Ecological models for gene therapy 1: models for intraorganismal ecology
In this paper, we discuss the perspective of intra-organismal ecology by investigating a family of ecological models. We consider two types of models. First order models describe the population dynamics as being directly affected by ecological factors (here understood as nutrients, space, etc). They might be thought of as analogous to Aristotelian physics. Second order models describe the population dynamics as being indirectly affected, the ecological factors now affecting the derivative of the growth rate (that is, the population acceleration), possibly through an impact on non-genetically inherited factors. Second order models might be thought of as analogous to Galilean physics.
In the joint paper, we apply these ideas to a situation of gene therapy.
Keywords: Intra-organismal ecology
This perspective draws back to the speculations of Roux (1881) and Weismann (1904)) on selection occurring inside the organism. More recently, eco-evolutionary processes between cells within an organism have been considered, both to explain the existence of protection mechanisms against the proliferation of cancer cells within an organism Cairns (1975); Nowak et al ( 2003), to predict the spread of resistant phenotypes within cell populations during cancer treatments Nowell (1976); Merlo et al (2006), to describe intra-host dynamics of infectious diseases or cancers as predator-prey interactions between viruses and the immune system Nowak and May (2000); Merlo et al (2006), or to apply the neutral theory of biodiversity Hubbell (2001) to gut or skin flora communities (e.g. Turnbaugh et al (2007); Roth and James (1988, 1989)).
This paper takes place in such an intra-organismal eco-evolutionary perspective. The topic we are interested in is the study of a gene therapy. Gene therapy aims at correcting a physiological dysfunction whose origin is the inadequate expression of a defective gene. In practice, patient cells are genetically modified in vitro by inserting a given gene, and then re-injected to the patient, with the aim that these modified cells replace the resident cells, or at least that they durably persist within patient’s body: that is, that cells be successfully engrafted Aiuti and Giovannetti ( 2003); Cavazzana-Calvo et al (2005). From an ecological point of view, the prospective replacement of a cellular strain by another is similar to a competitive exclusion or a drift, while modification of the cellular environment, for instance by producing a missing enzyme, is similar to ecosystem engineering Jones et al (1994) or, in other terms, to niche construction (Odling-Smee et al ( 2003), chap.5). Successful or unsuccessful engraftment qua species invasion will then depend on the details of the ecological interaction Gonzalez et al ( 2008). The aim of the present work is to determine the conditions of successful engraftment.
To this aim, we will study a family of ecological models describing the dynamics of cell populations within an organism. In the first part, we will present the family of models of competition between cells in the general, non-pathological, case. This will enable us to more easily discuss those aspects of the models that are not limited to the particular case of gene therapy.
General model of cell population dynamics
We consider that cells proliferate inasmuch as limiting constraints enable them to. More specifically, we will assume here that the population dynamics directly depends from a limiting factor
1 First order model
We consider that the limiting factor
Equation 1 admits one (stable) equilibrium:
The system relaxes towards the equilibrium with a characteristic time
From then on, we consider only the case where
2 Second order model
We now turn to a different type of modeling, following the works of Ginzburg and colleagues on demographic inertia Ginzburg et al (2004). We consider that the per capita growth rate now shows a certain inertia (comparable to inertia in Newtonian physics) that is perceptible at the scale of population dynamics. Put it differently, we will not separate the time-scales of the per capita growth rate dynamics and of the population dynamics.
From the biological point of view, such an inertia in the per capita growth rate can result from a dynamics in cell quality (e.g., available amount of intracellular resources or organization quantity, sensu Bailly and Longo (2009)). Then, if environmental conditions worsen, intracellular resources lead to a delay in the demographic response; conversely if living conditions get better the cells first rebuild their intracellular resources before their demographic parameters (division and mortality) get affected. Individual quality can also be transmitted to offspring, a phenomenon known in ecology as maternal effects (e.g. Mousseau and Fox (1998)).
In this model, it is the change of the per capita growth rate that depends on the per capita limiting factor
The model is formally similar to the first order model, but notice that the dimensionality and the meaning of the variables has now changed. The equilibrium of equation 2 results when
2.1 A note on the analogy with physics
In this model, demographic factors (
According to equation 2, in the idealized case where
Our interpretation of demographic inertia departs from Ginzburg’s here Ginzburg et al (2004) (chap.6), who consider that the default state of dynamics is the absence of limiting factors (i.e.
By contrast, we consider here that the metabolism
Contrary to Newtonian physics however where the default state is general, the idealized equation 2 where all factors are put to zero can make sense only in very special cases where cell quality is fully heritable during divisions (e.g. some environment sensitive epigenetic marks), that is, it does not hold for intracellular resources that are shared among daughter cells. Below we will consider the impact of resource sharing among offspring, and see how it radically modifies the dynamics
2.2 Differences between inertial and non-inertial dynamics: accelerated death, overshoot
In the inertial model (equation 2), death is accelerated when the limiting factor is the strongest (i.e. set to zero): the per capita growth rate decreases and can tend towards
An inertial dynamics also allows to overshoot the demographic equilibrium value of the population (figure 2.2, figure 2.2). Overshoot leads to demographic oscillations around the equilibrium value with a pulsation
2.3 Friction, antifriction
In the model described by equation 2, oscillations around the equilibrium value are neither damped nor amplified. Such a behavior is structurally unstable: small modifications of the model lead to the convergence toward stable equilibrium or to divergence May (1973); Nowak and May ( 2000).
From a biological point of view, oscillations are damped when good quality cells (
Equation 3 cancels out when
In particular, in the case of a free fall (
Ginzburg et al (2004) p90 also modeled population dynamics with a second order equation (that is, with
- pseudoperiodical regime with damped oscillations (
): the pulsation is given by :
The period is given by
. The relaxation time is given by :
- critical regime (
): the system shows no oscillations and relaxes with a characteristic time .
- aperiodical regime (
): the systems returns to equilibrium with a relaxation time :
Note that relaxation is slower than in the damped oscillation regime, because now friction also opposes to the return to equilibrium.
We will see how, from a theoretical point of view, the existence of friction in cell population dynamics can have therapeutic implications.
3 Model with two species
In this section we derive the models of sections 1.1 and 1.2 to describe the case of an interaction between two species (in our case, two cellular strains). These are the models we will use in the next part. Though we describe the dynamics from a general point of view in this section (that is, without making any symmetry assumption about the species in presence) we will be able to drastically reduce the number of parameters in the following part assuming that the cellular strains (genetically modified and non-modified) are identical in most respect.
3.1 First order model
We suppose that the two species interact in a competitive way via their dependency to the limiting factor
The behavior of the system, and in particular the stability of the equilibra in the space of parameters is shown in figure 3.1. The system has three equilibria: two equilibria correspond to the loss of a least one species and reduce to the monospecific case, one corresponds to the coexistence between populations
(b) the equilibrium corresponding to coexistence is given by the couple
The equilibrium is a coexistence iff
3.2 Second order model
We similarly extend the second order monospecific model to the two-species case by doing the same assumptions about the competitive interaction between 1 and 2:
The equilibrium of this system obtains for the same equilibrium values than the first order system (steady
3.3 Second order model with friction
We now add friction to the two-species model:
From then on, we will only consider cases where
A model of population dynamics should exhibit three essential behaviors: decline in absence of resources, growth in non-limiting situations (
It is impossible to represent the possible actions on these three independent behaviors with only two parameters (e.g.,
Under these constraints, and to favor parsimony which is essential to our application on gene therapy (next section), we have sacrificed the behavior of the population far from the
or in the version of Lotka (1925):
We did not choose this model for the following reasons:
- The difficulty of interpreting the parameters Olson (1992). First,
, or , both represent and have an impact on the density-dependence ( in Verhulst’s equation and thus depends on ; conversly, and is independent from in Lotka’s equation but now the density parameter, that is, the amount to which the population is sensitive to itself, is ). Second, should not be interpreted as a carrying capacity but as an equilibrium value Berryman ( 1992). In other terms, in the logistic equation the inflexion point is a center of symmetry between growth when the population is small, and growth near the equilibrium, which does not seem to have any obvious biological basis Winsor (1932).
- The unrealistic form of the density dependence Getz (1996), when
McCarthy (1997); Courchamp et al (1999); Etienne et al (2002); Kent et al (2003), but also when : in this case the per capita death rate is proportional to the ratio , and not to a property of the biological system in the absence of resources (death by food shortage for instance). This behavior comes from the fact that Verhulst’s equation is a truncated Taylor series. We find this same behavior in the inertial model of Ginzburg et al (2004) p90 which has the same form (but at the second order).
We have chosen here to model the dynamics for situations where
In this work, we limited ourselves to the ecological dimension of the cellular niche, that is, to the impact of density on competition. However, in intra-organismal ecology density-dependence has effect that are unknown in organism ecology. Physical constraints, in particular, are known to affect the differentiation of stem cells in given niches Gerecht-Nir et al (2004); Mohr et al (2006); Stevens et al (2007) as well as to affect the malignant phenotype and the response to treatments in the case of cancer Ingber and Jamieson (1985); Huang and Ingber ( 2005); Paszek et al (XXXX) and Schwartz ( 2004) chap.15. This is a new behavior by comparison with organism ecology, where the most similar behaviors would be migration and metamorphosis.
Last, in classical population ecology, populations, once lost, do not reappear if there is no migration nor dormant propagule bank in the environment. Thus,
The model being simple and basically describing a relaxation toward an equilibrium (at the first order, or at the second order with friction), some structural homogeneity is expected with existing models in the literature. We can notice in particular a certain formal homology (partial, except in cases where we introduce a
In this work, we focused on the structural stability of our modeling, by introducing a friction term. A strong friction makes the system tend towards a first order behavior: inertia loses its dynamical importance. In the general models presented above, friction affects relaxation but not the equilibrium stability. This will not be the case in the next section anymore.
Our model shows how the same equational form can be interpreted at the first or the second order (keeping in mind that the dimension and the meaning of the parameters change according to the order). At the first order, the system describes the growth of an organ, or, in the model with two species, the potential invasion of an organ by a cellular strain. At the second order, our model is structurally identical to that of Ginzburg et al (2004) p44 modeling the quality of individuals. This structural homology between first and second order enables to study the importance of the time-scale separation hypotheses between the individual’s quality (
The diversity of empirical results in population dynamics makes it difficult to a priori choose between the first and second order models. Qualitative results of the first order model are in concordance with some empirical results as regards the growth of an organ or of the quality of a cell (see e.g. resp. Kooijman (2000) p33:fig.2.5 and p2:fig.1.1). However the second order model is in concordance with demographic oscillations (damped, amplified, or not) and accelerated death observed in organism ecology (see the review by Ginzburg et al (2004) p92-93), and in intra-organismal ecology (Corbin et al ( 2002), see also companion paper).
Acknowledgements: This work is based on notes written up by Regis Ferrière in 2004 after a project started at the CEMRACS 2004, whose participants were Antonio Cappucio, Etienne Couturier, Michel de Lara, Regis Ferriè re, Olivier Sester, Pierre Sonigo, Christian et Carlo. The authors also whish to thank the organizers and participants of the StabEco workshop, held at the Laboratory Ecology and Evolution, University of Paris 6, on the 17/12/2010. Philippe Huneman and Minus van Baalen provided invaluable comments on earlier versions of the manuscript.
This work consists in an update of a previous work in French Pocheville ( 2010), chap. 3, realized while both A.P and M.M. were benefiting from a funding from the Frontiers in Life Sciences PhD Program and from the Liliane Bettencourt Doctoral Program. Manuscript was written while A.P was benefiting from a Postdoctoral Fellowship from the Center for Philosophy of Science, Uni- versity of Pittsburgh. M.M. is currently benefiting from a Postdoctoral Fellowship from the Region Ile-de-France, DIM-ISC.
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5.1 Model of a limiting maximal per capita growth rate
Let’s start from equation 1:
Here the per capita growth rate tends toward infinity when N(t)/
where b is a scale constant (number of cells by limiting factor units) introduced to describe the behavior of the per capita growth rate at small cell densities.
The equation for the maximal per capita growth rate
This equation describes
The population tends towards an equilibrium value
Thus a condition on
The behavior of model (1 bis) is very similar to the classical logistic model (figure 5.1).
5.2 Linearized monospecific first order system
See the two species system, with
5.3 Linearized monospecific second order system
We can write the equation in function of
The equilibrium obtains:
We consider the behavior near this equilibrium, that is
Changing the variable:
The pulsation is given by
The pseudo-pulsation is given by
5.4 Linearized two species first order system
Near the equilibrium, we write:
Rearranging, we get:
We seek for the eigenvalues of this system. They are the roots of the characteristic polynomial:
With this parameters, we get:
Thus the eigenvalues are :
It turns out that
In the case where
5.5 Linearized two species second order model (without friction)
The calculus is identical to the first order system, but the interpretation differs.
5.6 Linearized two species second order system (with friction)
We consider the case where