the epthelal tssue. Prelmnary numercal smulatons of the spatally extended problems wthn the crcadan model I (15) and model II (28-29) have shown that when the number of cells s large enough, a complete synchronsaton,.e. the total algnment of the oscllaton phases n all cells, cannot be acheved when the

16 couplng s small (see the numercal examples n the next Secton). Instead, the cells are organsed n collectve spato-temporal patterns ncludng clusters of cells oscllatng wth almost the same phase. The thn layer of cells between clusters exhbt oscllatons wth ntermedate phase. Thus, we ntroduce a new varable whch characterses the local dephasng defned as the phase dfference of the crcadan rhythm of an th cell and the average phase of the adjacent cells: t) =< ( t) ( t) >, (3) ( k kadj() The oscllaton phase can be defned n dfferent ways but we choose a physcal rather than formal defnton and characterse t by the concentraton dfference I ( t) Model I : ( t) =, (31) max( ) II ( t) ( t) Model II : ( t) =, (32) max(, ) where both expressons are normalzed by ts maxmal value. If the -th cell s n a fully synchronsed feld, the value of, obvously, wll be zero. On the other hand, the dephasng (3) ncreases near the boundary of the clusters. The maxmum value of the dephasng would occur n case of an solated cell wthn a phase cluster. Accordng to our hypothess, ths cell s most at rsk to become a cancer cell. Apparently, n a dynamcally changng pattern where cluster boundares are varyng n tme, the cell transformaton should occur much less frequently. (a) (b) Fgure 5: (a) The phase portrat of Eq. (33) wthout nose (top) and wth a non-zero stochastc nput (bottom); (b) A tme seres for a normal cell (blue) and a cell transformatng nto the cancerous state (red). At ths stage, we need to determne more accurately what we mean by cancer n our work. Generally speakng, the metamorphoss of healthy cell nto cancer looks lke a chan reacton caused by ntal errors, whch result n more severe errors, each ncreasngly enablng the cell to avod the controls that lmt normal tssue growth. Ths mples that the cell undergoes several transformatons n sequence, untl t fnally becomes a full-fledged cancer cell, part of the nvasve tumour. Snce n ths paper we focus on the dynamcs of a large number of cells enterng nto the chemo-mechancal nteracton, we

17 restrct ourselves to the smultaneous transformaton of a healthy cell nto a cancer cell, whch allows the use of smple model of bstablty. Bstablty s known to be mportant phenomenon for understandng man events of cellular functonng, such as decson-makng processes durng cell cycle, dfferentaton, apoptoss and so on. Ths concept s also extensvely used to descrbe the loss of cellular homeostass assocated wth early development of cancer [59]. More mportantly, the behavour of bstable systems exhbt hysteress. It means t can generate dfferent output values for the same nput value dependng on ts state, a key property for swtch-lke control of cellular processes. Ths s what we need n our case. Under these assumptons, the equaton for the state of th cell can be wrtten as dz = Z (1 Z )( Z ) N ( t), (33) dt where Z s the state functon, s the dampng parameter, (t) s an uncorrelated zero mean nose nput n the range [1;1], N s the ampltude of the nose, assumed to be multplcatve. Wthout nose, Eq. (33) has two stable statonary solutons: Z =, standng for the normal state of the cell, and Z = 1 correspondng to the cancer state (ndcated n Fg. 5a by black crcles). These solutons are separated by an unstable steady state whch separates ther domans of attracton (open crcle n Fg. 5a). The parameters of the model (see Table 1) are calbrated n such a way that transton would occur under the nfluence of nose, but only at a hgh value of the dephasng (3), and the reverse transton of already transformed cell s mpossble (Fg. 5b). Table 1: Lst of prncpal parameters. Spatal dmensons and force are measured n terms of arbtrary unts L and F respectvely. = 1. FL 1 K = 1. LF 1 h 1 = 1. FL 3 F =.2 F 4 p = 2 1 l =.15 L q = 1.4 = 1 h 1 =.1 L 1 h 1 =.15 = 6. h A = 3 3/2 L 2 = 5. nm 1 = 5. nm 1 k = 3 nm 1 h 1 A = 8. nm h 1 = 5. nm 1 = 5. nm 1 B =.3 h 1 B =.4 h 1 A = 4. nm h 1 N = 2.5 h 1

18 3 Numercal results 3.1 Numercal methods The ntal confguraton of the system s a regular hexagonal lattce comprsng 156 healthy cells and no cancer cells. The shape and locaton of each cell s defned by ts nodes. The tssue as a whole has the form of a strpe wth two free borders wth perodc boundary condtons appled there. The typcal values of the parameters for normal cells are gven n Table 1. Most of the parameter values of mechancal submodel are taken from our prevous work [21], where the values were calbrated so as to gve the realstc results. Parameter values for crcadan rhythms submodel have been taken from [47, 48], plus we add some our own, e.g. for dmerzaton and dffuson. The set of dfferental equatons descrbng the dynamcs of mechancal subsystem (1-5), the crcadan rhythms and sgnallng ((15) or (28-29) dependng on the crcadan model) and the state functon dynamcs (3-33) has been solved usng the explct Euler method, whose stablty was warranted by a suffcently small tme step t =.5. The tme step for the calculaton of the molecular processes n cells was synchronsed wth the step of calculatng the mechancal movement of the tssue cells. In order to avod storng a large amount of data for long tme delay ntervals, we have used a novel numercal smulaton algorthm [22] based on adaptve storng n the computer memory of some selected nodal data only, and a subsequent nterpolaton to calculate the ntermedate values. Ths adaptve numercal scheme for data storng can speed up the calculaton up to 8 tmes wthout vsble devatons from benchmark smulaton. Fgure 6: Evoluton of the proten concentraton (frst column), the dephasng (second column) and the state functon Z (thrd column) n the epthelum composed by 156 cells calculated for the Crcadan model I wth transformaton parameters = 1 h 1, =.15, N =.1 h 1. The evoluton of the system starts from random phase dstrbuton. The upper and lower rows of frames correspond to t = 35 h and 5 h respectvely.

19 Fgure 7: The number of transformed cells (left axs) and spatal dephasng functon R (t) (rght axs) as a functon of tme calculated for the Crcadan model I wth transformaton parameters = 1 h 1, =.15, N =.1 h Synchronsaton of crcadan rhythms Crcadan model I We take as the ntal condton a random phase dstrbuton. In the process of evoluton, the rhythms n cells try to synchronse through a weak nonlnear nteracton gven by the last term n Eqs. (15), generatng the varous spato-temporal patterns. From the perspectve of bology, an arbtrary dstrbuton of phases n cells taken as the ntal condtons looks artfcal, snce synchronsaton of rhythms happens at the stage of embryonc development. Nevertheless, the numercal study wth random ntal condtons allows us to better understand the self-organsaton propertes of the system and to evaluate the mechansms of pattern formaton. The typcal values of the parameters governng the crcadan rhythms n each cell wthn the Model I are as follows: A = 5 nm h 1, B = 5 h 1, =.1 nm 1, =.2 nm 1, = 8 h (see Fg. 4a). Fg. 6 presents the results of numercal smulaton of the temporal evoluton of the system (15). The patterns of the concentraton x, the dephasng and the state functon Z are shown for two consecutve moments of tme. We have found that the spatal dynamcs ncludes the slow development of quas-standng waves whch arse aganst the mean feld oscllatng wth the basc perod 27 h (Fg. 6, left column). The dephasng calculated accordng to (3,31) reaches a maxmum value on the boundary of the standng waves doman (Fg. 6, central column). The cells that fall nto ths regon, are at hgh rsk to transform nto cancerous state. The rght column n Fg. 6 confrms ths concluson: most of the transformed cells are grouped n a zone of uncertan oscllaton phase characterzed by the maxmal dephasng. The mportant pont s the small moblty of the pattern shown n the fgure. Snce the boundary of the standng waves change slowly, the same cells are always at I

20 rsk of the transformaton. Note that the parameter values for the transformaton submodel (33) have been calbrated so that the number of transformed cells was suffcent to demonstrate the effect ( = 1 h 1, =.15, N =.1 h 1 ). In order to characterse how the dephasng jump can speed up the cell transformaton we ntroduce a measure gven by the the dephasng computed n terms of the area of the dssynchronzaton zone (.e. the number of cells where (t) s larger than an arbtrary threshold ) normalsed by the total number of cells n the system versus tme. In other words, we compute as a functon of tme the quantty: M 1 R( t) =, M (34) =1 where = 1 f ( t ) > and = otherwse, M s the total number of cells. In fact, the spatal dephasng functon R (t) s the analogue of the spatal reacton rate calculated commonly n pattern formaton studes n the nonlnear chemstry. The typcal value we have used for the threshold s.2 max. The Fgure 7 present the number of transformed cells (left axs) and spatal dephasng functon R (t) (rght axs) as a functon of tme. It can be seen that the spatal dephasng functon oscllates wth the perod of the basc crcadan rhythm reachng a maxmum value once per day. The absolute majorty of rreversble cell transformatons occur just at ths perod of tme Crcadan model II We found that the spato-temporal dynamcs observed wthn the crcadan model II s more vared compared wth the model I. The prncpal dynamcal mode of the system here s a travellng spral wave pattern, smlar to a related contnuous system [22]. The waves arse from ntal dsturbances, and travel outward from the source, untl ths pattern spreads to the entre area. The wave front s orgnally well ordered, but secondary nstabltes appear due to numerous secondary exctaton nucle. Nonlnear nterplay between wave structures results n the development of a chaotc pattern va the mechansm of core breakup. Spral wave patterns are most common n two-dmensonal systems far from equlbrum [6, 61]. Spral waves are observed n a wde range of bologcal and chemcal systems, observed n lqud phase reacton-dffuson systems [62, 63], on catalytc surfaces [64], and n subcellular processes [65]. Fg. 8 presents the results of numercal smulaton of the temporal evoluton of the system (28,29) wth the effect of cell transformaton governed by Eqs. (3-33). The patterns of the concentraton x, the dephasng and the state functon Z are shown at the same tme t = 5 h. If the degradaton rate s low (the upper row n Fg. 8) the scenaro s as follows. After the evoluton of the system starts from random phase dstrbuton, the nonlnear dynamcs demonstrates two dfferent spatotemporal behavour: the mean feld oscllatons wth the perod about 22.6 h and a spral travellng wave pattern (Fg. 8, frst frame n the upper row).

21 Snce the spral wave moves rapdly n space, the number of cell transformaton occurrng due to dephasng s small (Fg. 8, second and thrd frames n the upper row). Fgure 8: Concentraton of the proten (frst column), the dephasng (second column) and the state functon Z (thrd column) n the epthelum composed by 156 cells calculated for the Crcadan model II shown at t = 5 h. The evoluton of the system started from random phase dstrbuton. The frames correspond to the case of small degradaton wth B =.3 h 1, B =.4 h 1 (the upper row) and large degradaton B = 6 h 1, B = 8 h 1 (the lower row). Please check the Supplementary multmeda fles to see the anmaton. When the rate of degradaton of protens ncreases sharply, we observe, nstead of the travellng waves, quas-steady clusterng. The cells are self-organsed n two large groups wth a partcular phase of oscllatons separated by cells fluctuatng wth ntermedate phases (Fg. 8, the lower row). Cluster formaton n the systems wth a large number of dscrete elements whch exchange sgnals has been also recently observed n nteractng synthetc genetc oscllators [66]. Ths knd of clusterng seems to be a sgnfcant feature of dverse communtes. Therefore t may strongly affect dfferentaton of cells n the varous organs. The effect of the transton to a quas-steady clusterng pattern wth a sharp ncrease of the lnear proten degradaton n cells may be also responsble for ageng of tssues. Our numercal experments show that the clasterng regme, lke n the case of the crcadan model I, alternaton of healthy cells (Fg. 8, second and thrd frames n the lower row) s promoted n the stagnant zone between the clusters. Thus, a sharp rse of proten degradaton and change of the spatal dynamcs can be seen as the key event n the transformaton process.

22 Fgure 9: Evoluton of a non-nvasve tumour n epthelal tssue n tme: the colour code shows the relatve cell sze wth respect to the average sze of a normal cell ( left) and the state functon Z where Z = 1 and Z = stand for cancer and normal cells, respectvely ( rght). The frames arranged from top to bottom correspond to tmes t = 2,4,6,8 h respectvely. The parameters defnng the propertes of a cancer cell are A = 3 3/2 L 2, = 1.2 FL 1 = 1. FL 3, l =.15 L, T = 12 2 h. Please check the Supplementary multmeda fles to see the tumour growth n dynamcs.,

23 Fgure 1: Evoluton of an nvasve cancer tumour n epthelal tssue n tme: he colour code shows the relatve cell sze wth respect to the average sze of a normal cell ( left) and the state functon Z where Z = 1 and Z = stand for cancer and normal cells, respectvely ( rght). The frames arranged from top to bottom correspond to tmes t = 2,4,6,8 h respectvely. The parameters defnng the propertes of a cancer cell are A = 3 3/2 L 2, = 1.2 FL 1, = 1. FL 3, l =.4 L, T = 12 2 h. Please check the Supplementary multmeda fles to see the tumour growth n dynamcs. 3.3 Tumour development The entre set of nformaton avalable to date about the cancer cells suggests that ther chemo-mechancal and bologcal propertes are

24 substantally modfed compared to normal cells. The most mportant dfferences have been underlned n the Introducton. Whle a regular cell stops dvdng f some genetc alternaton occurs, cancer cell contnues to dvde. As a result of contnued cell dvson, many daughter cells wth abnormal DNA emerge. They consttute a separate speces of cells that form ther own medum tumour. To model ths transton, we have ntroduced two dstnct set of parameters charactersng the physcal and chemo-mechancal propertes of the two type of cells, normal and cancerous. The proposed model ncludes the followng specal propertes of cancer cells: the averaged sze of cancer cells s governed by the same reference cell area A (cancer cells can be ether larger or smaller normal tssue cells); changes of the permeter and the area of cancer cells are governed by dfferent values of elastctes and ; the mode of a cancer cell dvson does not follow the probablty relaton (4) dependng on the shape of the cell, but s forced nstead to a specfc perod of dvson T wth a weak scatterng; the mechansm of cell transformaton prevents the reverse transton of cancer cells nto normal state; the hgh moblty (nvasvty) of cancer cells s set by the parameter l, the hgher values of whch ntate the process of ntensve ntercalaton n the ensemble of cancer cells; snce the crcadan genes n cancer cells mutate, and the crcadan rhythm does not work properly, the crcadan dynamcs defnng the rhythm s suspended n cancer cells; the cancer cells do not exchange sgnals wth normal cells and do not partcpate n the synchronsaton of the crcadan rhythm throughout tssue. At the same tme, the cancer cells contnue to nteract mechancally wth tssue, and, n turn, put pressure on the healthy cells. It s natural that each descendant of cancer cell nherts all propertes of ts parent. Fgures 9 and 1 present numercal smulatons of two dfferent types of cancer. Fg. 9 shows the evoluton of a non-nvasve tumour consstng of farly large transformed cells. The parameters defnng the propertes of cancer cells are A = 3 3/2 L 2, = 1.2 FL 1, = 1. FL 3, l =.15 L, T = 12 2 L. In order to focus upon the development of the tumour, the alteraton process has been stopped durng the smulaton run just after the frst cell turned nto a cancer cell. Ths moment of tme was fxed to t = h.