A hybrid mathematical model for selforganizing ce

Authors: E Di Costanzo R Natalini L Preziosi

Publish Date: 2014/07/26

Volume: 71, Issue: 1, Pages: 171-214

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Abstract

In this paper we propose a discrete in continuous mathematical model for the morphogenesis of the posterior lateral line system in zebrafish Our model follows closely the results obtained in recent biological experiments We rely on a hybrid description discrete for the cellular level and continuous for the molecular level We prove the existence of steady solutions consistent with the formation of particular biological structure the neuromasts Dynamical numerical simulations are performed to show the behavior of the model and its qualitative and quantitative accuracy to describe the evolution of the cell aggregateWe thank Andrea Tosin for some useful discussions and suggestions The research leading to these results has received funding from the European Union Seventh Framework Programme FP7/20072013 under Grant Agreement N 257462 HYCON2 Network of excellence This work has also been partially supported by the PRIN project 20082009 “Equazioni iperboliche non lineari e fluidodinamica” The authors Ezio Di Costanzo and Luigi Preziosi are members of the Gruppo Nazionale per la Fisica Matematica GNFM Roberto Natalini is member of the Gruppo Nazionale per l’Analisi Matematica la Probabilità e le loro Applicazioni GNAMPA of the Istituto Nazionale di Alta Matematica INdAMAbout the choice of the parameters of the model we point out that while some values can be found or estimated from the biological or modelling literature the others have been obtained by numerical data fitting or using some relations provided by the stationary modelTables 2 3 summarize respectively the values of the dimensional and nondimensional parameters In the case of a range of variability for a parameter the selected value used in the simulations is put in brackets Finally the last column in Table 2 specifies the references for the provided dataNow we will make some comments in this regard Firstly cell radius R is fixed to 10mu text m starting from the experimental data in Lecaudey et al 2008 Radii barR R 1 are chosen to be equal to 20mu text m taking into account a possible effect of cell extensions Radius R 2 is chosen to be equal to 20mu text m considering the lateral inhibition activated when two cells start to be in touch Radius R 3 concerning with the range of production or degradation of a chemical signaling is set to be equal to R because we think to a source or a drain defined by the dimension of a single cell For R 4 and R 5 we fix respectively the values 20mu text m and 25mu text m First value provides a repulsion force when two cells start to be overlapped see Eq 10 1 second values implies an adhesion force in the spatial radial range 20–25 mu text m The values of varGamma 0 alpha beta mathrmL beta mathrmF gamma omega mathrmadhF mu mathrmF delta /lambda and sigma are obtained by a numerical data fitting on the respective dimensionless values in order to obtain in the simulations a cell migration velocity and a neuromast formation consistent with the experimental results This values are marked as “data fitting” in Table 2 In particular varGamma 0 controls the slope of the function 12 and its value has been chosen to mark a sharp difference in 12 between a cell in the centre of the primordium and one on the boundary However we have found that changes in the value of varGamma 0 do not influence significantly the behaviour of the system Then by numerical tests we have seen that alpha the coefficient related to the SDF1a haptotactic effect influences almost linearly the velocity of the cell migration in the first few hours before the transition leadertofollower occurs data not shown This is expected from Eq 16 1 if we consider a regime of uniform velocity The value for alpha is then fixed to have a cell velocity of approx 69mu mathrmm/mathrmh according to Lecaudey et al 2008 The values of beta mathrmL and beta mathrmF influence cell alignment Without the alignment effect we have tested that the repulsion alone is not sufficient to ensure a distance between the cells consistent with the experimental observations Values too small of beta mathrmL about 5times 1017mathrmh1 imply a large transverse compression of the primordium with distances between the centres of the cells less than 40  of the cell diameter causing even a crossing over of the cells data not shown The range of variability of beta mathrmF omega mathrmadhF mu mathrmF and gamma suggested in Table 2 have been taken in order to have the neuromast detachment Outside these ranges we can not reproduce a complete neuromast formation Precisely we find data not shown that beta mathrmF which is related to the alignment effect for follower cells is to be fixed remaining in the range of beta mathrmL at least two orders of magnitude smaller than beta mathrmL see Table 2 Similarly omega mathrmadhF which represents the coefficient of the elastic adhesion for the followers is to be chosen in the range of omega mathrmadhL about two orders of magnitude smaller than omega mathrmadhL The parameter mu mathrmF that is the damping coefficient for a follower cell can be fixed about one order of magnitude larger than mu mathrmL Moreover for gamma the coefficient related to the FGF chemotactic effect we find the range indicated in Table 2 numerical data fitting not shown The parameter delta /lambda influences the switch variable varphi it and its increasing values produce at the same time t a number of leadertofollower transition gradually decreasing data not shown The range proposed in Table 2 ensures after about 10mathrmh after the migration begins a remaining number of leaders from 25  to about 55  over the total cell number in the primordium These percentages are reasonable in view of the results presented in Lecaudey et al 2008 although it would be interesting to quantify experimentally the number of leader and follower cells Finally the parameter sigma related to the degradation of the SDF11a signal affects the gradient of the chemoattractant and then the cell velocity during migration Its value in Table 2 has been fixed to have in the first few hours before leadertofollower transition a velocity consistent with the data in Lecaudey et al 2008 An interesting aspect that has come out from our tests in Sect 5 is that the cells of the primordium can selfgenerate their own gradient Evidence for this has recently be obtained by Donà et al 2013 Even fixing a constant initial data for the SDF11a along the xaxis with non zero values for sigma we are able to reproduce the collective motion though with a reduced velocity for the same values of sigma Comparing the case with initial gradient of SDF1a and the case of zero initial gradient we find for the value of sigma used in Table 2 a decreased velocity of about 35  data not shown Anyway an initial gradient for the SDF1a signal seems to be necessary since for too large values of sigma we observe a detachment of the head of the primordiumAbout the information on the parameters arising from the stationary model we refer to formulas 47 48 53 Table 1 Figs 3 4 The first two relations give us a limitation for k mathrmF=k mathrmF/lambda and k mathrmL=k mathrmL/lambda the ratios of the coefficient of sensitivity to FGF signal and the coefficient of lateral inhibition for a leader and for a follower cell while the third one provides a value of omega mathrmrep related to the repulsion coefficient when we have fixed gamma by a numerical data fitting as mentioned before Namely the right hand side of these equations depend on N and d 1 once the other parameters are chosen So to obtain the values in Tables 2 3 we have fixed for an example N=8 and d 1=3/2 R They represent reasonable values under the experimental observations in Nechiporuk and Raible 2008 and Lecaudey et al 2008Then a value for xi the parameter of FGF production is obtained from the respective nondimensional value xi in order to have the maximum nondimensional value of FGF f max =1 in our domain Finally other constants are estimable from data available in literature s max the maximum concentration of SDF1a from Kirkpatrick et al 2010 f max the maximum concentration of FGF from Walshe and Mason 2003 omega mathrmadhL the elastic adhesion for a leader cell from Bell et al 1984 mu mathrmL the damping coefficient for a leader cell from Rubinstein et al 2009 D the diffusion coefficient of FGF from Yeh et al 2003 Filion and Popel 2005 and a phenomenological formula in He and Niemeyer 2003 eta the degradation constant of FGF from Beenken and Mohammadi 2009 and Lee and Blaber 2010 using the FGF halflife estimatesThe methods used in the numerical simulations employ a 2D finite difference scheme We consider the spatial domain varOmega =abtimes cd and the spatial steps varDelta x varDelta y such that ab is divided in M=fracbavarDelta x intervals and cd in N=fracdcvarDelta y intervals with M N integers Then we introduce a Cartesian grid consisting of grid points x my n where x m=mvarDelta x and y n=nvarDelta y The same can be done for the time interval 0T in this case if varDelta t is the time step t k will be the nth temporal step ie t k=kvarDelta t With the notation u mnk we denote the approximation of a function uxyt at the grid point x my nt k

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