A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix
aa r X i v : . [ q - b i o . CB ] M a y A mathematical insight in the epithelial–mesenchymal-liketransition in cancer cells and its effect in the invasion of theextracellular matrix
Nadja Hellmann ∗ , Niklas Kolbe † , Nikolaos Sfakianakis ‡ Abstract
Current biological knowledge supports the existence of a secondary group of cancer cells within thebody of the tumour that exhibits stem cell-like properties. These cells are termed cancer stem cells ,and as opposed to the more usual differentiated cancer cells , they exhibit higher motility, they aremore resilient to therapy, and are able to metastasize to secondary locations within the organism andproduce new tumours. The origin of the cancer stem cells is not completely clear; they seem to stemfrom the differentiated cancer cells via a transition process related to the epithelial-mesenchymaltransition that can also be found in normal tissue.In the current work we model and numerically study the transition between these two types ofcancer cells, and the resulting “ensemble” invasion of the extracellular matrix . This leads to thederivation and numerical simulation of two systems: an algebraic-elliptic system for the transitionand an advection-reaction-diffusion system of Keller-Segel taxis type for the invasion.
Cancer research aims to understand the causes of cancer and to develop strategies for its diagnosis andtreatment. In this effort, mathematics contributes with the modelling of the biological processes, and thecorresponding analysis and numerical simulations. As a research field, it has been very active since the1950s; see e.g. [23, 3, 8] and has spanned over a wide range of applications from intracellular bio-chemicalreactions to cancer growth and metastasis, see e.g. [25, 24, 2, 11, 9, 21, 30, 22, 13, 32, 12].Intrinsically, cancer cells exhibit higher proliferation rates than normal cells. Recent evidence thoughhave revealed the existence of a subpopulation of cancer cells that posses lower proliferation rates thanthe rest of cancer cells, exhibits stem-like properties such as self-renewal and cell differentiation, andare able to metastasize i.e. to detach from the primary tumour, invade the vascular system, and afflictsecondary sites [4, 19]. These are termed
Cancer Stem Cells (CSCs) and constitute a small part ofthe tumour, while the bulk of the tumour consists of the more proliferative
Differentiated Cancer Cells (DCCs), [31, 27]The origin of CSCs is not completely known; a current theory states that they are de-differentiated cancercells originating from the more usual DCCs [12, 27]. This type of transition of cancer cells, influencestheir cellular potency and seems to be related to a type of cellular differentiation program that can befound also in normal tissue, the
Epithelial-Mesenchymal Transition (EMT) [19]. The EMT takes placein the first step of metastasis, the invasion of the
Extracellular Matrix (ECM) by the primary tumour,cf. [31, 14].The EMT is triggered by the micro-environment of the cell [26]. The protein, for example,
EpidermalGrowth Factor (EGF) and its corresponding cellular receptors EGFR plays pivotal role in this triggering[33, 28]. This process can be briefly described as follows: the EGFs function as “on”/“off” switchesthat stimulate cellular growth and proliferation. In certain pathological mutations, they get stuck in the“on” position and cause unregulated cell growth, untimely EMT, and over-expression of EGFRs. Theover-expression though, of EGFRs has been connected to the tumor formation in mammary epithelialcells, see e.g. [10]. ∗ Institute of Molecular Biophysics, University of Mainz, Germany [email protected] † Institute of Mathematics, University of Mainz, Germany [email protected] ‡ Institute of Mathematics, University of Mainz, Germany [email protected] haptotaxis . In haptotaxis, the cells move biased, along directions being dictated by theavailability or the gradients of either extracellular adhesion sites (which are located on the ECM), or ofECM-bound chemoattractants.From the macroscopic point of view haptotaxis (as the other forms of taxis) is typically modelled underthe paradigm of Keller-Segel systems which was first introduced in [15] to describe the aggregations of slime mold . Since their first derivation, they have been successfully applied in a wide range of biologicalphenomena spanning from bacterial aggregations to wound healing, and cancer modeling. In cancergrowth modelling in particular, Keller-Segel type systems were extended to include also enzymatic in-teractions/reactions, yielding
Advection-Reaction-Diffusion (ARD) models, see for instance the generichaptotaxis system proposed in [2] (promoting ECM degradation by
Matrix Metalloproteinases (MMPs)),or the chemo-hapto-taxis system proposed in [5] that promotes the role of the enzyme plasmin .From a numerical point of view, the treatment of this type of problems is a challenging task, since theirsolutions develop (generically) heterogeneous spatio-temporal dynamics in the form of merging/emergingclusters of cancer cell concentrations, see for instance Figure 3. To treat these dynamics we employa second order Finite Volume method equipped combined with a third order Implicit-Explicit timeintegration scheme, see Section 2 and our previous works [17, 16] for more details.In this work, our objective is twofold; first we model the EMT transition from DCCs to CSCs, triggeredby the availability of free EGF molecules in the extracellular environment, and of the correspondingEGFR cellular receptors, and secondly we embed this EMT model in an ECM invasion system thataddresses the “ensemble” CSCs and DCCs. We subsequently perform some numerical experiments toinvestigate properties of the combined system.In more detail, we construct a parabolic system describing the concentrations of EGFs and EGFRsand their attachment/detachment dynamics. These processes are assumed to be much faster than thecellular motility and invasion of the ECM, hence we rescale the previously developed system and deducean elliptic system that addresses the free and bound EGF. The rescaled elliptic system serves as thelarge time asymptotic limit of the parabolic system.To model the ECM invasion, we employ an ARD haptotaxis model that we adjust to include the CSC-DCC “ensemble” and the EMT between them. In particular we assume that the cancer cells diffuse inthe environment, move responsively to the gradients of the ECM, proliferate in a logistic manner, andmost notably undergo EMT. We also include the ECM degradation by a matrix degenerating MMP,which in turn is assumed to be produced by both families of cancer cells.With the combination of these models we are able to reproduce the current biomedical understandingthat the CSCs a) stem from the bulk of DCCs, b) they separate themselves from the main body of thetumour, c) they invade the ECM while exhibiting highly dynamic clustering phenomena.Similar ECM invasion systems that can be found in the literature, take into account multiple subpopula-tions of cancer, can be found (see e.g. [1, 7]). These systems however control the transition between thesubpopulations in terms of time dependent Heavyside step functions, while in our approach the suggestedmodels connect the microscopic dynamics of EGF with the macroscopic dynamics of EMT and ECMinvasion.
Typical modeling of the ECM invasion by a single type of cancer cell includes terms for diffusion, taxis,reaction, and proliferation. For two types of cancer cells, the corresponding equations apply with anaddition of a coupling term, which in our case describes the transition from DCC to CSC via EMT, i.e. ∂c CSC ∂t = diffusion + taxis + µ EMT c DCC + reaction + proliferation ∂c DCC ∂t = diffusion + taxis − µ EMT c DCC + reaction + proliferation . (1)2here c CSC , c DCC represent the densities of the CSCs and the DCCs respectively. The coefficient µ EMT describing the probability of the EMT transition is given by Michaelis-Menten kinetics (see also [34]) µ EMT = µ [ g ] DCC b µ / + [ g ] DCC b , (2)where [ g ] DCC b represents the concentration of the EGF molecules that are bound on the correspondingEGFR receptors on the membrane of the DCC cells, and where µ is the maximum EMT rate, and µ / is the Michaelis constant, i.e. the bound EGF needed to generate half maximal EMT.In addition to [ g ] DCC b , we set [ g ] f to represent the concentration of free EGF molecules (in the extra-cellular environment), [ g ] the total (free and bound on both DCCs and CSCs) concentration of EGF.Accordingly, [ r ] , [ r ] b , and [ r ] f denote the total, occupied, and free receptors on the surface of both typesof cancer cells.Throughout this work we assume that all [ g ] and [ r ] concentrations depend on time and on space;this is so since the free EGF molecules are able to translocate (for example diffuse) in the extracellularenvironment, and the bound EGFs and the EGFRs are to be found on the surfaces of the (also time andspace dependent) CSCs and DCCs. We assume moreover that the total EGFs and EGFRs obey localmass conservation properties, i.e. [ g ] ( t, x ) = [ g ] f ( t, x ) + [ g ] b ( t, x ) (3)[ r ] ( t, x ) = [ r ] f ( t, x ) + [ r ] b ( t, x ) (4)for every t ≥ x ∈ Ω ⊂ R bounded. The parabolic system.
The dynamics dictate that free EGF molecules bind onto free EGFR receptorswith rate k + , while at the same time bound EGF molecules detach with rate k − (see also [34]). It isassumed that the k + and k − rates are the same for both types of cancer cells. That is, the time evolutionof both the free and the bound EGF molecules is given by the system: ( ∂ τ [ g ] b = k + [ g ] f [ r ] f − k − [ g ] b ,∂ τ [ g ] f = D f ∆[ g ] f − k + [ g ] f [ r ] f + k − [ g ] b . (5)where τ is the corresponding time variable. We have also assumed that the free EGF molecules diffusein the extracellular environment.Recalling now the mass conservation of the EGFRs (4), we can deduce that:[ r ] f = [ r ] − [ r ] b = λ CSC c CSC + λ DCC c DCC − [ g ] b , (6)where λ CSC , λ
DCC represent the total “number” of EGFR receptors per cell per family, and where wehave used that the bound EGF molecules coincide with the bound EGFR receptors.
The elliptic system.
The EGF dynamics (i.e attachment/detachment onto EGFRs and diffusion) aremuch faster than the dynamics of EMT and ECM invasion. We rescale hence the time variable τ inSystem (5), to match the time scale t of the systems (1), (19). To this end, we first rewrite (5) in theform ∂ τ u¯ = D ∆u¯ + p (u¯) , where u = ([ g ] b , [ g ] f ) T , D = (0 , D f ) T , and where p denotes the nonlinear reaction terms in (5). Weemploy now the change of variables v¯( t, x ) = u¯( τ ( t ) , x ) , (7)with the “much faster” time variable τ given by τ ( t ) = tε , < ε << . (8)3elation (7) yields after ∂∂t differentiation as ε∂ t v¯ = D ∆v¯ + p (v¯) , (9)since ∆v¯( t, x ) = ∆u¯( τ ( t ) , x ) , p (v¯( t, x )) = p (u¯( τ ( t ) , x )) , ∂τ ( t ) ∂t = 1 ε , ∂x∂t = 0 . We denote the above introduced variables in the t timescale as follows (cid:18) g b ( t, x ) g f ( t, x ) (cid:19) = (cid:18) [ g ] b ( τ ( t ) , x )[ g ] f ( τ ( t ) , x ) (cid:19) = v¯( t, x ) . Taking now in (9) the formal limit as ε →
0, we deduce the elliptic system that corresponds to (5): ( k + g f ( λ CSC c CSC + λ DCC c DCC − g b ) − k − g b , D f ∆ g f − k + g f ( λ CSC c CSC + λ DCC c DCC − g b ) + k − g b . or k D g f ( λ CSC c CSC + λ DCC c DCC − g b ) − g b , g f . (10)where k D = k − k + represents the attachment/detachment ratio of the EGFs onto the EGFRs. Though (10)does not include time derivatives, both concentrations g b and g f depend on time t through the densitiesof the cancer cells c CSC and c DCC . Neumann and additional conditions
We augment the system (10) with homogeneous Neumannboundary conditions, ∂g f ( x ) ∂ n = 0 , x ∈ ∂ Ω . (11) Remark 1.
The system (10) - (11) has no unique solution. This can be seen by making the ansatz g b = K ( λ CSC c CSC + λ DCC c DCC ) , K ∈ (0 , . For this choice the constant function g f = k D K − K , satisfies the algebraic equation of (10) , the boundary condition (11) , and ∆ g f = 0 . Thus for every K , aclassical solution of (10) , (11) exists. To recover uniqueness in the system (10)-(11) we impose an additional condition, namely we set the totalmass of EGF to be constant (in time t ), M = Z Ω g ( t, x ) dx = Z Ω g b ( t, x ) dx + Z Ω g f ( t, x ) dx. (12)Condition (12) comes as a natural consequence of the system (5) endowed with zero Neumann boundaryconditions. The total mass of EGF in this system does not change with time τ since, after integrationof the second equation of (5) over Ω, we see that ddτ Z Ω [ g ] b dx + ddτ Z Ω [ g ] f dx = Z Ω ∂ τ [ g ] b dx + Z Ω − ∂ τ [ g ] b + D f ∆[ g ] f dx = 0 . In effect, the total mass of EGF in the elliptic system (10) given by (12) will be equal to M = Z Ω [ g ] f dx + Z Ω [ g ] b dx, in the parabolic system (5) for given initial concentrations [ g ] b and [ g ] f .4 C S C / DCC c on c en t r a t i on µ E M T µ EMT
Figure 1:
Plot of µ EMT for the system (19) - (22) at time t = 6 . Note that it is space dependent and that it“lives” over the support of the DCCs. Solution of the elliptic system.
We note at first that the Laplace equation in System (10) has aclassical solution for which the gradient ∇ g f is unique. We can thus deduce that g f = ˜ g f is constant.Accordingly, the first equation of the system (10) implies g b = k D ˜ g f k D ˜ g f (cid:0) λ CSC c CSC + λ DCC c DCC (cid:1) , = ˜ g f k D + ˜ g f (cid:0) λ CSC c CSC + λ DCC c DCC (cid:1) . (13)Similar, but slightly more involved, is the derivation for the DCC-bound EGF. We repeat the assumptionthat the attachment/detachment rates k + , k − are the same for both families of cancer cells, and that thesuperposition principle [ r ] f = [ r ] CSC f + [ r ] DCC f with [ r ] CSC f , [ r ] DCC f the free EGFRs on the CSCs, DCCsrespectively, holds. Accordingly the DCC-bound EGF is given by g DCC b = ˜ g f k D + ˜ g f λ DCC c DCC . (14)On the other hand, the free EGF ˜ g f (which is constant) satisfies the quadratic equation0 = | Ω | k D ˜ g f + (cid:18) | Ω | + 1 k D Z Ω λ CSC c CSC + λ DCC c DCC dx − M k D (cid:19) ˜ g f − M, (15)obtained after integration of (10) with respect to x and taking (12) into account. The equation (15) hasa single positive root since,∆ = (cid:18) | Ω | + 1 k D Z Ω λ CSC c CSC + λ DCC c DCC dx − M k D (cid:19) + 4 1 k D | Ω | M ≥ − k D M | Ω | ≤ . The root itself i.e. the free EGF, is given by˜ g f = − (cid:16) | Ω | + k D R Ω λ CSC c CSC + λ DCC c DCC dx − M k D (cid:17) + √ ∆2 | Ω | k D = − (cid:0) | Ω | k D + R Ω λ CSC c CSC + λ DCC c DCC dx − M (cid:1) + k D √ ∆2 | Ω | (17) Remark 2.
We note that ˜ g f , though constant in space, is not constant in time since R Ω λ CSC c CSC + λ DCC c DCC dx varies with time t . In effect, g DCC b and µ EMT depend on time. Note also that although ˜ g f is constant in space, g DCC b (14) is not constant in space, so neither is µ EMT (2) constant in space. b [g] f g b g f (a) t = 0 b [g] f g b g f (b) t = 2 b [g] f g b g f (c) t = 40 Figure 2:
Numerical solutions of (5) converge towards the numerical solution of (10) as t increases, ifNeumann boundary conditions (11) are assumed for both systems and if system (10) is augmented by thecondition R Ω g b + g f dx = R Ω [ g ] f +[ g ] b dx . The parameters D f = 0 . , k + = 0 . , k − = 0 . and time independent λ CSC c CSC + λ DCC c DCC = e − x − for x ∈ Ω = (0 , were used. Combining now (2) with (14) we deduce that µ EMT = µ ˜ g f λ DCC c DCC µ / k D + µ / ˜ g f + ˜ g f λ DCC c DCC , (18)which is non constant in space. We refer to Figure 1 for a graphical representation of µ EMT in theexperimental scenario (19)-(22) at time t = 6. Comparison of the systems (5) and (10) . The benefit that stems from the lack of time variablein (10) is the instantaneous “sensing” of the cancer cell concentration as opposed to the “large” timeasymptotic convergence of the system (10).The solutions of the elliptic system (10), coincide with the large time asymptotic solution of the parabolicsystem (5) endowed with zero Neumann boundary conditions (11) if a selection criterion –in particular aprescribed total mass of EGF M = R Ω [ g ] f dx + R Ω [ g ] b dx in (10)– is employed. This can be also observednumerically in Figure 2.Hence, at every instance of the “large” time scale t , at which EMT and cell movement take place, wecan employ the elliptic system (10) to compute the density of the free EGF (17) and accordingly deducethe EMT coefficient from (18).From a numerical point of view, this process is cheap since no PDE solvers are needed. The ECM invasion model we propose is based on the “generic” invasion model of [2]. In this respectit includes haptotaxis, degradation of the ECM by MMPs, and production of the MMPs by the cancercells. In addition to this basic model, we assume a secondary family of cancer cells, EMT transitionbetween DCCs and CSCs, logistic proliferation for the cancer cells, and production of the MMPs by6 (a) t = 0 . (b) t = 1 . (c) t = 6 . (d) t = 10 . (e) t = 13 . (f) t = 15 . Figure 3:
Time evolution of the cancer invasion system (19) - (22) . (a) the CSCs are produced –via EMT–by the DCCs, (b)-(c) due to their higher motility, the CSCs escape the main body of the tumour and invadethe ECM faster than the DCCs. Clearly two propagation fronts have been formed, (d)-(f ) the CSCs presentdynamic merging and emerging concentrations; the DCCs propagate into the ECM slower, smoother, andwithout the wealth of phenomena of the CSC. both families of cancer cells. It reads itself as follows: ∂c CSC ∂t = D ∇ c CSC − χ ∇ (cid:0) c CSC ∇ v (cid:1) + µ EMT c DCC + µ c CSC (1 − c CSC ) ∂c DCC ∂t = D ∇ c DCC − χ ∇ (cid:0) c DCC ∇ v (cid:1) − µ EMT c DCC + µ c DCC (1 − c DCC ) ∂v∂t = − δmv∂m∂t = D m ∆ m + α c CSC + α c DCC − βm (19)For the rest of this work we assume that (19) is equipped with the initial conditions: c CSC0 ( x ) = 0 ,c DCC0 ( x ) = e − x . ,v (0 , x ) = 1 − . c DCC0 ( x ) ,m (0 , x ) = 0 . c DCC0 ( x ) , x ∈ Ω , (20)where Ω = [0 , . P = (cid:8) D = 3 . × − , D = 3 . × − , D m = 2 . × − , χ = 4 × − , χ = 4 × − ,µ = 0 . , µ = 0 . , α = 0 . , α = 0 . , δ = 2 , β = 1 (cid:9) , (21)and for computing µ EMT , we have used P µ EMT = (cid:8) µ = 0 . , λ CSC = 1 . , λ DCC = 1 , M = 1 , k D = 0 . , µ / = 2 (cid:9) , (22)7 −2 D C S C / DCC d i s t an c e −2 D t o t a l m a ss CSCsDCCs10 −2 C S C / DCC d i s t an c e −2 t o t a l m a ss CSCsDCCs10 −2 µ C S C / DCC d i s t an c e −2 µ t o t a l m a ss CSCsDCCs
Figure 4:
Propagating fronts-distance (left), and total masses (right) of the CSCs and the DCCs as functionsof k D = k − k + (upper), M (middle), and µ / (lower) in the case of the cancer invasion system (19) - (20) withthe rest EMT parameters as in P µ EMT , (22) ; the ECM parameters are as in P , (21) . An increase of both, theratio k D and the Michaelis constant µ / weakens the migration and the growth of CSCs. Increasing the EMTmass M on the other hand yields a larger CSC/DCC distance and a higher amount of CSCs. see also (18), (17), (16).The above parameters in P were chosen in the magnitudes in which they were used in similar models (cf.[1], [20]). Due to the properties of cancer stem cells, we chose their diffusion, and haptotaxis coefficientsto be larger than the corresponding numbers for the differentiated cancer cells. The parameters in (22)were extracted by numerical experimentation and are such that the system (19) exhibits the currentbiological understanding of the problem.The numerical method employed on this system is a second order Finite Volume (FV) method in spacecombined with a third order implicit-explicit time integration scheme (IMEX3). This numerical methodhas been developed for such type of cancer invasion models that exhibit merging/emerging concentrationphenomena. For more details we refer to our previous works [17, 16] and to [6, 18].In Figure 3 we present the dynamics of the system (19)-(22). After a short period of time, some DCCsat the ECM degenerating tumor front undergo de-differentiation and become CSCs. They exhibit highermotility and invasiveness than the DCCs and escape the main body of the tumor, they also exhibit moredynamic phenomena than the DCCs. 8e further study the influence of modifications of the EMT coefficient on the solution of the system.Therefore we compute numerical solutions of the system (19)-(21) and various P µ EMT , and measure thedistance (in x ) from the propagating CSC front to the slower propagating DCC front at t = 20 as wellas the total mass of CSCs and DCCs on Ω at the same time instance.In Figure 4 we first present the effect that various values of k D have in the afore mentioned propagating-front-distance and CSCs, DCCs masses. An increase of the detachment/attachment ratio yields a de-crease of the front distance and the mass of CSCs, but an increase of the total mass of DCCs. Similarly,Figure exhibits the effect that M has on the distances of the fronts and the masses. Both, the frontdistance and the CSC mass increase with M but the ascent stagnates for large M . The opposite is truefor the mass of DCCs. Finally, Figure 4 shows that the model proposes less DCCs and a smaller distanceof the fronts for larger values of µ / .This behavior matches our biological understanding of the EMT system. A larger amount of EGF (large M ) strengthens EMT and thus proliferation and invasiveness of CSCs. An increase of detachment overattachment of EMT molecules ( k D ) decreases the number of bound EGF and weakens the EMT process,which is also the case when heightening the Michaelis constant µ / . The saturating behaviour at high M , and low k D and µ / , respectively, reflects that maximal receptor saturation and µ EMT , respectively,is reached. Thus, further increase/decrease of parameters does not change the dynamics, and distanceand proliferation of CSC any more.
In this work we give a mathematical insight to the EMT and its impact on the process of cancer invasion.In particular, we develop an algebraic-elliptic model (10) to compute the amount of free EGF moleculesin the vicinity of a tumour. We are then able to evaluate the non-constant µ EMT coefficient that drivesthe de-differentiation from DCCs to CSCs. Subsequently, we propose a Keller-Segel type system (19) forthe ECM invasion by both types of cancer cells.The numerical simulations that follow, although they serve here as proof of concept, exhibit clearly thatthe “combined” system is able to reproduce the current bio-medical understanding regarding the EMTfirst step of cancer metastasis.Our future work includes the extension of the EMT and the ECM invasion models to 2-dimensionalspaces, the inclusion of further bio-chemical interactions of the cancer cells with the environment, andthe use of more biologically relevant parameters.The effort invested in this work, and its importance is justified by the fact that in the evolution ofcancer, the regulation of EMT is seen as potential drug target in cancer medicine [29, 10]. A betterunderstanding, hence, of these dynamic transitions and their relation to the cancer invasion of the ECMwill be beneficial to optimize drug targeting.
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