Adhesion-driven patterns in a calcium-dependent model of cancer cell movement
Katerina Kaouri, Vasiliki Bitsouni, Andreas Buttenschön, Rüdiger Thul
mmanuscript No. (will be inserted by the editor)
Adhesion-driven patterns in a calcium-dependent model of cancer cellmovement
Kaouri, K · Bitsouni, V · Buttenschön, A · Thul, R
Received: date / Accepted: date
Abstract
Cancer cells exhibit increased motility and proliferation, which are instrumental in the forma-tion of tumours and metastases. These pathological changes can be traced back to malfunctions of cellularsignalling pathways, and calcium signalling plays a prominent role in these. We formulate a new model forcancer cell movement which for the first time explicitly accounts for the dependence of cell proliferationand cell-cell adhesion on calcium. At the heart of our work is a non-linear, integro-differential (non-local)equation for cancer cell movement, accounting for cell diffusion, advection and proliferation. We also employan established model of cellular calcium signalling with a rich dynamical repertoire that includes experimen-tally observed periodic wave trains and solitary pulses. The cancer cell density exhibits travelling fronts andcomplex spatial patterns arising from an adhesion-driven instability (ADI). We show how the different cal-cium signals and variations in the strengths of cell-cell attraction and repulsion shape the emergent cellularaggregation patterns, which are a key component of the metastatic process. Performing a linear stabilityanalysis, we identify parameter regions corresponding to ADI. These regions are confirmed by numericalsimulations, which also reveal different types of aggregation patterns and these patterns are significantly af-fected by Ca . Our study demonstrates that the maximal cell density decreases with calcium concentration,while the frequencies of the calcium oscillations and the cell density oscillations are approximately equal inmany cases. Furthermore, as the calcium levels increase the speed of the travelling fronts increases, whichis related to a higher cancer invasion potential. These novel insights provide a step forward in the design ofnew cancer treatments that may rely on controlling the dynamics of cellular calcium. Keywords
Cancer cells · Non-local model of cancer · Calcium · Cell-cell adhesion · Travelling wave · Aggregation patterns · Adhesion-driven instability · Oscillatory signalling pathway
K. KaouriSchool of Mathematics, Cardiff University, CF24 4AG, UKTel.: +442920875259E-mail: KaouriK@cardiff.ac.ukV. BitsouniSciCo Cyprus, Nicosia 1700, Cyprus & School of Mathematics, Cardiff University, CF24 4AG, UKE-mail: [email protected]. ButtenschönDepartment of Mathematics, University of British Columbia, Vancouver V6T 1Z2, BC, CanadaE-mail: [email protected]. ThulSchool of Mathematical Sciences & Centre for Mathematical Medicine and BiologyUniversity of Nottingham, Nottingham, NG7 2RD, UKE-mail: [email protected] a r X i v : . [ q - b i o . CB ] M a r Kaouri, K et al.
Mathematics Subject Classification (2000)
MSC 35B36 · MSC 35Q92 · MSC 35R09 · MSC 70K50 · MSC 92C15 · MSC 92C17 · MSC 92-08
Cell-cell adhesion and cellular proliferation are fundamental features of multicellular organisms, along withcell division, migration and apoptosis. These processes are orchestrated and coordinated by a multitude ofcellular signalling pathways (Alberts et al, 2000). When these signalling cascades are disturbed, numerouspathologies ensue, including cancer. Amongst the many molecular changes that characterise cancer, alter-ations of intracellular calcium (Ca ) signalling have been identified as a crucial driver (Colomer and Means,2007). In particular, Ca has been reported as a key factor in cellular proliferation (Roderick and Cook,2008; Shapovalov et al, 2013) and in cellular adhesion (Weinberg, 2013). Here, we formulate and analyse forthe first time a model that describes the evolution of a cancer cell density incorporating the effects of Ca in the adhesion and proliferation processes.Rising levels of intracellular Ca have been shown to increase the proliferation of cancer cells in variouscancer types such as breast and prostate cancer, melanoma, hepatocellular and non-small-cell lung carcinoma(Prevarskaya et al, 2014, 2018). Experiments (Simpson and Arnold, 1986; Taylor and Simpson, 1992) haveshown that increasing extracellular Ca levels increased intracellular calcium Ca levels, which increasedthe cell number and the DNA synthetic ability of cell lines.Cellular adhesion is mediated through cadherins, which are transmembrane proteins and belong to theclass of calcium-dependent cell adhesion molecules (CAMs) (Weinberg, 2013). As an example, considerepithelial cells, which bind to each other by linking the extracellular domains of E-cadherins (Morales et al,2002). The cytosolic domain of E-cadherin binds to β − catenin, which in turn binds to the cytoskeleton.Changes in the function of β − catenin result in the loss of the ability of E-cadherin to sustain sufficientcell–cell adhesion (Makena and Rao, 2020; Wijnhoven et al, 2000), while alterations in any type of cadherinexpression may affect cell adhesion and signal transduction (Cavallaro et al, 2002). Intracellular Ca directlyimpacts on the dynamics of both cadherins and catenins (Ko et al, 2001). Moreover, Hills et al (2012) haveshown that activation of extracellular Ca -sensing receptors leads to an increase in E-cadherin expressionand an increase in the binding of β − catenin. In cancer, disrupted cell-cell adhesion due to abnormal expressionof cadherins and their associated catenins has been linked to metastasis (Morales et al, 2002). For instance,(Byers et al, 1995; Cavallaro and Christofori, 2004) have shown a reduced expression of cadherins in variouscancer types, including melanoma, prostate, breast cancer, invasive carcinomas and carcinoma cell lines, andcancers of epithelial origin, when Ca levels are increased. This results in a reduced force between cells andconsequently to cell migration. These results are in line with findings that show that altering CAM functionin metastatic cancer cells blocked their ability to invade healthy tissue and move to secondary sites (Kotteaset al, 2014; Naik et al, 2008; Slack-Davis et al, 2009; Zhu et al, 1992). Taken together, the combined changesin cell-cell adhesion and the increase in the proliferation rate and their dependence on Ca are importantmechanisms in cancer and enhance the formation of cancer cell clusters/aggregations that can migrate in acollective manner, a process critical for cancer progression (Friedl et al, 2004; Glinsky et al, 2003; Knútsdóttiret al, 2014).Ca signalling uses an extensive molecular repertoire of signalling components termed the Ca signalling“toolkit" (Berridge et al, 2000). A key feature of Ca signalling is Ca release from the EndoplasmicReticulum (ER) to the cytosol through inositol-1,4,5-trisphosphate (InsP ) receptors (InsP Rs). Togetherwith Ca resequestration from the cytosol through sarco-endoplasmic Ca ATPase (SERCA) pumps, aprocess known as calcium-induced-calcium release can give rise to intracellular Ca oscillations (Berridgeand Galione, 1988; Berridge et al, 2000; Parekh, 2011; Dupont and Combettes, 2016; Thul et al, 2008; Dupontet al, 2011a, 2016a; Schuster et al, 2002; Uhlén and Fritz, 2010; Powell et al, 2020; Sneyd et al, 2017). Inaddition, Ca can spread across a population of cells, forming an intercellular Ca wave (Bereiter-Hahn athematical model of cancer and calcium 3 (2005); Charles et al (1993, 1991, 1992); Deguchi et al (2000); Narciso et al (2017); Sanderson and Sleigh(1981); Yang et al (2009); Young et al (1999)).Mathematical models of intracellular Ca oscillations vary substantially in their complexity, ranging fromtwo coupled nonlinear ordinary differential equations (ODEs) to three-dimensional hybrid partial differentialequations (PDEs) — see (Dupont et al, 2016a; Falcke et al, 2018) for recent perspectives. In the presentstudy, we employ the model developed in (Atri et al, 1993), which for simplicity we will call the ‘Atri model’.The Atri model is a so-called ‘minimal’ model consisting of only two ODEs that can generate non-linearrelaxation oscillations at constant InsP concentrations (Dupont et al, 2016b; Keener and Sneyd, 2009a,b).Importantly, the Atri model most consistently described hormone-induced Ca oscillations in HeLa cells(an immortal cell line derived from cervical cancer cells), compared to seven other minimal models forintracellular Ca oscillations (Estrada et al, 2016). In addition, the mathematical structure of the Atrimodel allows us to determine analytically the parameter range sustaining calcium oscillations and otherbifurcations of the system — see (Atri et al, 1993; Kaouri et al, 2019). Despite its simplicity, the Atri modelgenerates prototypical Ca signals such as Ca oscillations and action potentials which correspond toperiodic wave trains and solitary pulses, respectively, when Ca diffusion is taken into account. The Atrimodel is, hence, sufficient for our modelling framework since our focus is on studying cancer cell movementwith Ca signals as input.We base our model for the cancer cell density on previously published work (Armstrong et al (2006);Bitsouni et al (2017, 2018); Bitsouni and Eftimie (2018); Chaplain et al (2011); Dyson et al (2016); Domschkeet al (2014); Eftimie et al (2017); Gerisch and Chaplain (2008); Gerisch and Painter (2010); Green et al(2010); Hillen and Buttenschön (2019); Painter et al (2015); Shuttleworth and Trucu (2019); Szymańska et al(2009)). These models include nonlinear PDEs with reaction terms for cell growth/proliferation and a non-local advection term, describing cell-cell adhesion. The latter is expressed as an integral term that describeshow a cell at position x adheres to other cells at position x ± s , for some s > -dependent. It is worth noting that additional molecular components and processescould be included. For instance, integrins and TGF- β proteins are explicitly represented in (Bitsouni et al,2018; Engwer et al, 2017) and (Bitsouni et al, 2017; Eftimie et al, 2017), respectively. Moreover, collagen-controlled cell-matrix adhesion, where Ca is considered as constant, has been developed in (Shuttleworthand Trucu, 2019), while (Ramis-Conde et al, 2008, 2009) studied cadherin-dependent cellular adhesion in anindividual-cell-based multiscale model. However, since our study explores the impact of intracellular Ca on cancer cell movement, we focus on diffusion, cell-cell adhesion and proliferation, the core components ofcancer cell behaviour.The structure of the paper is as follows. In Section 2 we formulate a new model that captures thecrucial role of Ca signalling in cancer by incorporating Ca -dependent adhesion and proliferation effects.In Section 3 we perform a linear stability analysis and show the ability of the model to generate ADIsand hence cell aggregations. In Section 4 we solve the model numerically. We present various types ofaggregation patterns, as well as travelling wave patterns. Taken together, our work provides new insights intothe connection between Ca signalling and cancer cell movement, and suggests a mechanistic approach thatcan contribute to developing Ca -transport-targeting tools for cancer diagnosis and treatment (Prevarskayaet al, 2013, 2014). Kaouri, K et al.
We denote by u ( x, t ) the cancer cell density, by c ( x, t ) the cytosolic Ca concentration and h ( x, t ) is thefraction of InsP Rs on the ER that have not been inactivated by Ca . Then the model takes the form ∂c∂t = D c ∂ c∂x + J ER − J pump , (2.1a) τ h ∂h∂t = k k + c − h , (2.1b) ∂u∂t = D u ∂ u∂x − ∂∂x ( uF [ c, u ]) + f ( c, u ) , (2.1c)where J ER = k f µ ([InsP ]) h bk + ck + c and J pump = γck γ + c . Equations (2.1a) and (2.1b) are the spatially extended Atri model for Ca signalling. In equation (2.1a)the term J ER is the flux of Ca from the ER into the cytosol through InsP Rs, where the constant k f is the calcium flux when all InsP Rs are open and activated, b is a basal current through the InsP Rs,and µ ([InsP ]) = [InsP ] / ( k µ + [InsP ]) is the fraction of the InsP Rs that have InsP bound and is anincreasing function of [InsP ]. In the spatially clamped Atri model relaxation oscillations can be sustained atconstant [InsP ], and µ is a bifurcation parameter (see Atri et al (1993); Kaouri et al (2019) for representativebifurcation diagrams). J pump is the Ca flux through the SERCA pumps where γ is the maximal pumprate and k γ is the Ca concentration at which the pump rate is at half-maximum. In equation (2.1b) thetime constant τ h >
1s represents the slower time-scale of the inactivation of the InsP R by Ca comparedto its activation (Atri et al, 1993; Dupont et al, 2016b). Equations (2.1c) is a non-local, non-linear PDE forthe cell density that combines diffusion, cell-cell adhesion (advection) and proliferation (see Domschke et al(2014) and references therein). All parameter values can be found in Tables 1 and 2.2.1 Effect of Ca on cell proliferationThe role of Ca signals in the proliferation of cancer cells is cancer type specific due to differences in thebehaviour of the Ca -conducting channels and pumps (Monteith et al, 2017). Here, we assume that Ca enhances the proliferation rate since it has been shown that InsP Rs are upregulated in cancer (Monteithet al, 2007, 2017), leading to an enhanced proliferation and survival in all types of cancer (Cárdenas et al,2016; Prevarskaya et al, 2018; Rezuchova et al, 2019; Tsunoda et al, 2005). Moreover, assuming that cancercells proliferate in a logistic manner (to describe the observed slow-down in tumour growth following the lossof nutrients (Laird, 1964)), we choose the growth function f ( c, u ) as f ( c, u ) = r (cid:18) g ( c ) (cid:19) u (cid:18) − uk u (cid:19) , (2.2)where r is the basal growth rate of u and k u is the carrying capacity. The Ca -dependent function g ( c )describes the enhanced proliferation of cancer cells that is associated with a major re-arrangement of Ca pumps, Na + /Ca exchangers and Ca channels (Capiod et al, 2007; Simpson and Arnold, 1986; Taylorand Simpson, 1992); we assume that it is given by g ( c ) = r c r + c , (2.3)i.e. it saturates as c increases and vanishes at c = 0. We choose r , r and r based on experimental evidence,as follows. For r it was shown that doubling times for cancer cells range from 1 −
10 days (Cunningham athematical model of cancer and calcium 5 and You, 2015; Morani et al, 2014). In Panetta et al (2000) the doubling time for breast and ovarian cancerranges between 0 . − r , the highest reaction rate that canbe achieved at saturating Ca concentrations, and the half-maximal Ca concentration constant, r , arebased on experimental evidence in (Simpson and Arnold, 1986; Taylor and Simpson, 1992). All parametervalues can be found in Tables 1 and 2.2.2 Effect of Ca on cell-cell adhesionCancer cells often show a decrease in cell-cell adhesion compared to healthy cells, which correlates withtumour invasion and metastasis (Cavallaro and Christofori, 2001; Makena and Rao, 2020). When adhesivebonds are formed and broken a cell-cell adhesion-mediated directed cancer cell migration occurs as a resultof cellular attraction and repulsion. The cell-cell adhesion forces are created through the binding of adhesivemolecules such as cadherins (Byers et al, 1995; Kim et al, 2011; Panorchan et al, 2006), see Section 1. Thus,we consider a calcium-dependent adhesion term in a bounded domain Ω = [0 , R s ] in the cell density equation(2.1c) where the non-local cell-cell interactions are described by a function that depends on cell density andCa , F [ c, u ] = S ( c ( x, t )) R s Z R s K int ( r ) (cid:18) u ( x + r, t ) − u ( x − r, t ) (cid:19) d r, where K int ∈ L ∞ ( Ω ) is the interaction kernel between cancer cells, with ∂ x K int ∈ L ∞ ( Ω ), and S ( c ) theadhesion strength function, which depends on Ca . R s > R s equalsfive times the length of an average cell (Armstrong et al, 2006; Gerisch and Chaplain, 2008). Biologicallythis represents the extent of the cell’s protrusions, e.g. filopodia. We define an attraction-repulsion kernel(see (Eftimie et al, 2007, 2017)) as K int ( x ) = q a K a ( x ) − q r K r ( x ) , with q a and q r describing the magnitude of attractive and repulsive interactions, respectively, and K a ( x )and K r ( x ) denoting the spatial range over which these interactions take place. We will take the kernel tobe attractive at medium/long ranges (i.e. at the edges of the cell) ensuring cell cohesion, and repulsiveat very short ranges (i.e. over the cell surface) to represent cell volume-exclusion effects and thus preventunrealistically high cell densities (Palachanis et al, 2015). Throughout the rest of this study, we considerGaussian attraction and repulsion kernelw (Eftimie et al, 2007) so that K int ( x ) = q a p πm a e − ( x − sa )22 m a − q r p πm r e − ( x − sr )22 m r , (2.4)where s a and s r represent the location of maximal attraction and repulsion, respectively, with s r < s a < R s .The constants m j = s j / , j = a, r , represent the widths of the interaction kernels, respectively. They arechosen such that the support of more than 98% of the mass of the kernels is inside the interval [0 , ∞ )As discussed in Section 1, expression of Ca -dependent cell-cell adhesion molecules is reduced in severalhuman cancer types when Ca levels are increased (Byers et al, 1995; Cavallaro and Christofori, 2004),which leads to a decreased adhesive force between the cells. A biologically realistic choice for the adhesionstrength function is thus S ( c ) = s ? (cid:18) − a ca + c (cid:19) (2.5)an inverse Hill function for c that tends to zero for large c values. We estimated the parameters a , a and s ? so that the adhesive force exhibits a biologically sensible response to Ca variations (for parameter valuessee Table 2). Kaouri, K et al. t = tτ h , ˜ x = xL , ˜ c = ck , ˜ u = uk u , ˜ R s = R s L , ˜ q a = k u q a , ˜ q r = k u q r , ˜ S (˜ c ) = τ h L S ( k ˜ c ) . The length scale, L , is defined as the typical cell size/diameter of an average cancer cell. Cancer cells canbe smaller or bigger than healthy cells depending on several factor including the cancer type. HeLa cells,for example, are around 40 µ m in diameter, while they measure 20 µ m in their naturally compressed state(Boulter et al, 2006; Puck et al, 1956). Generally, the average cancer cell diameter is between 20 − µ m (Haand Bhagavan, 2011). Here, we choose L = 20 µ m, while we set the time scale as τ h = 2s (Kaouri et al, 2019).In addition, we rescale the cell density with the cell carrying capacity, k u , taken to be ∼ . · cell/volume(Gerisch and Chaplain, 2008). We obtain the dimensionless parameters: e D c = D c τ h L , K = k f τ h k , Γ = γτ h k , K = k γ k , K = k k , e D u = D u τ h L , ˜ r = r τ h , ˜ r = r k . We also briefly discuss the choice of the diffusion coefficients. It has been shown in (Allbritton et al, 1992) thatthe diffusion coefficient of free cytosolic Ca is 2 . · − cm s − . The action of omnipresent Ca bufferscan be subsumed into an effective Ca diffusion coefficient, which we here set to D c = 0 . · − cm s − .Assuming that the delay of Ca propagation through gap junctions joining cells is negligible, we arriveat e D c = 0 .
1. The diffusion coefficient of cancer cells is in the range of 10 − − − cm s − (Bray, 1992;Chaplain and Lolas, 2006; Franssen et al, 2019). This corresponds to dimensionless values of e D u between5 · − − · − . We choose e D u = 0 . e K a,r ( e r ) = L K a,r ( L ˜ r ) = L K a,r ( r ) so that e K int (˜ r ) = L k u ( q a K a ( r ) − q r K r ( r )) . Therefore, we have for the non-local term F [ c, u ] ( x, t ) == L τ h ˜ R s e S (˜ c ) ˜ q a Z ˜ R s (cid:18) e K a (˜ r ) − e q r e q a e K r (˜ r ) (cid:19) (cid:0) ˜ u (cid:0) ˜ x + ˜ r, ˜ t (cid:1) − ˜ u (cid:0) ˜ x − ˜ r, ˜ t (cid:1)(cid:1) d r = L τ h ˜ R s ˜ s ? e q a ˜ F [˜ c, ˜ u ] (cid:0) ˜ x, ˜ t (cid:1) = F ˜ F [˜ c, ˜ u ] , where F = L ˜ s ? ˜ q a / ( τ h ˜ R s ) is the typical cancer cell speed.Clark and Vignjevic (2015) showed that cancer cell speeds cannot exceed 10 µ m / min. We Consider thetypical cancer cell speed, F , to vary between 1 µ m / min and 10 µ m / min to account for various cancer typeswhich are characterised by slower or faster cells (e.g. for A375M2 human melanoma the speed ranges between0 . − − µ m / min, and for MDA-MB-231 breast cancer it ranges between 0 . − − . µ m / min). We find thatthe ratio τ h F /L is in the range 0 . ≤ τ h L F ≤ . , leading to 0 . ≤ ˜ s ? ˜ q a ≤ . . (2.6) athematical model of cancer and calcium 7 This provides bounds for the value of ˜ s ? ˜ q a we are going to choose in Sections 3 and 4.After dropping the tildes for notational convenience, we obtain the following non-dimensional system: ∂c∂t = D c ∂ c∂x + µK h b + c c − Γ cK + c , (2.7a) ∂h∂t = 11 + c − h, (2.7b) ∂u∂t = D u ∂ u∂x − τ h L F ∂∂x ( u ( F [ c, u ])) + r (cid:18) r c r + c (cid:19) u (1 − u ) . (2.7c)Although D u = 0 . Table 1: Model parameters, dimensional values, non-dimensional values and relevant references.
Param. Description Dim. value Non-dim.value Reference D c Diffusion coefficient of Ca µ m s − . b Fraction of activated InsP Rs recep-tors when [Ca ]=0 - 0.111 Atri et al (1993) k K m (Michaelis constant) for activa-tion of InsP Rs receptors by Ca . µ M 1 Atri et al (1993);Kaouri et al (2019) k f Ca flux when all InsP Rs recep-tors are open and activated 16 . µ Ms − K =324 / k µ K m (Michaelis constant) for bindingof InsP to its receptor 0 . µ M 1 Atri et al (1993);Kaouri et al (2019) γ Maximum rate of pumping of ERCa µ Ms − Γ = 40 / k γ [Ca ] c at which the rate of Ca pumping from the cytosol is at half-maximum 0 . µ M K = 1 / k K m (Michaelis constant) for inacti-vation of InsP receptors by Ca . µ M K = 1 Atri et al (1993);Kaouri et al (2019) D u Diffusion coefficient of cancer cells 0 . µ m s − . R s Sensing radius 100 µ m 5 Armstrong et al(2006); Gerisch andChaplain (2008) Continued on next page
Kaouri, K et al.
Table 1 –
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Param. Description Dim. value Non-dim.value Reference k u Carrying capacity of the cancer cellpopulation 6 . · cells / cm r Growth rate of the cancer cell pop-ulation 7 days (doublingtime) 0 . Table 2: Estimated model parameters, non-dimensional values and relevant references.
Param. Description Non-dim.value Reference q a Magnitude of attraction 0 − .
44 Guided by linear sta-bility analysis (Section3.2) and the range(2.6), based on Clarkand Vignjevic (2015) q r Magnitude of repulsion 0 − .
44 Guided by linear sta-bility analysis (Section3.2) s a Attraction range 1 Bitsouni and Eftimie(2018) s r Repulsion range 0 .
25 Bitsouni et al (2017,2018); Bitsouni and Ef-timie (2018) m a Width of attraction kernel 1 / m r Width of repulsion kernel 1 /
32 Bitsouni et al (2017,2018); Bitsouni and Ef-timie (2018) s ? Magnitude of cell-cell adhesion forces of the cancercell population 1 Armstrong et al(2006); Bitsouni et al(2017, 2018); Gerischand Chaplain (2008) a Lowest value of cell-cell strength due to increase in[Ca ] 0 . a Half-minimum ( K m ) [Ca ] 0 . r Largest reaction value at saturating [Ca ] 1 . r Half-maximal [Ca ] 4 Simpson and Arnold(1986); Taylor andSimpson (1992) athematical model of cancer and calcium 9 c ? , h ? , u ? ) of the system (2.7) are given by (cid:18) c ? ,
11 + c ? , (cid:19) and (cid:18) c ? ,
11 + c ? , (cid:19) , (3.1)with c ? ≥ c ? + c ? + (cid:18) − µ K Γ (cid:19) c ? + (cid:18) − µ K Γ (cid:18) K + b (cid:19)(cid:19) c ? − µ K Γ Kb = 0 . We seek conditions for a steady state ( c ? , h ? , u ? ) to become unstable due to ADI. We thus consider smallperturbations to the steady state, (cid:0) ¯ c, ¯ h, ¯ u (cid:1) , such that c ( x, t ) = c ? + ¯ c ( x, t ) , h ( x, t ) = h ? + ¯ h ( x, t ) , u ( x, t ) = u ? + ¯ u ( x, t ). Substituting these into (2.7), linearising around the spatially uniform steady state, and usingthe notation ¯ y = (¯ c, ¯ h, ¯ u ), we obtain ∂ ¯ y∂t = D ∂ ¯ y∂x − ∂J a ∂x + J r ¯ y , where D is a diagonal matrix with entries ( D c , , D u ) and J a = (0 , , α ), with α = u ? R s S ( c ? ) Z R s K int ( r ) (¯ u ( x + r, t ) − ¯ u ( x − r, t )) d r , and J r = J ( c ? , h ? ) 00 r u ? (1 − u ? ) 2 r r c ? r + c ? r (1 − u ? ) r c ? ) r + c ? ! , where J = µK h − b (1 + c ? ) − Γ K ( K + c ? ) µK b + c ? c ? − c ? (1 + c ? ) − , is the Jacobian of the linearised Atri model. We seek solutions of the form ¯ y = we iξx + λt , where w =( A c , A h , A u ) with | A c | , | A h | , | A u | (cid:28)
1. The wave number and frequency of the perturbations are denoted by ξ and λ , respectively. We then find λw = ( J d + J r ) w , with J d = − D c ξ − D u ξ + 2 ξu ? b K s int ( ξ ) S ( c ? ) /R s , where b K s int ( ξ ) = R R s K int ( r ) sin( ξr )d r is the Fourier sine transform of K int ( r ).Since the cell density equation (2.7c) is not coupled to the Atri equations (2.7a) and (2.7b), the matrix M = J d + J r has a block structure and the eigenvalues of M are split into those of the (linearised) Atri modeland that of the (linearised) cancer cell density equation. Hence, to identify ADIs we only need to study thelinear stability of the cell density equation, i.e. the eigenvalue (dispersion relation) λ u ( ξ, c ? ) = − D u ξ + 2 ξu ? R S ( c ? ) ˆ K s int ( ξ ) + r (1 − u ? ) (cid:18) r c ? r + c ? (cid:19) , which for the Gaussian attraction and repulsion kernels given in (2.4) becomes λ u ( ξ, c ? ) = − D u ξ + 2 ξu ? R S ( c ? ) (cid:18) e − ( ξma )22 sin( ξs a ) − q r q a e − ( ξmr )22 sin( ξs r ) (cid:19) + r (1 − u ? ) (cid:18) r c ? r + c ? (cid:19) . (3.2)Solutions with λ u > u ? = 0 and ξ = 0, weobtain λ u (0 , c ? ) = r (cid:18) r c ? r + c ? (cid:19) > , In contrast, for u ? = 1 and ξ = 0, we find λ u (0 , c ? ) = − r (cid:18) r c ? r + c ? (cid:19) < . Here, we use the superscript to indicate the value of u ? . Note that λ u ( ξ,
0) and λ u ( ξ,
0) are the eigenvalues ofthe linearised Fisher’s equation when q a = q r = 0 . For positive q a and q r and no calcium, λ u ( ξ,
0) becomespositive for some ξ > q a and q r , we present the non-negative contour plots of λ u ( ξ,
0) in Fig. 1, where negative values are mappedto zero for better visualisation. In Figs. 1(a)–1(d) we set q a = 0 . q a = 0 . q a = 0 .
33 and q a = 0 . q r varies from 0 to 0 .
44. We observe extended regions with λ u >
0, which indicate patternformation in the nonlinear system via ADI. In Figs. 1(b)–1(d) we observe disjoint parameter regions, whichgrow larger as q a increases. Note that we do not need to plot for larger values of ξ since λ u ( ξ,
0) tends to −∞ as ξ tends to ∞ .We next establish the effect of Ca on ADI. In Fig. 2 we display contour plots corresponding to non-negative values of λ u ( ξ, c ? ), for nine different combinations of q a and q r : (0.14, 0.01), (0.16, 0.01), (0.22,0.01), (0.33, 0.01), (0.01, 0.22), (0.14, 0.22), (0.22, 0.22), (0.33, 0.33) and (0.44, 0.44). We observe that theADI regions vanish at sufficiently large values of c ? for all figures except Figs. 2(d), 2(h) and 2(i). Note thatwe choose 0 ≤ c ? ≤ . models may achieve higher c ? levels but we expect a qualitatively similar behaviour. Also, as c ? increasesthe range of ξ in the ADI regions decreases. In Figs. 2(c),2(d), 2(h) and 2(i) we observe disjoint parameterregions for positive λ u ( ξ, c ? ).The stability of the spatially homogeneous Atri model (determined by the matrix J ) has been covered indetail in (Atri et al, 1993; Kaouri et al, 2019). Hopf bifurcations occur at µ = 0 .
289 and µ = 0 . µ . Including diffusion leads to the emergence of periodic wave trains and solitary pulses when the Atrimodel exhibits limit cycles and action potentials, respectively (see Fig. 4). athematical model of cancer and calcium 11 (a) ξ q r (b) ξ q r (c) ξ q r (d) ξ q r Fig. 1: The contours of non-negative λ u ( ξ, c ? = 0, u ? = 1,which enclose parameter regions corresponding to adhesion-driven instabilities, for: (a) q a = 0 .
14 (b) q a = 0 .
22 (c) q a = 0 . q a = 0 .
44. In (a)–(d) q r varies from 0 to 0 .
44, respectively. The remaining parameter values are given in Tables 1 and 2.Negative values of λ u ( ξ,
0) have been set to zero for better visualisation..2 Kaouri, K et al. (a) ξ c * (b) ξ c * (c) ξ c * (d) ξ c * (e) ξ c * (f) ξ c * (g) ξ c * (h) ξ c * (i) ξ c * Fig. 2: Contour plots of the dispersion relation λ u ( ξ, c ? ) as c ? varies for: (a) q a = 0 . q r = 0 .
01; (b) q a = 0 . q r = 0 . q a = 0 . q r = 0 .
01; (d) q a = 0 . q r = 0 .
01; (e) q a = 0 . q r = 0 .
22; (f) q a = 0 . q r = 0 .
22; (g) q a = 0 . q r = 0 . q a = 0 . q r = 0 .
33; (i) q a = 0 . q r = 0 .
44. All other parameter values are given in Tables 1 and 2. Negative values of λ u ( ξ,
0) have been set to zero for better visualisation.athematical model of cancer and calcium 13
In this section we numerically solve model (2.7) using a method-of-lines approach. The domain [0 , L ] isdiscretized into a cell-centered grid with uniform length h = 1 /N , where N = 100 is the number of gridcells per unit length. All simulations are performed with L = 120 and with periodic boundary conditions.The diffusion terms are discretized using a second order centered difference scheme. The advection termis discretized using a third order upwind scheme, augmented with a a flux-limiting scheme to ensure thesolution’s positivity. The non-local term in equation (2.7c) presents challenges regarding its efficient andaccurate evaluation. Here we employ the scheme based on the Fast Fourier Transform introduced in (Gerisch,2010), using the trapezoidal rule to pre-compute the integration weights. The resulting system of ODEs isintegrated using the ROWMAP integrator introduced in Weiner et al (1996). We use the implementation provided in (Weiner et al, 1996). The integrator (written in Fortran) was wrapped using f2py into a scipyintegrate class (Virtanen et al, 2019). The spatial discretisation (right hand side of ODE) is implementedusing NumPy. The integrator’s error tolerance is set to v tol = 10 − . For the full details of the numericalmethods we refer to (Gerisch, 2001; Hundsdorfer and Verwer, 2003).The initial conditions of the system are taken to be narrow Gaussian functions as follows: c ( x,
0) = c ? + 1 . e − . ( x − L ) , (4.1a) h ( x,
0) = 11 + c ? , (4.1b) u ( x,
0) = e − . ( x − L ) . (4.1c)4.1 Adhesion-driven instability, pattern formation and cell aggregationsEach term in the cancer cell density equation (2.7c) critically affects the behaviour of cancer cells. Thus,below we examine the effect of each term in turn and compare the results with those of the linear stabilityanalysis in the absence of Ca , in Section 3.2. We explore a wide range of values for q a (magnitude ofattraction) and q r (magnitude of repulsion), guided by Fig. 1. For q a we also take into account the range of q a reported in (2.6)), based on measurements of the speed of cancer cell movement. No experimental evidencewas found for q r and we consider the same range as for q a . We thus examine several possible scenarios andidentify various types of patterns and aggregations.In Fig. 3(a) we plot the cell density for non-zero diffusion and advection but zero proliferation; thisrepresents cells with very slow doubling time. We take q a = 0 . q r = 0 .
01, i.e. attraction much larger thanrepulsion. We see that the cancer cells form a single stationary pulse. In Fig. 3(b), we add proliferation, buttake zero adhesion (Fisher’s equation). The cancer cells exhibit a travelling front that propagates in oppositedirections at a constant speed, as expected (Murray, 2003). In Fig. 3(c) we include diffusion, advection andproliferation, with q a = 0 .
14 and q r = 0 .
01. We still see a Fisher-like travelling front, consistently withFig. 1(a) which predicts no ADI for these choices of q a and q r .In Fig. 3(d), we further increase the strength of attraction to q a = 0 .
22 while keeping q r = 0 .
01, and apattern emerges behind the travelling front due to ADI, as predicted by Fig. 1(b). It is a “mixed" pattern,featuring merging and emerging peaks; some cancer cells form stationary pulses, while others organise intotravelling pulses. This behaviour can be explained by the strong attractive forces that make cells form largeaggregations. This type of pattern has been identified in previous work (see Andasari et al (2011); Bitsouniet al (2017); Hillen and Painter (2009); Loy and Preziosi (2019); Eftimie et al (2017); Wang and Hillen(2007)),In Fig. 3(e), we lower attraction to q a = 0 .
14 and increase the magnitude of repulsion to q r = 0 . u ( x, t ), for no Ca effect ( a = a = r = r = 0), governed by equation (2.7c). The initialconditions are given in (4.1c). (a) q a = 0 . , q r = 0 .
01, no proliferation; (b) q a = 0 , q r = 0; (c) q a = 0 . , q r = 0 .
01; (d) q a = 0 . , q r = 0 .
01; (e) q a = 0 . , q r = 0 .
22 ; (f) q a = 0 . , q r = 0 .
22. All other parameter values as in Tables 1 and 2.athematical model of cancer and calcium 15 the strong repulsive forces leading to a larger number of smaller aggregations than those in the case whereattraction is larger than repulsion, as in Fig. 3(d).This behaviour again agrees with the linear stabilityanalysis (see Fig. 1(b)). Finally, in Fig. 3(f) we take equal attraction and repulsion, q r = q a = 0 .
22. Thepattern is similar to that in Fig. 3(e). Note: in order to see the more detailed features of the Figures thereader is encouraged to follow the electronic version of the paper.4.2 Calcium signalsHere, we investigate the behaviour of the spatially extended Atri model (2.7a) and (2.7b). The four panelsin Fig. 4 display the behaviour of the Ca concentration as we increase µ , which is equivalent to increasingthe InsP concentration. For µ = 0 .
1, for which the spatially clamped Atri model possesses a linearly stablefixed point (Atri et al, 1993; Kaouri et al, 2019), Fig. 4(a) illustrates that the initial Gaussian conditiondecays to this fixed point. Setting µ = 0 .
288 leads to a solitary travelling pulse (Fig. 4(b)), while a value of µ between the two Hopf bifurcations results in a periodic wave train (Keener and Sneyd, 2009a,b); in Fig. 4(c)we take, as an example, µ = 0 .
3. Finally, for larger values of µ the Atri model is linearly stable again and wefind a similar pattern to Fig. 4(a), in that the initial condition decays to the steady state, but in a periodicmanner. In Fig. 4(d) we take µ = 0 . signalsemerge in almost all Ca models. Here, we use them as input to the cancer cell density equation (2.7c).4.3 The effect of Ca on the cell densityWe now examine the effect of the Ca signals on the cancer cell density. We fix the attraction and repulsionmagnitudes, q a and q r , and vary µ . Fig. 5 (top panel) ( q a = 0 . q r = 0 .
01) shows a Fisher-like travellingfront in all Figs.(a)–(d), irrespective of the InsP and Ca levels; this is consistent with the linear stabilityanalysis that predicts no ADI. These results are in line with Fig. 2(a). In contrast, when we increase q a to0 .
22 in Fig. 5 (bottom panel) small InsP concentrations ( µ = 0 . µ = 0 .
3, respectively) induce a pattern,due to ADI. As we increase the InsP concentration, the pattern vanishes, as illustrated in Figs. 5(c ) and5(d ) which are for µ = 0 .
45 and µ = 0 .
6, respectively. These results are in line with Fig. 2(c).In Figs. 6 we see that for larger values of q a ( q a = 0 . q r = 0 .
01) patterns emerge behind the Fisher-likefront for all values of µ . This is consistent with the linear stability analysis — see Fig. 2(d). For small valuesof µ , µ = 0 . µ = 0 . µ , and consequentlylarger values of Ca (see Figs. 6(c) and 6(d)) the patterns are thin stripes (stationary pulses).In Figs. 5 and 6 attraction dominates over repulsion. In Fig. 7 we plot the cancer cell density whenrepulsion is stronger than attraction ( q a = 0 . q r = 0 . concentration( µ = 0 . µ = 0 . µ ,patterns vanish — see Figs. 7(c) and 7(d), respectively for µ = 0 .
45 and µ = 0 .
6. These results are consistentwith Fig. 2(f). Finally, for large and equal values of q a and q r , Figs. 2(g) and 2(h) predict that ADI patternsexist for all Ca concentrations within the physiological range of the Atri model. This is confirmed in Fig. 8,where we observe ADI patterns for any InsP (and Ca ) level when q a = q r = 0 . varies. Furthermore,below we will summarise the effect of Ca on three important characteristics of the solution: the wave speedof the Fisher-like front, and also the amplitude and frequency of the cancer cell density. (a) (b)(d) (c) Fig. 4: Patterns of the Ca concentration, c ( x, t ), generated by the Atri model (2.7a)- (2.7b) for (a) µ = 0 . c ? = 0 . µ = 0 .
288 ( c ? = 0 . µ = 0 . µ = 0 . c ? = 1 . c ? for all µ when Ca is oscillatorythe steady state is linearly unstable Wave speed:
In Figs. 5–8 we see that as µ increases (fixed q a and q r ) the speed of the travelling frontincreases. This can be linked to a higher invasion and hence metastatic potential of the cancer cells. On theother hand, for fixed µ the wave speed does not change much as q a and/or q r vary. Amplitude:
Comparing Figs. 5 and 6 we see that the maximal cell density increases as q a , the attractionmagnitude, increases from 0 .
14 to 0 .
33. Also, comparing Figs. 6 and 8 we see a significant increase in themaximal cell density as q r increases from 0 .
01 to 0 .
33 (and q a fixed to 0 . q r increases from 0 .
01 to 0 .
33 (while q a is fixed to0 . q a and q r as µ increases the maximal cell density decreases, as we can see in Figs. 5–8. Frequency:
Moreover, we investigate how Ca signalling affects the temporal frequency of cancer cell densityoscillations. In Fig. 9 we fix x = 55 and plot c ( x, t ) and u ( x, t ) for two choices; at the top panel we have q a = 0 . q r = 0 .
01 (attraction much larger than repulsion) and in the bottom panel we have q a = 0 . q r = 0 .
22 (attraction comparable to repulsion). From the frequency bifurcation diagram of the Atri model athematical model of cancer and calcium 17 q a =0.14, q r =0.01 q a =0.22, q r =0.01 (a) (d)(c) (b)(a') (b')(c') (d') Fig. 5: Cancer cell density, u ( x, t ), governed by equation (2.7c), as q a increases (top panel for q a = 0 .
14 and bottom panel for q a = 0 . q r = 0 .
01. The initial conditions are given by (4.1). For (a), (a ) µ = 0 . c ? = 0 .
016 (non-oscillatory Ca ); (b),(b ) µ = 0 . c ? = 0 . ); (c), (c ) µ = 0 .
45 ( c ? = 1 . ); (d), (d ) µ = 0 . c ? = 1 . ). The rest of model parameters are given in Tables 1 and 2. As predicted from the linear stability analysis(see Fig. 2(b)), when q a increases ADI emerges for small values of µ . Note that although we report c ? for all µ when Ca isoscillatory the steady state is linearly unstable.8 Kaouri, K et al. (a) (b) (c) (d) Fig. 6: Cancer cell density, u ( x, t ), governed by equation (2.7c), for q a = 0 . q r = 0 .
01. The initial conditions are given in(4.1). (a) µ = 0 . c ? = 0 .
016 (non-oscillatory Ca ); (b) µ = 0 . c ? = 0 .
556 (oscillatory Ca ); (c) µ = 0 . c ? = 1 . ); (d) µ = 0 . c ? = 1 . ). The rest of model parameters are given in Tables 1 and 2.Note that although we report c ? for all µ when Ca is oscillatory the steady state is linearly unstable. (see Fig. 2 in Kaouri et al (2019)) we choose four values of µ that sufficiently ‘sample’ the variation ofthe frequency as µ increases. We see that the frequency of Ca oscillations is approximately equal to thefrequency of cell density oscillations, if the cell density is oscillatory. We have verified this observation byalso computing the frequency spectra for t ∈ (1900 , q a and q r , the effect of Ca oscillations on the celldensity is similar, and thus other figures are not included for brevity. Since cell proliferation and cell-cell adhesion, which play a critical role in invasion and cancer metastasis,are Ca -dependent, here we have developed and analysed a new model for Ca signalling in cancer. TheCa dynamics have been described by the spatially extended Atri model (Atri et al, 1993), which consistsof a reaction-diffusion equation for the Ca concentration, coupled with an ODE for the fraction of InsP receptors on the ER that have not been inactivated by Ca . This model, although simple enough, generates athematical model of cancer and calcium 19 (a) (b) (c) (d) Fig. 7: Cancer cell density, u ( x, t ), governed by equation (2.7c), for q a = 0 .
14 and q r = 0 .
22. The initial conditions are givenin (4.1). (a) µ = 0 . c ? = 0 .
016 (non-oscillatory Ca ); (b) µ = 0 . c ? = 0 .
556 (oscillatory Ca ); (c) µ = 0 . c ? = 1 . ); (d) µ = 0 . c ? = 1 . ). The rest of model parameters are given in Tables 1 and 2.Note that although we report c ? for all µ when Ca is oscillatory the steady state is linearly unstable. four ‘prototypical’ Ca signals as many other excitable Ca models; periodic wavetrains (which correspondto limit cycles in the spatially clamped Atri model), solitary pulses (which correspond to action potentials),decaying wavetrains and solutions decreasing monotonically with time. The cancer cell density evolution isdescribed by a non-local PDE that incorporates diffusion, cell-cell adhesion (advection) and proliferation.We have modelled the dependence of the adhesion and proliferation terms on the Ca dynamics, motivatedby experimental evidence, and we have considered cancer types where the adhesion strength decreases withCa (Byers et al, 1995; Cavallaro and Christofori, 2004), while proliferation increases with Ca (Cárdenaset al, 2016; Prevarskaya et al, 2018; Rezuchova et al, 2019; Tsunoda et al, 2005). The model, assumptionsand parameter values are presented in Section 2. As much as possible, the model parameters were chosenfrom experimental studies (see Tables 1 and 2).In Section 3 we linearised the model (2.7) and determined the parameter range for which an adhesion-driven instability (ADI) forms, while varying systematically the magnitudes of cell-cell attraction and repul-sion, q a and q r , respectively. In the absence of Ca (Fig. 1) we showed that ADIs may arise for sufficientlylarge values of either q a and q r (or both). ADIs correspond to cell aggregations which are critical for cancer (a) (b) (c) (d) Fig. 8: Cancer cell density, u ( x, t ), governed by equation (2.7c),for q a = q r = 0 .
33. The initial conditions are given in (4.1). (a) µ = 0 . c ? = 0 .
016 (non-oscillatory Ca ); (b) µ = 0 . c ? = 0 .
556 (oscillatory Ca ); (c) µ = 0 . c ? = 1 .
195 (oscillatoryCa ); (d) µ = 0 . c ? = 1 . ). The rest of model parameters are given in Tables 1 and 2. The resultsare consistent with Fig. 2(g). Note that although we report c ? for all µ when Ca is oscillatory the steady state is linearlyunstable. invasion and metastasis. Then, in Fig. 2 we investigated the effect of Ca on the cell aggregations and foundthat they change qualitatively and eventually vanish as the Ca level increases.In Section 4 we solved the full non-linear model (2.7) numerically and systematically investigated arange of attraction and repulsion magnitudes, guided by the linear stability analysis. Firstly, we validatednumerically the results of the linear analysisin the absence of Ca (Fig. 3). We subsequently examinedthe effect of four types of Ca signals on the cancer cell density, paying special attention to the periodicwave trains (Figs. 5-8). We found that as Ca levels increase the maximal cell density decreases due to thedecreased cell-cell adhesion strength preventing the formation of clusters of high density levels. Moreover,as Ca levels increase the speed of the travelling wave fronts increases which is linked to a faster spread ofcancer. An other important result from our numerical investigations is that the frequency of Ca oscillationsis approximately equal to the frequency of the cancer cell density oscillations, when the cell density isoscillatory. Moreover, cellular aggregations vanish for sufficiently large Ca levels, as it was predicted bythe linear analysis. Our results demonstrate that accounting for the dependence of cell-cell adhesion andproliferation on Ca signalling we can reveal the conditions for which cancer cell aggregations appear as athematical model of cancer and calcium 21Fig. 9: Cancer cell density and Ca oscillations. Each plot shows a cross-section (i.e. u ( t ) = u (55 , t )) of a solution of model (2.7)with initial conditions given in (4.1) for selected increasing values of µ . (Top) q a = 0 .
22 and q r = 0 .
01. (Bottom) q a = 0 .
22 and q r = 0 .
01. The rest of model parameters are given in Tables 1 and 2. The cell density u (55 , t ) picks up the oscillations in theCa concentration. Indeed the frequencies (computed using the Fourier transform) match.2 Kaouri, K et al. Ca varies. This allows us to study the dependence of the cancer invasion potential on Ca and paves theways for new therapies based on controlling Ca .Our model provides a general framework for cancer cell movement under the effect of any oscillatorysignalling pathway dynamics and paves the way for treatments that are based on controlling these pathways,and in particular Ca signalling. It, however, has various limitations which outline avenues for future work.The assumption that the adhesion strength function is decreasing with Ca is not appropriate for all cancertypes; an increase of cell-cell adhesion with Ca has been observed in some cancers. Additionally, therepulsion magnitude has been taken over a wide range since there is no experimental evidence supporting itsvalue. New experiments could investigate this. Another limitation of the model is that it includes cell–cellinteractions; it would be useful to incorporate the interaction of the cancer cells with the extracellular matrix(ECM) in future work as this would allow to study cancer invasion in more detail. Additionally, the waycell-ECM interactions are dependent on Ca could be also modelled. Finally, the delay of the Ca wavesin the gap junctions between cells has been considered negligible; a cell-based model accounting for thesegap junctions could be developed. Moreover, as we are now equipped with the insights generated by theone-dimensional geometry, we plan to develop the model to two and three dimensions.A main focus of this study was to unravel the impact of the cellular Ca signalling on the behaviour ofcancer cells. As such, a key component of our model is the description of the cellular Ca dynamics. Wechose the Atri model as a typical representative for a minimal framework that captures essential featuresof the dynamics of the cellular Ca concentration such as Ca oscillations. This naturally raises thequestion about how robust our results are with respect to the Ca model that we employed. The answerto this question combines two main lines of argument: the specific model for the InsP R and whether Ca oscillations are deterministic or stochastic. For the first point, we note that there exist a substantial numberof InsP R models, see e.g. (Atri et al, 1993; De Young and Keizer, 1992; Li et al, 1994; Li and Rinzel, 1994b;Meyer and Stryer, 1988; Sneyd and Dufour, 2002; Siekmann et al, 2012; Sneyd and Falcke, 2005; Shuai et al,2007; Ullah et al, 2012). While they differ in their complexity, the overall range of the Ca concentration andthe frequency of the Ca oscillations are comparable amongst them. Consequentially, exchanging the Atrimodel for any of the other InsP R models will most probably not change our conclusions. A more contentiouspoint is whether Ca oscillations should be described within a deterministic or stochastic framework. Bothapproaches have been used extensively to date as e.g. in (Dupont et al, 2011b; Falcke et al, 2018; Gasperset al, 2014; Kummer et al, 2000; Li and Rinzel, 1994a; Politi et al, 2006; Powell et al, 2020; Shuai andJung, 2002; Skupin et al, 2008; Sneyd et al, 2017; Sun et al, 2017; Tang et al, 1996; Thul et al, 2009; Thuland Falcke, 2007, 2006, 2004a,b; Thurley et al, 2011; Tilunaite et al, 2017; Thul, 2014; Thurley et al, 2012;Tsaneva-Atanasova et al, 2005; Voorsluijs et al, 2019; Weinberg and Smith, 2014; Wieder et al, 2015) — seealso the book by (Dupont et al, 2016b) for a detailed discussion. As this study is the first to explore the roleof Ca in a mathematical model of cancer cell propagation, we opted for a deterministic approach. Thisprovides us with a baseline against which we can test future models in which the Ca dynamics will bedescribed stochastically. Acknowledgements
The authors would like to thank Dr. A. Athenodorou for his valuable technical support. VB acknowledgessupport from the European Union’s H2020 Research and Innovation Action under Grant Agreement No 741657 (SciShops.eu).AB was partially supported by an NSERC (Natural Sciences and Engineering Research Council) post-doctoral fellowship, andis grateful to the Pacific Institute for Mathematical Sciences for providing space and resources for AB’s postdoctoral research.
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