Ecological models for gene therapy 2: niche construction, non-genetic inheritance, and ecosystem perturbations
In this paper, we apply the perspective of intra-organismal ecology by investigating a family of ecological models suitable to describe a gene therapy to a particular metabolic disorder, the adenosine deaminase deficiency (ADA-SCID). The gene therapy is modeled as the prospective ecological invasion of an organ (here, bone marrow) by genetically modified stem cells, which then operate niche construction in the cellular environment by releasing an enzyme they synthesize. We show that depending on the chosen order (a choice that cannot be made on a priori assumptions), different kinds of dynamics are expected, possibly leading to different therapeutic strategies. This drives us to discuss several features of the extension of ecology to intra-organismal ecology.
Keywords: Intra-organismal ecology; niche construction; ecosystem engineering; non-genetic inheritance; inertial dynamics; gene therapy; adenosine deaminase deficiency; severe combined immunodeficiency; ADA-SCID
In this second article, we will study the impacts of niche construction by genetically modified cells on cell populations dynamics. This will enable us to propose some recommendations and new empirical questions to the practitioner. We will show that the conclusions will depend on sensitive hypotheses on time-scale separation (or entanglement) of the considered cellular processes. We will then discuss how ecological concepts can be implemented in cell biology in general, and how extending ecology to cell biology can in turn enrich ecological questions.
Our biological model is the deficiency in adenosine deaminase (ADA), a rare monogenic disease (occurrence between 1:300 000 and 1:1 000 000) Cavazzana-Calvo et al (2004, 2005). From a biochemical point of view, ADA deficiency causes a dysfunction of the metabolism of purines characterized by the accumulation of metabolites in intra- and inter-cellular compartments, which results in particular in a premature apoptosis of lymphocytes Cavazzana-Calvo et al (2005). The lymphocyte anomaly is still not completely elucidated Gaspar et al (2009). ADA deficiency leads to severe anomalies in the immune system (that is, SCID: severe combined immunodeficiency), as well as to other systemic problems, and without treatment the disease is fatal within the first year. Three kinds of treatments are possible: hematopoietic stem cell transplantation (HSCT), enzyme injection, and gene therapy. Hematopoietic stem cell transplantation represents a good option if and only if a related and compatible donor is available (88% survival after one year, vs. 29% and 67% in the case of non-compatible or non-related donors respectively). Enzyme injection takes place every one or two weeks, using polyethylene-glycol bovine adenosie deaminase (PEG-ADA), and enables to maintain a high level of plasmatic ADA, but the restoration of the immune function is suboptimal in the long term. Last, gene therapy enables to restore the immune and metabolic function, with a complete restoration in the best cases, even without preliminary myeloablation Aiuti and Giovannetti (2003); Aiuti et al (2007)
When a patient, who is initially receiving enzyme therapy, receives an injection of modified cells within a gene therapy, there is a trade-off between continuing the enzyme treatment that enables to maintain a high level of plasmatic ADA, and discontinuing the enzyme treatment with the aim of procuring a selective advantage to modified cells, possibly due to their own enzyme production, to favor engraftment Aiuti et al (2002b). The modeling aim of the present work is to investigate the conditions for engraftment from an intra-organismal point of view and the impact enzyme injections can have on cell population dynamics.
We now modify the general first order system (Equations 4-5, first article) to model in particular the intra-organismal ecological dynamics of a ADA-SCID gene therapy (see introduction). We consider that non-modified cells (i.e. the autochthons, hereafter noted
G cells are supposed to have a normal dynamics Cassani et al (2009), as described by Equation 4 (first article). To facilitate model interpretation, we separate the cost of enzyme production (noted c) from the intrinsic mortality (m),
A cells follow the same dynamics than G cells, but they do not pay the cost of enzyme production. In contrast, when the enzyme is missing in the environment, A cells undergo an additional mortality d due to the accumulation of intracellular metabolites. The presence of the enzyme lowers the additional mortality, times a scale factor denoted b. We consider that A cell detoxification by the enzyme depends on the enzyme plasmatic concentration, and not on the per capita quantity, which is assumed to be negligibly modified by the interaction with A cells. Notice that now, the model is not scale-independent anymore: multiplying the quantity G, A, E by a given factor will lower the additional mortality of A due to the increase in enzyme concentration. The dynamics for A becomes:
The enzyme undergoes an intrinsic decay (with a characteristic time
This system admits several equilibria (the stability conditions are discussed below):
- In absence of both
This biological equilibrium describes the state of the system in case of a myeloablation, before the injection of modified cells.
, and :
This is the hypothetical equilibrium before gene therapy or after
This is the therapeutic target equilibrium, where
, , then:
Substituting the value of
We can notice that at this equilibrium, the enzyme concentration
When there is coexistence, the system follows a regime of exponential relaxation, or a pseudoperiodical regime with damped oscillations (figure 1), or is unstable (see appendix 5.1). These oscillations result from the coupling of
In this section, we modify the first-order model into a second-order model following the reasoning exposed for the general model, to be able to investigate the therapeutic impacts of a potential time-scale non-separability between the per capita growth rate dynamics (
The equation for enzyme dynamics is as before:
This system admits the same equilibria as the first order system, but its behavior in the neighborhood of the equilibria is different.
If the enzyme dynamics is fast compared to the population dynamics, we may approximate
We can investigate the structural stability of the second order model by adding a friction term. (Recall that this term represents also the share of intracellular resources between daughter cells when the limiting factor has non-heritable effects, see our discution on friction in the previous article.)
The equation for enzyme dynamics is as before:
We will treat this case by numerical experimentation (figure 2, 2, 2). It turns out that friction counteracts the inertia introduced by the enzyme decay rate (
This work seeks to investigate the possible perturbations of cell population dynamics with a therapeutic aim. From the therapeutic point of view, such a modeling work can be considered as a mean to optimize the research of relevant empirical variables.
When considering the first order model, it appears that enzyme injection is counterproductive for ADA-SCID gene therapy, in the sense that within a stable coexistence equilibrium, injections do not increase the quantity
Coexistence at the equilibrium entails that
The equilibrium is stable when
We would like to draw attention to the fact that this reasoning holds at the equilibrium only (and, also, if there is coexistence, which seems to be the case at the scale of several years). From a therapeutic point of view, it can be unavoidable to resort to sporadic enzyme injections when the patient is in bad condition Aiuti et al (2002b), to temporarily increase the enzymatic level above its equilibrium value. Notice however, that the injections should then take place on a period as short as possible (in approximately one month, the plasmatic enzyme seems to go back to its pre-injection equilibrium value despite the injections, Aiuti et al (2002b) Fig.1). In addition to the observed decrease in
In this study, we focused on a permanent gene therapy. However, the perspective of gene therapies also concerns transitory gene therapies. For instance, in case of a bone fracture, it is possible to modify cells in such a way that they produce osteogenic growth factors and thus improve bone welding. This kind of therapy enables a more spatially targeted treatment than the injection of an exogenous protein Baltzer and Lieberman (2004). In this case, the aim is not the invasion of a tissue by
A similar model could be used for cancer gene therapies using mesenchymal stem cells. These cells have a positive tropism for tumors and their metastases, and can be transformed to deliver oncolytic viruses of therapeutic proteins specifically on tumoral sites, and their persistence in the patient’s organism is not necessarily wanted (see the review by Dwyer et al (2010)).
To our knowledge, the available data do not enable to conclude with respect to the importance of demographic inertia in the case of intraorganismal cell population dynamics. Demographic inertia, in particular inertia due to maternal effects, has been discussed in ecology (e.g. Ginzburg and Taneyhill (1994), but see Berryman (1995)). We know of no such a discussion in intra-organismal ecology.
In case of demographic inertia, the populations have their own pulsation and can oscillate around the equilibrium, in particular when they experience perturbations. Oscillatory behaviors, and fluctuations in general, are numerous in intraorganismal ecology (e.g. Wagner et al (1996); Perazzo et al (2000). Notably, some hematological disorders (some leukemias and neutropenias in particular) result in oscillatory behaviors of cell population dynamics with periods ranging from ten to hundred days depending on the disease; in this case, the temporal pattern is even part of the clinical description Birgens and Karle (1993); Haurie et al (1999); Hirase et al (2001); Hirayama et al (2003); Xiao et al (2003); Colijn and Mackey (2005).
As for ADA-SCID gene therapy, the lymphocyte number fluctuates (from a factor one to six, see Aiuti et al (2002a)) but the data are insufficient to assess the potential intrinsic period of the population dynamics. Given the state of current knowledge, we thus cannot ascribe oscillatory behaviors to an intrinsic demographic inertia, in contrast with an external forcing. However, the search for oscillatory mechanisms could answer this question, in a potentially easier way than for macro-organisms, thanks to the shorter characteristic time of demographic renewal. Notice however that in intraorganismal ecology, we could expect that the organism exerts a friction on potential oscillations of cell populations to be able to maintain a certain homeostasis. This friction would result from cell relationships with their environment and would be thus difficult to isolate in vitro.
The demographic inertia, if proven, could be of importance for cell population management (that is, in our case, for managing non-modified
Generally speaking, even in the absence of external perturbations like enzyme injections, it turns out that demographic inertia can be a source of supplementary instability leading to engraftment failing.
The ecological cell niche can be a source of demographic inertia (in our model, because of the enzyme’s characteristic time
It appears that
As regards the effect of the enzyme on the dynamics, we have chosen a simple model where
In this paper, we investigated the impact of a cell niche construction activity (the synthesis of a missing enzyme) that would not be time-scale separable from the cell population dynamics. This drove us to focus on the importance of transitory dynamics (oscillations, relaxation times) that are due to niche and demographic inertia.
Two different perspectives emerge, given that we consider that ecological interactions should be described as first or second order systems. First order systems describe demographic dynamics that are directly affected by demographic factors (resources, sources of mortality), while second order systems describe demographic dynamics that are indirectly affected, through the interplay at the individual level between metabolism and resource acquisition. At the second order, demographic factors are analoguous to forces in the Newtonian mechanics, that impact the acceleration of a movement and not its speed. In mathematical terms, a supplementary dynamic dimension is added to the definition of the niche.
Ginzburg et al (2004) (p102-103), in a programmatic conclusion, urged ecologists to identify the ecological “forces”, force here having to be understood as a cause inducing a modification of the energetic state of the individuals or of the corresponding growth rate. Among such forces, Ginzbug & Colyvan see energetics, maternal effects, and predator-prey relationships. The niche as a substrate for demographic inertia, and niche construction as an ecological force, could be added to this program.
In this paper, we have chosen to describe the lymphocyte population with the minimal model of an unstructured population. However, the populations of interest (lymphocyte strains) are, from an ecological point of view, metapopulations that are structured in source-sink populations because of cell differentiation, which might also be correlated to cell age (e.g., hematopoietic stem cell → lymphoid progenitor → lymphoblast → prolymphocyte → lymphocyte → T-lymphocyte, to mention just one strain). The fact that differentiated cells can de-differentiate could be modelled in a structurally homologous way than migration occuring from a sink population in ecology.
Most probably, such a highly structured cell fate, contrained by the cellular environment posed by the organism, should have major dynamical effects. Then “the organism [would be] an ecosystem, but [would be] more than an ecosystem”, and intra-organismal ecology would have to be not just an application, but also a true extension of classical ecology.
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 Ferrière, 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 a partial update of a previous work in French (Pocheville 2010:chap.3), realized while both authors were benefiting from a funding from the Frontiers in Life Sciences PhD Program and from the Liliane Bettencourt Doctoral Program. A.P is currently benefiting from a Postdoctoral Fellowship from the Center for Philosophy of Science, University of Pittsburgh. M.M. is currently benefiting from a Postdoctoral Fellowship from the Region Ile-de-France, DIM-ISC.
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After linearization, we seek for the eigenvalues of the system (of dimension 3). These eigenvalues are roots of the characteristic polynomial:
It turns out that in the case where there is coexistence, two scenarii are possible:
- The polynomial has three negative roots: in this case the system converges exponentially toward equilibrium
- or the polynomial has a negative root and two conjugated complex solutions. In this case the system is stable or unstable according to the sign of the real part of the roots. In the case where the system is stable, the regime is pseudoperiodical with damped oscillations.
The literal resolution of the polynomial can be easily realized, but the literal expression of the solutions is too complicated to be informative.
5.2 Linearized second order system with two species, enzyme construction, and time-scale separation on the enzyme dynamics
The following characteristic polynomial obtains:
The polynomial discriminant is
If there is coexistence then we have
In this case the eigenvectors have a dynamics in a
When the quantity