A General Model of Structured Cell Kinetics
aa r X i v : . [ q - b i o . CB ] F e b A GENERAL MODEL OF STRUCTURED CELL KINETICS
MICHAEL GRINFELD, NIGEL MOTTRAM, AND JOZSEF FARKAS
Abstract.
We present a modelling framework for the dynamics of cells structured by the con-centration of a micromolecule they contain. We derive general equations for the evolution of thecell population and of the extra-cellular concentration of the molecule and apply this approach tomodels of silicosis and quorum sensing in Gram-negative bacteria.
Keywords: structured populations, transport equations, integro-differential equations, silicosis Introduction
Many physical systems involve the interaction of micro-scale objects and macro-scale objects withina region. For instance, in biology, the micro-scale objects could be molecules of a particular chemicalwith the macro-scale-objects being cells and the region a Petri dish or an organ. This region, adomain of volume W , may be of fixed size or change with time but we assume that the micro-scaleobjects, of a species X (which from now we call molecules for brevity), have no volume while themacro-scale objects (which from now on we call cells) each have volume V . The X molecules maybe present inside or outside the cells, with the concentration of X varying between the cells. Wealso assume that the molecules of X can participate in any subset of the following processes: theycan be injected into or be removed from the domain, they can enter and exit cells and they canbe produced, processed and destroyed by cells. Suppose also that the fate of a cell is dependenton the amount of X that it contains. The goal of the present paper is to introduce, using ideas ofMetz and Diekmann [10] and of Brown [2], a modelling framework for such situations.Below we describe in detail two specific examples of systems where our modelling approach isappropriate, the dynamics of silicosis, the biological background for which can be found in [13], andquorum sensing in Gram-positive bacteria following Brown [2]. However, our framework is suitablefor many other situations, some of which are briefly considered in Section 5. We are confidentthere are many other examples, in both biological and non-biological systems, where the proposedphilosophy may be useful.The structure of the paper is as follows. In Section 2 we introduce the biological background ofsilicosis and argue that the mathematical model of [13], which is couched in terms of coagulation-fragmentation equations, does not give a correct description of the dynamics; this is the originalmotivation for developing the present approach. In Section 3 we show how to derive the requiredequations in the general case. In Section 4 we complete the specification of the silicosis model andbriefly consider quorum sensing in Gram-positive bacteria, for which our framework is in fact an“overkill”, but which may prove useful in the process of testing numerical methods. Finally, inSection 5 we suggest other areas of application of our framework, discuss mathematical issues anddraw conclusions.Note that the present paper is purely methodological and that all results on existence, uniqueness,or asymptotic behaviour of solutions of the type of equations we derive here, are left to future work. . Silicosis
Let us summarise the 1995 silicosis model of Tran et al. [13], which should be consulted for ref-erences. The biological background is as follows: quartz particles are ingested and arrive in thelung. There they may be picked up by macrophages, with the intent of being removed togetherwith their quartz load via the muco-ciliary escalator. However, if a macrophage accumulates toolarge a quartz load, it becomes immobile and eventually dies by apoptosis in the lung, releasing itsquartz load.The variables in the model of Tran et al. are: free quartz dust in the lungs in concentration x ( t )and concentrations of macrophages M k ( t ) containing k particles of quartz. They write down anequation for the evolution of free quartz particle concentration and for M k , equations of the form(1) dM k dt = α k − xM k − − α k xM k + · · · , where α k − and α k are the kinetic constants for the process of macrophages with k − k particles of quartz, respectively, ingesting one additional quartz particle.In eq. (1), the · · · stand for the two different “death” processes: the disappearance of cells togetherwith their quartz load via the the muco-ciliary elevator; or cell apoptosis accompanied by the releaseof the quartz load into the lungs. To make this model fully specified, it must be complemented byan equation for the production of na¨ıve macrophages, M ( t ); this rate in general depends on thequartz load in the lungs. See Section 4.1 below for a reasonable form of such an equation.This model was later considered in [5]; mathematically it is interesting and falls in the frameworkof coagulation-fragmentation equations (see [1] for an up-to-date exposition of this area of infinite-dimensional dynamical systems). The global existence of its solutions has been proven in laterwork by da Costa and coauthors [6], and many mathematical questions connected with the modelof Tran et al. are still open.To understand our objections to the modelling of silicosis as a coagulation–fragmentation system,consider a typical coagulation–fragmentation reaction scheme, c k + c ⇋ c k +1 . Here c k , c , c k +1 are concentrations of k -mers, monomers, and ( k +1)-mers of some chemical species,respectively. The equation for the evolution of c k +1 corresponding to this reaction scheme is dc k +1 dt = α k c k c − β k +1 c k +1 , where α k is the kinetic constant for the coagulation reaction between k -mers and monomers and β k +1 is the kinetic constant of the fragmentation of a molecule of ( k + 1)-mer into a monomer and a k -mer. This equation is simply mass-action kinetics, and α k and β k +1 are assumed to be functionsof k only.If we now compare this coagulation–fragmentation process with eq. (1), we would suppose that theunderlying reaction scheme is M k + x ⇋ M k +1 . This situation is subtly different in that the reaction here is between the molecules of quartzoutside the cells and the content of the cells. Though M k , x , M k +1 have units of concentration,the reaction encoded in this scheme in general involves the concentration of quartz, and not thenumber of quartz particles, inside the cells. As an example, consider the case of exchange of quartz etween the outside and the inside of a cell driven by passive (Fickian) diffusion. Then the rate ofexchange of molecules is proportional to ( x − q I ), where q I is the internal concentration of quartz.However, the k in M k denotes the number of quartz molecules in a cell, not their concentration. Todefine internal concentration of quartz, we cannot assume that a cell is a point object, and haveto endow it with a volume, say V . But if cells now have finite volumes, it becomes clear that theconcentration of free quartz is also a function of the number of cells which is not the case in thecoagulation–fragmentation setup. This is precisely the kind of confusion of units that is avoided inthe type of model proposed below.3. Derivation of Equations Ω (a) (b)(c) (e)(d)(f) Figure 1.
The situation being modelled: the domain Ω contains species X (reddots) and cells (black circles). The possible processes involved are: (a) na¨ıve cellsenter the domain; (b) species X enters the domain; species X (c) enters or (d) exitscells; (e) X -laden cells exit the domain and (f) cells release molecules of X in Ω;also present, but not indicated in this figure, are synthesis and degradation of X .We are modelling the situation sketched in Figure 1 and described as follows. The system consistsof a domain Ω of volume W that contains molecules of X (denoted by red dots in Fig. 1) withextracellular concentration x , and a population of cells (the black circles in Fig. 1). We assumethat the volume of each cell is V and treat the molecules as having no volume.The cells differ in their X content, and we define the time-dependent density of the cells withinternal concentration of X being y to be M ( y, t ). That is, the number of cells with internalconcentrations of X between y and y , with y < y , is Z y y M ( y, t ) d y. Then V ( t ) = V Z ∞ M ( y, t ) d y is the total volume occupied by the cells and we assume that for all time t , V ( t ) < W . By this wemean that if V (0) < W , the dynamics of the system ensures that V ( t ) < W . urthermore, inside each cell we have a process that involves the substance X , which can bedescribed by an ordinary differential equation such as(2) dydt = f ( x, y ) = J ( x, y ) + g ( y ) , where we have separated the transport term J ( x, y ), that may depend on both the internal andexternal concentration of X , and the intracellular synthesis and degradation term g ( y ). This typeof equation is called an i -equation (for individual) by Metz and Diekmann [10].Metz and Diekmann [10] also show (in many different ways) how to derive the equation governingthe evolution of M ( y, t ), what they call the p -equation (for population),(3) M t ( y, t ) + ( f ( x, y ) M ( y, t )) y = P ( M ( y, t ) , y ) + Q ( M ( y, t ) , y ) , where in the right-hand side the terms P and Q encode all the population level processes (suchas birth and death). Specifically, we denote by Q ( M ( y, t ) , y ), the population level processes thatfeed back into the x dynamics (e.g. when cells die in Ω releasing their contents). The model isthen closed by specifying the x dynamics, adding suitable initial conditions x (0) and M ( y, f ( x ( t ) , M (0 , t ) = s ( · ) , where the function s can depend on a variety of variables. Thus the unknown dependent variablesare x ( t ) and M ( y, t ).Deriving the equation governing the evolution of x ( t ) is algorithmic, by keeping account of the totalnumber of molecules of X in the extracellular space. In general, the result is an integro-differentialequation. The derivation of the equation for x ( t ) crucially uses the thinking of Brown [2], whichwe now discuss.Brown [2] considers the system described above but with the simplification that every cell containsthe same number of molecules, so that the concentration of X in all cells is y , a constant. If K ( t ) isthe number of cells at time t , then the total volume occupied by cells is V ( t ) = K ( t ) V , the cell-freevolume is W − V ( t ) and the total number of molecules outside of cells is N E = ( W − V ( t )) x ( t ).We will work in two stages. First we write equations for the transport of X molecules between theinterior and exterior of the cells, assuming that the flux is given by J ( x, y ) and that it is positivewhen molecules enter the cells. We have(4) dN E dt = − V J ( x, y ) . An obvious example of a flux would be J ( x, y ) = D ( x − y ), with D a diffusion constant, but inexamples of interest, mechanisms involving facilitated transport or phagocytosis should be consid-ered.The proportionality to V in eq. (4) comes from the consideration that the rate of change shouldbe proportional to the available surface area of the cells, which, given we assume a fixed individualcell volume and surface area, is proportional to the number of cells and thus proportional to thetotal volume of cells.Now let us rewrite these equations in terms of concentrations only. Since N E = x ( W − V ), we have dxdt = − VW − V J ( x, y ) + xW − V dVdt . ow we add to this molecular transport equation terms involving synthesis and degradation termswhich we collect in one term, H ( x, y ); we have(5) dxdt = − VW − V J ( x, y ) + xW − V dVdt + H ( x, y ) . Here H ( x, y ) incorporates all extracellular production and degradation of X and all the populationlevel processes that feed back into the extracellular concentration x ( t ).We now consider the more general case in which the cells may have different internal concentrationsof X . This necessitates a number of changes. First of all, the total number of molecules beingreleased per unit time by cellular processes, i.e Q in eq. (3), is V Z ∞ yQ ( M ( y, t ) , y ) dy and hencewe can write H ( x, y ) = h ( x ) + V W − V Z ∞ yQ ( M ( y, t ) , y ) dy, where h ( x ) is the rate of extracellular production and degradation of X , which depends on theparticular modelling context.Secondly, the term V ( t ) J ( x, y ) = V K ( t ) J ( x, y ) is replaced by the integral V Z ∞ M ( y, t ) J ( x, y ) dy .With these changes, the equation for the evolution of x ( t ) becomes(6) dxdt = h ( x ) + 1 W − V (cid:20) − V Z ∞ J ( x, y ) M ( y, t ) dy + V Z ∞ y Q ( M ( y, t ) , y ) dy + x dVdt (cid:21) . Therefore our modelling framework consists of the equation governing the cell population, the p -equation, eq. (3); the equation governing the concentration of X external to the cells, the x -equation, eq. (6); and the data for any particular model set through the biologically determinedterms J ( x, y ), g ( x ), P ( M ( y ) , t ) , y ), Q ( M ( y ) , t ) , y ), h ( x ) and the boundary condition function s ( · ).4. Applications
In this section we formulate in detail a new model of silicosis and discuss a quorum-sensing modelin Gram-positive bacteria. In Section 5 we will mention some other situations that are covered byour framework.4.1.
Silicosis.
As in Tran et al. [13], we assume that quartz is being ingested at a constant rate.We set M ( y, t ) to be the density of macrophages having internal quartz concentration y at time t . As in [13], we assume that new macrophages are produced at rate s determined by the quartzload, L ( t ) := Z ∞ yM ( y, t ) dy , such that(7) s ( L ( t )) = s + u ( L ( t )) , where s is a background level of recruitment of na¨ıve cells into the domain when quartz is notpresent in any cells and u ( · ) is a bounded function, with u (0) = 0.Cells with internal concentration of quartz y are removed by the muco-ciliary escalator at a rate p ( y ),where p ( y ) is a decreasing function of y since, as their quartz content increases, cells are increasinglyimmobile. Cells are also more liable to die by apoptosis as their quartz content increases, and sothe rate of them releasing their contents inside the lungs, q ( y ), is an increasing function of y . s there is no intracellular processing of quartz, so that g ( y ) = 0 in eq. (2), we only need to specifythe transport mechanism. Phagocytosis of quartz particles cannot be described by simple diffusion,so we set(8) dydt = J ( x, y ) , where the function J is non-negative, bounded, increasing in x and decreasing in y , as is alsoassumed in [13]. An example would be J ( x, y ) = γxx + y + x / , where γ is the flux when x → ∞ and x / at which the value of x when the flux for na¨ıve cells (i.e., y = 0) is half the maximum value, both positive constants.So far we have all the information needed to specify the p -equation for M , which is therefore(9) M t ( y, t ) + ( J ( x, y ) M ( y, t )) y = − ( p ( y ) + q ( y )) M ( y, t ) . Now we need to formulate the equation for x ( t ). From (6) it follows that all we need to do is tospecify the function h ( x ), the rate of change of concentration of quartz particles in the extracellularregion due to introduction from outside the domain. If we assume that A particles of quartz areingested per unit time, we have h ( x ) = AW − V , and hence(10) dxdt = 1 W − V (cid:20) A − V Z ∞ J ( x, y ) M ( y, t ) dy + V Z ∞ yq ( y ) M ( y, t ) dy + x dVdt (cid:21) . We can derive an expression for dV /dt in terms of the variables M ( y, t ) to substitute in (10). Since dV /dt is given by dVdt = V Z ∞ M t ( y, t ) dy, integrating the p -equation by parts and assuming that that for all times t > y →∞ J ( x, y ) M ( y, t ) = 0 , we obtain(11) dVdt = V (cid:18) s ( L ( t )) − Z ∞ ( p ( y ) + r ( y )) M ( y, t ) dy (cid:19) . In addition, we must specify suitable initial conditions x (0) and M ( y,
0) as well as the boundarycondition at y = 0, which, using eq. (7), is(12) J ( x, M (0 , t ) = s + u ( L ( t )) . We note that the resulting system is a linear transport equation (for the population variable M )coupled to an nonlinear integro-differential differential equation for the extracellular quartz con-centration x .Having assumed that the load L ( t ) = Z ∞ yM ( y, t ) dy s finite for all time, the correct setting for the theory is a space of positive Radon measures witha finite first moment. Particular choices of the functions p ( y ) and r ( y ) and the input function s ( · )must ensure that if V (0) < W , then V ( t ) < W . For example, this can be shown to be the case ifwe assume that p ( y ) + r ( y ) is bounded below and that s ( · ) is bounded above by constants.4.2. Quorum sensing in Gram-negative bacteria.
For background on quorum sensing in bac-teria and a number of models that assume that the internal concentration of the signal molecule X , i.e., y , is the same in all cells, see [2]. The case of Gram-negative bacteria is an interesting testcase of our approach, as it involves a significant simplification: we can assume that the number ofcells is constant, so V ≡ const is now a parameter and dV /dt = 0. Fickian diffusion of X betweenthe cells and the extracellular region is assumed, so that the i -equation is dydt = D ( x − y ) + g s ( y ) + g d ( y ) , where D is a diffusion constant, and we included terms modelling intracellular synthesis g s ( y )(which in [2] is taken to be a constant term plus a Hill-type term) and intracellular degradation g d ( y ) (which in [2] is taken to be linear, so that g d ( y ) = − m I y , parameterised by the intracellulardegradation rate m I ). The p -equation is M t ( y, t ) + ([ D ( x − y ) + g s ( y ) + g d ( y )] M ( y, t )) y = 0 . The boundary condition is M (0 , t ) = 0 and the x equation becomes dxdt = V DW − V Z ∞ ( y − x ) M ( y, t ) dy − m E x, where m E is the extracellular degradation rate.5. Remarks
We start by briefly mentioning other possible applications. The list is clearly incomplete. • X can be a drug that interacts with specific cells, so this is a suitable framework forchemotherapy modelling; • X may be bacteria that are ingested by neutrophiles, and can multiply inside the cell. Thisframework is therefore a possible model of bacterial infection and resistance [9]; • X can be a mitogen, giving a model of stem cell number maintenance in which cells thatingest enough mitogen will multiply, and divide their mitogen label among the daughtercells, while cells that do not have enough mitogen, die [8]. • As mentioned in the Introduction, this framework can also be used to model non-biologicalsystems, one example of which is the decontamination of a chemostat using zeolites. Thereare many examples of this type of system, showing that it is not necessary to interpret“macro-scale objects” as biological cells.Clearly the i -equation for y could have been an stochastic differential equation; then the x -equationwould have been an integro-stochastic differential equation, and the M equation a stochastic partialdifferential equation. Furthermore, extensions to multidimensional x and y are straightforwardalthough incorporating spatial structure seems to us much more challenging (as it is for coagulation–fragmentation equations).To summarise, we have presented a modelling framework that seems to cover a vast number ofpossible modelling contexts beyond the reach of coagulation–fragmentation equations. Such aframework necessitates the analysis of complicated mathematical objects and so work on existence, tructure of equilibria, convergence to equilibria and their regularity, etc., is required when specificexamples are considered. The search will be for measure-valued solutions, and relevant work inthis direction has been undertaken by Carrillo, Gwiazda and co-workers; see, for example, [3, 7]. Interms of possible numerical solutions to the governing equations, there are no off-the-shelf numericalmethods that we know of, although it seems that escalator box train (EBT) methods could beadapted to the problem (see related work of Carrillo, Gwiazda and Ulikowska [4]).Finally, we note that in many of the biological settings for which this framework could be used,deeper understanding of the active transport of molecules in and out of cells may be needed.Good models of transport across membranes (facilitated transport, phagocytosis, pumps etc.) arerelatively sparse in the literature (though see, e.g., [11, 12]) and further work on such models wouldbe of significant benefit. References [1] J. Banasiak, W. Lamb, and P. Lauren¸cot,
Analytic Methods for Coagulation-Fragmentation Models , Vols. I andII, CRC Press, 2019.[2] D. Brown, Linking molecular and population processes in mathematical models of quorum sensing, Bull. Math.Biol. (2013), 1813–1839.[3] J. A. Carillo, R. M. Colombo, P. Gwiazda, and A. Ulikowska, Structured populations, cell growth and measure-valued balance laws, J. Diff. Eqns (2012), 3245–3277.[4] J. A. Carrillo, P. Gwiazda, and A. Ulikowska, Splitting-particle methods for structured population models:convergence and applications, Math. Models Meth. Appl. Sci. (2014), 2171–2197.[5] F. P. da Costa, M. Drmota and M. Grinfeld, Modelling silicosis: the structure of equilibria, Eur. J. Appl. Maths, (2020), 950–967.[6] F. P. da Costa, J. T. Pinto and R. Sasportes, Modelling silicosis: existence, uniqueness and basic properties ofsolution, arXiv:2005.10042.[7] P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperolic Diff. Eqns (2010), 733–773.[8] Y. Kitadata et al. , Competition for mitogen regulates spermatogenetic cell homeostasis in an open niche, CellStem Cell (2019), 79–92.[9] R. Malka, E. Shochat, and V. Rom-Kedar, Bistability and bacterial infections, PLoS One (2010), e10010.[10] J. A. J. Metz and O. Diekmann, eds., The Dynamics of Physiologically Structured Populations , Springer-Verlag,Berlin 1986.[11] R. J. Naftalin, Reassessment of models of facilitated transport and cotransport, J. Membrane Biol. (2010),75–112.[12] R. Rea, M. G. de Angelis, and M. G. Baschetti, Models for facilitated transport membranes: a review, Membranes (2019), 26.[13] C.-L. Tran, A. D. Jones and K. Donaldson, Mathematical model of phagocytosis and inflammation after theinhalation of quartz at different concentrations, Scand. J. Work Environ. Health (1995), 50–54. Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, GlasgowG1 1XH, UK
Email address : [email protected] School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ,UK
Email address : [email protected]
Division of Computing Science and Mathematics, University of Stirling, Stirling FK9 4LA, UK
Email address : [email protected]@stir.ac.uk