Modelling experimental uveitis: barrier effects in autoimmune disease

Abstract

Objective and design

A mathematical analysis of leukocytes accumulating in experimental autoimmune uveitis (EAU), using ordinary differential equations (ODEs) and incorporating a barrier to cell traffic.

Materials and subjects

Data from an analysis of the kinetics of cell accumulation within the eye during EAU.

Methods

We applied a well-established mathematical approach that uses ODEs to describe the behaviour of cells on both sides of the blood–retinal barrier and compared data from the mathematical model with experimental data from animals with EAU.

Results

The presence of the barrier is critical to the ability of the model to qualitatively reproduce the experimental data. However, barrier breakdown is not sufficient to produce a surge of cells into the eye, which depends also on asymmetry in the rates at which cells can penetrate the barrier. Antigen-presenting cell (APC) generation also plays a critical role and we can derive from the model the ratio for APC production under inflammatory conditions relative to production in the resting state, which has a value that agrees closely with that found by experiment.

Conclusions

Asymmetric trafficking and the dynamics of APC production play an important role in the dynamics of cell accumulation in EAU.

Appendix: Details of the model

Our modelling approach is based on the principles of immune-response models that have been proposed and analysed in earlier literature. The model is expressed in terms of the customary mass action equations, where it is assumed that concentration heterogeneities are unimportant; a significant distinction from earlier models is that different concentrations can exist on opposite sides of the barrier. The possible dynamic processes are characterised by rate constants controlling production, proliferation, death (apoptosis for cells), activation or transformation (differentiation) of one species to another produced by contact with another species in the system, and permeation through the barrier.

One advantage of single-compartment models is that, being limited to only three or four cell populations, they lend themselves well to the analytical techniques of dynamical systems. With the introduction of a barrier and the consequent effects on the population dynamics, our model requires at least seven variables, and rigorous mathematical analysis becomes intractable. While there is a clear interest in keeping the number of model variables to a minimum, we have introduced a further three cell populations in order to represent important aspects of the cell dynamics. These are: the immunisation concentration of APCs that do not permeate the barrier; ‘mother’ and ‘daughter’ activated T-cell populations constructed to represent the proliferation/differentiation process during clonal expansion; and a population of memory T cells whose existence may be significant for the longer-term response. Investigations of a model that omits memory cells suggest that sustained oscillatory behaviour similar to that observed experimentally does not occur when T _o is maintained at the concentration level of about 5 cm⁻³ indicated by experimental data [27]. After an immune response, T cells that become memory cells increase the total concentration of antigen-specific T cells that can then be reactivated by activated APCs increases.

The basic mechanisms operating in our model are summarised as follows:

• Naïve T-cell numbers (T _o) in the resting state are maintained through a homeostatic combination of cell production, cell division and apoptosis. In contact with activated APCs, they become a first generation of ‘mother’ effector T cells (T ^*_m ). The unactivated T _o cells do not permeate the barrier [28].

• Mother effector T cells (T ^*_m ) either undergo apoptosis, or divide into N generations of daughter effector T cells (T ^*_d ). The mother effector T cells are not allowed to permeate the barrier—this restriction reflects the observation that activated T-cell proliferation inside the eye is inhibited by the local environment. In the circulation, activated T-cell division is not inhibited [14, 15]. Neglecting permeation of this population is justified by the relatively small proportion of mother cells to daughter cells.

• Daughter effector T cells are activated (T ^*_d ) and can permeate the barrier, undergo apoptosis, or differentiate into (unactivated) memory T cells. The population is bounded, due to spatial and nutritional factors.

• Memory T cells (T _M) can also be activated into mother cells (T ^*_m ), or undergo apoptosis. The memory T cells do not permeate the barrier. This population is also bounded due to spatial and nutritional factors.

• The immunisation population of APCs that initiate the immune response (A _oo) only undergoes apoptosis. It is considered to be an immobilised population of APCs in the periphery (likely to be dendritic cells), and cannot permeate the barrier.

• APCs outside the retina (A _o) in the resting state are maintained through a homeostatic combination of cell production, cell division, apoptosis and permeation through the barrier.

• APCs inside the retina (A _i) in the resting state are maintained through a homeostatic combination of entry (permeation) through the barrier and apoptosis. They can acquire retinal antigen and be activated on contact with activated T cells, otherwise they undergo apoptosis.

• Activated APCs inside the retina (A ^*_i ) have acquired retinal antigen and can permeate through the barrier, or undergo apoptosis.

• Activated APCs outside the retina (A ^*_o ) can permeate the barrier, undergo apoptosis and activate naïve T cells.

The following set of equations defines the time evolution, during EAU, of the species listed in Table 2, according to the system described above. Each equation is constructed as the sum of components representing the basic mechanisms listed above.

$$ \frac{{{\text{d}}T_{\text{o}} }}{{{\text{d}}t}} = a_{\text{T}} + p_{\text{T}} T_{\text{o}} \left( {1 - \frac{{T_{\text{o}} }}{{T_{\text{L}} }}} \right) -\delta_{\text{T}} T_{\text{o}} - \alpha_{\text{T}} T_{\text{o}} (A_{\text{o}}^{*} + A_{\text{oo}}^{*} ) $$

(1)

$$ \frac{{{\text{d}}T_{\text{m}}^{*} }}{{{\text{d}}t}} = - \delta_{\text{T}}^{*} T_{\text{m}}^{*} + \alpha_{\text{T}} (T_{\text{o}} + T_{\text{M}} )(A_{\text{o}}^{*} + A_{\text{oo}}^{*} ) - H_{ 1}(t) p_{\text{T}}^{*} T_{\text{m}}^{*} $$

(2)

$$ \frac{{{\text{d}}T_{\text{M}} }}{{{\text{d}}t}} = - \delta_{\text{T}} T_{\text{M}} + H_{2}(t) \beta T_{\text{d}}^{*} \left( {1 - \frac{{T_{\text{M}} }}{{T_{\text{ML}} }}} \right) - \alpha_{\text{T}} T_{\text{M}} (A_{\text{o}}^{*} + A_{\text{oo}}^{*} ) $$

(3)

$$ \frac{{{\text{d}}T_{\text{d}}^{*} }}{{{\text{d}}t}} = - (K_{{T_{\text{d}} }}^{*} T_{\text{d}}^{*} - K_{{T_{\text{i}} }}^{*} T_{\text{i}}^{*} )f(T_{\text{i}}^{*} )h(t) + H_{1}(t) (N - 1)p_{\text{T}}^{*} T_{\text{m}}^{*} \left( {1 - \frac{{T_{\text{d}}^{*} }}{{T_{\text{dL}} }}} \right) - \delta_{\text{T}}^{*} T_{\text{d}}^{*} - H_{2}(t) \beta T_{\text{d}}^{*} \left( {1 - \frac{{T_{\text{M}} }}{{T_{\text{ML}} }}} \right) $$

(4)

$$ \frac{{{\text{d}}T_{\text{i}}^{*} }}{{{\text{d}}t}} = v(K_{{T_{\text{d}} }}^{*} T_{\text{d}}^{*} - K_{{T_{\text{i}} }}^{*} T_{\text{i}}^{*} )f(T_{\text{i}}^{*} )h(t) - \delta_{\text{T}}^{*} T_{\text{i}}^{*} $$

(5)

$$ \frac{{{\text{d}}A_{\text{o}}^{{}} }}{{{\text{d}}t}} = a_{\text{A}} + p_{\text{A}} A_{\text{o}} \left( {1 - \frac{{A_{\text{o}} }}{{A_{\text{L}} }}} \right) - (K_{{A_{\text{o}} }} A_{\text{o}} - K_{{A_{\text{i}} }} A_{\text{i}} )f(T_{\text{i}}^{*} )h(t) - \delta_{\text{A}} A_{\text{o}} $$

(6)

$$ \frac{{{\text{d}}A_{\text{i}} }}{{{\text{d}}t}} = v(K_{{A_{\text{o}} }} A_{\text{o}} - K_{{A_{\text{i}} }} A_{\text{i}} )f(T_{\text{i}}^{*} )h(t) - \delta_{\text{A}} A_{\text{i}} - \alpha_{\text{A}} T_{\text{i}}^{*} A_{\text{i}} $$

(7)

$$ \frac{{{\text{d}}A_{\text{o}}^{*} }}{{{\text{d}}t}} = - (K_{{A_{\text{o}} }}^{*} A_{\text{o}}^{*} - K_{{A_{\text{i}} }}^{*} A_{\text{i}}^{*} )f(T_{\text{i}}^{*} )h(t) - \delta_{\text{A}}^{*} A_{\text{o}}^{*} $$

(8)

$$ \frac{{{\text{d}}A_{\text{i}}^{*} }}{{{\text{d}}t}} = v(K_{{A_{\text{o}} }}^{*} A_{\text{o}}^{*} - K_{{A_{\text{i}} }}^{*} A_{\text{i}}^{*} )f(T_{\text{i}}^{*} )h(t) - \delta_{\text{A}}^{*} A_{\text{i}}^{*} + \alpha_{\text{A}} T_{\text{i}}^{*} A_{\text{i}} $$

(9)

$$ \frac{{{\text{d}}A_{\text{oo}}^{*} }}{{{\text{d}}t}} = - \delta_{\text{A}}^{*} A_{\text{oo}}^{*} $$

(10)

The following constants are used in the rate equations with appropriate subscripts/superscripts to denote the species to which they apply:

α, activation of T _i → T ^*_i (α _T) or A _i → A ^*_i (α _A); β, differentiation of activated (effector) daughter cells to memory cells; δ, apoptosis (cell death); K, permeability constant; a, production of T cells (a _T) or A cells (a _A); p, division of naïve T cells (p _T), proliferation of mother cells to daughter cells(p ^*_T ), and division of A _o cells (p _A); f(T ^*_i ), permeation rate dependence on T ^*_i (see below); h(t), permeation rate dependence on time (see below); H _n , Heaviside switching functions (H _n  = 0 below a time threshold and H _n  = 1 above the threshold) that account for the delay in proliferation of daughter cells [29] (n = 1) and the time lag between cell recognition and cell division and differentiation [30] (n = 2); v, the ratio of the volume occupied by populations within the retina to the volume occupied by populations in the peripheral volume; T _L, T _dL, T _ML and A _L, limiting concentrations in the production and proliferation terms for the respective species in Eqs. (1), (3), (4) and (6). The general properties of the production and proliferation terms are discussed more fully below [Eq. (13)]; (N−1), the maximum number of daughter cells generated by a mother cell.

For the remainder of this section, we examine some of the terms appearing in Eqs. (1)–(10) in further detail.

Barrier permeation

The permeation rate through the barrier is assumed to be proportional to the concentrations on either side of the barrier. The principle of mass balance underlying our equations requires the introduction of the factor v for permeation-related terms in the evolution equations for species inside the eye. The unperturbed barrier is permeable to activated cells, and the permeability increases with the concentration T ^*_i up to a saturation limit [18]. We account for this by introducing a function f(T ^*_i ) which modifies the barrier permeability in response to the increase in leukocyte concentration inside the eye:

$$ f(T_{\text{i}}^{*} ) = \frac{{\mu + \sigma T_{\text{i}}^{*} }}{{\mu + T_{\text{i}}^{*} }} $$

(11)

In Eq. (11), σ is a dimensionless constant and μ has the dimensions of concentration. f(T ^*_i ) approaches the limit σ when T ^*_i  → ∞, is unity when T ^*_i  = 0 and has the value (1 + σ)/2 when the concentration T ^*_i  = μ. If T ^*_i oscillates, then f(T ^*_i ) also does so.

Permeability is also modified at barrier breakdown after about 10 days [19, 31]. Here we introduce a sigmoid function h(t) having the form:

$$ h(t) = \frac{{\left( {1 + \exp (bt_{\text{D}} )} \right)^{\gamma } }}{{\left( {1 + \exp [ - b(t - t_{\text{D}} )]} \right)^{\gamma } }} $$

(12)

with h(0) = 1; h(∞) = [1 + exp (bt _D)]^γ = ϕ. We use input values for t _D, the amplification of the permeability, and $\dot{h}(t_{\text{D}})/\gamma$ to determine the parameters for h(t): γ is found to be in the range [0.5, 1.5] and b is in the range [0.2, 0.5].

In the equations, provision has been made for each of the initial permeability rates to be different. This allows, for example, that the barrier may be asymmetrical, or that biological modifications to the cells on either side may alter their rate of transport. The model makes no allowance for spatial variations in cell concentration on either side of the membrane.

Homeostasis

In mice and humans, homeostasis in the populations of naïve T cells and APCs is maintained by replenishment from the thymus and bone marrow, respectively. To represent homeostatic processes for a concentration c we use logistic equations [16, 17] with a production rate a and proliferation (division) constant p:

$$ \frac{{{\text{d}}c}}{{{\text{d}}t}} = a + pc\left( {1 - \frac{c}{{c_{\text{L}} }}} \right) $$

(13)

The merits of this and other possible forms have been discussed in several places [13, 16, 32], and we have chosen this approach as it is widely applied in related models. As pointed out by May [16], Eq. (13) suffers from the disadvantage that proliferation rates can become negative when $c > c_{\text{L}}$. We return to this point below. An alternative form for the homeostatic limitation of memory cells specifically has been proposed in which the population is controlled by Heaviside switching functions [12].

Species ‘transformation’: activation or proliferation

The product terms: $\alpha_{\text{A}} T_{\text{i}}^{*} A_{\text{i}}$ and $\alpha_{\text{T}} T_{\text{o}} (A_{\text{o}}^{*} + A_{\text{oo}}^{*} )$ represent the activation of APCs by activated T cells, and the activation of naïve T cells (into ‘mother’ activated T cells) by activated APCs, respectively. The proliferation process results in (N − 1) daughter cells: however, this process is mediated by the availability of space and nutrition for activated cells. To reflect this upper bound in daughter numbers, we employ a logistic term. With similar justification, a logistic term is also included for the conversion of daughter cells into memory cells (in the outer compartment).

Parameters

Where suitable values could be found, we have used parameters estimated from data in the literature, or from our own experiments. Some of the parameters in the above equations were not available, or could only be estimated within orders of magnitude. This applies in particular to the permeability constants, and we have simplified these considerably by reducing their number to only two.

Some constraints on parameters emerge as a consequence of the resting behaviour predicted by the model. We investigated the effects of varying several of the parameters in order to explore the range of behaviour accessible to our set of equations.

Individual rate processes such as apoptosis, population growth by cell doubling, or permeation across the barrier, driven by a species concentration gradient, can be expressed in a general form dc/dt = ±kc(t) for a population concentration c(t). In the absence of other effects, the resultant concentration evolves as c = c(0)exp(± kt). Thus we can estimate rate constants from the rate at which these individual processes double or halve a population.

Concentrations are expressed in number of cells per mL (cm⁻³). There are approximately 10⁷ white blood cells in the murine circulation [33, 34]; of these approximately 5 % are monocytes, which we identify with A _o in our equations, equivalent to a resting state concentration of 5 × 10⁵ cm⁻³. The resting state concentration of naïve T cells (T _o) specific for a unique determinant, at about 5 cm⁻³, is very much lower [27]. The resting state concentration of A cells inside the eye (A _i) is 2.9 × 10⁵ cm⁻³, based on experimental cell counts [3] and a mouse eye retinal volume of 2.29 × 10⁻³ cm³ (H. Xu, personal communication).

The doubling time for activated T cells has been estimated to be between 8 and 12 h [12, 30]. The naive T cell population replenishes itself at a rather slower rate of less than once per day [35]. The division and maintenance of a stable population, which is the major factor in maintaining T-cell homeostasis [36], has recently been considered in detail [37].

The half-life for naïve T cells is quite long [37], while that of activated T cells has been estimated as between 2 and 6 days [30, 38]. The dendritic cells that initiate the immune response (A ^*_oo ) and do not pass the BRB have an estimated half-life of 3 days [30, 39].

Activation of T (α _T) cells by dendritic cells has been given as 10⁻³ cm³ day⁻¹ [30] as an ad hoc parameter. The subsequent process of differentiation of effector cells to memory cells has been widely discussed [40–44] and several models have been proposed recently [42, 44–46]. In some models CD8⁺ rather than CD4⁺ cells have been modelled; however, Kallies [43] has argued that comparable dynamics apply to both types. Our equations correspond most closely to the linear differentiation model discussed (with others) by Ganusov [44]. Data fitting from the publications cited suggest a half-life of between 600 and 900 h for the differentiation process.

The maximum increase in the barrier permeability due to T ^*_i [σ in Eq. (11)] is of the order of 10³ [18]. Calculations are not very sensitive to the other parameter in this equation, μ, which was set at 1,000 cm⁻³. We choose a similar order of magnitude for the increase in permeability due to barrier disruption (ϕ = 500) from Eq. (12).

In the next section we deduce values for the permeability constant and show that a _T, the rate of production of naïve T cells from the thymus per unit volume, and a _A, the rate of production of A _o cells from the bone marrow, are bounded by the model.

Equilibrium

Equations (1)–(10) define the evolution in a phase space of accessible states for which the resting state represents the equilibrium state in the absence of disease. The only non-zero concentrations in this state are T ^e₀ , the concentration of naïve T cells in the resting state, and A ^e₀ and A ^e_i , the resting-state concentrations of the (unactivated) APCs outside and inside the retina, respectively. As the resting state is an equilibrium point, T ^e₀ is given by the equation

$$ \frac{{2p_{\text{T}} }}{{T_{\text{L}} }}T_{0}^{\text{e}} = (p_{\text{T}} - \delta_{\text{T}} ) + \left[ {(p_{\text{T}} - \delta_{\text{T}} )^{2} + \frac{{4p_{\text{T}} a_{\text{T}} }}{{T_{\text{L}} }}} \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$

(15)

Equation (15) can be re-arranged as an expression for T _L, the limiting concentration that will sustain this concentration in the absence of any other perturbation:

$$ \frac{{T_{\text{L}} }}{{T_{0}^{\text{e}} }} = \frac{{p_{\text{T}} T_{0}^{\text{e}} }}{{a_{\text{T}} + (p_{\text{T}} - \delta_{\text{T}} )T_{0}^{\text{e}} }} $$

(16)

We define $q = T_{\text{L}} /T_{0}^{\text{e}}$, and impose the constraint q > 1: this ensures that the cell division term (p _T ) in Eq. (1) is positive in the resting state, where a negative rate of T-cell division would not have any biological meaning. This does not of course imply that dT _o/dt cannot be negative; this can occur if the rate of cell death is high enough.

A further consequence of the constraint q > 1 is that the rate of cell production a _T ≤ δ _T T ^e₀ , since the numerator in Eq. (16) must be greater than the denominator, which therefore sets an upper bound for a _T. Estimates for the parameters show that the half-life for naive T cells is much longer than the doubling time for production and consequently that p _T > δ _T. Using the doubling time and half-life from Table 3, p _T = 2.9 × 10⁻² h⁻¹ and δ _T = 5.07 × 10⁻⁴ h⁻¹. q has an approximately linear dependence on a _T, with a maximum value of q = p _T/(p _T − δ _T) = 1.018 and a minimum value of 1.0 (when a _T = δ _T T ^e₀  ≈ 2.5 × 10⁻³ h cm⁻³).

The maintenance of homeostasis according to the logistic equation therefore imposes an upper bound on the rate of cell production from the thymus. With suitable parameter and time-step choices, and in the absence of other influences, the analytical and numerical solutions of Eq. (15) return a constant value of T ^e_o with time.

The only other species present in the resting state are A _o and A _i; the rates of change of their concentrations in this state (when the factors modifying the permeability are unity) are given by

$$ \frac{{{\text{d}}A_{\text{o}} }}{{{\text{d}}t}} = a_{\text{A}} + p_{\text{A}} A_{\text{o}} \left( {1 - \frac{{A_{\text{o}} }}{{A_{\text{L}} }}} \right) - (K_{{A_{\text{o}} }} A_{\text{o}} - K_{{A_{\text{i}} }} A_{\text{i}} ) - \delta_{\text{A}} A_{\text{o}} $$

(17)

$$ \frac{{{\text{d}}A_{\text{i}} }}{{{\text{d}}t}} = v(K_{{A_{\text{o}} }} A_{\text{o}} - K_{{A_{\text{i}} }} A_{\text{i}} ) - \delta_{\text{A}} A_{\text{i}} $$

(18)

The steady-state values A ^e_o and A ^e_i , determined by the zeroes of Eqs. (17) and (18), can be expressed in terms of the ratio θ = A _L/A ^e_o [in analogy to the definition of q in Eq. (16)]

$$ \theta = \frac{{A_{\text{L}} }}{{A_{\text{o}}^{\text{e}} }} = \frac{{A_{\text{o}}^{\text{e}} p_{\text{A}} }}{{a_{\text{A}} - A_{\text{o}}^{\text{e}} B}} $$

(19)

in which B is defined by the expression

$$ B = - p_{\text{A}} +\delta_{\text{A}} +K_{{A_{\text{o}} }} \delta_{\text{A}} /(vK_{{A_{\text{i}} }} + \delta_{\text{A}} ) $$

(20)

and the ratio of A ^e_i and A ^e_o determined from Eq. (18):

$$ \frac{{A_{\text{i}}^{\text{e}} }}{{A_{\text{o}}^{\text{e}} }} = A =\frac{{vK_{{A_{\text{o}} }} }}{{vK_{{A_{\text{i}} }} + \delta_{\text{A}} }} $$

(21)

Equation (21) shows that the equilibrium populations of APCs inside and outside the retina are not equal if the barrier permeability is the same in each direction. If we choose a single permeability constant $K_{\text{A}} = K_{{A_{\text{o}} }} = K_{{A_{\text{i}} }}$, then Eq. (21) shows that A ^e_o  > A ^e_i , as supported by experimental observation.

Equation (19) can be rearranged as an expression for a _A

$$ a_{\text{A}} = A_{\text{o}}^{\text{e}} \left[ {\frac{{B\theta + p_{\text{A}} }}{\theta }} \right] $$

(22)

As for T _o, we note from its definition that if θ < 1, the rate of production of APCs [the second term in Eq. (17)] would be negative at equilibrium (i.e. A ^e_o /A _L > 1), which does not have any meaning, and therefore θ cannot be less than unity.

Substitution of Eq. (21) into Eq. (20) gives

$$ B = - p_{\text{A}} + \delta_{\text{A}} + A\delta_{\text{A}} /v $$

(23)

which, with Eq. (22), yields an expression for the rate of production of A cells:

$$ a_{\text{A}} = A_{\text{o}}^{\text{e}} \left[ {\delta_{\text{A}} (1 + A/v) - \frac{{p_{\text{A}} (\theta - 1)}}{\theta }} \right] $$

(24)

Since a _A > 0, Eq. (24), leads to an upper limit for θ:

$$ \theta < \frac{1}{{[1 - \delta_{\text{A}} (1 + A/v)/p_{\text{A}} }} $$

(25)

In most calculations we have used the ratio of retinal to vascular volumes of 61.9 (Xu, personal communication) as an estimate for v. The estimated values for δ _A and p _A (Table 1) give an upper limit of 1.313 for θ when v is given a value of 61.9. We have remarked that the peripheral volume is uncertain and may be larger, in which case v would be smaller than 61.9. Equation (25) sets a lower limit on v of 0.180, since θ _max then becomes < 0, but increases to a large positive value close to this limit. The maximum value permissible for the peripheral volume on this basis is 12.7 × 10⁻³ cm³. Equation (24) yields the fractional turnover rate of A _o cells of approximately 1/100 every hour. Calculations show the value of θ has to be very close to its maximum value for oscillations to occur; typically, if θ/θ _max is below ~0.995, the system does not oscillate. This upper limit on θ restricts the production limit A _L in Eq. (6) and consequently the number of A _o cells that can be generated in this way.

From Eq. (21) we can make a first-order estimate for the permeability constants by assuming that they are identical and equal to K (say). Using the concentrations in Table 1, with v = 61.9 and δ _A  = 9.627 × 10⁻³ h⁻¹ from the half-life of 3 days in Table 3, yields K = 2.14 × 10⁻⁴ h⁻¹. $K_{{A_{\text{i}} }}$ can be calculated from Eq. (21) using this as an order of magnitude estimate for $K_{{A_{\text{o}} }}$. Since $K_{{A_{i} }} > 0$, Eq. (21) requires that $K_{{A_{\text{o}} }} > {\text{A}}\delta_{A} /\nu$ (= 9.02 × 10⁻⁵ h⁻¹ with the above parameter estimates) which sets a lower limit for $K_{{A_{\text{o}} }}$. An upper limit depends on several parameters which may themselves have a range of possible values. Trial calculations showed that this upper limit is approximately 2.6 × 10⁻⁴ h⁻¹, and that the calculations could not be completed with higher permeabilities.

The single-compartment model

The single-compartment model is created by removing the barrier, and summing the pairs of Eqs. (4) + (5), (6) + (7) and (8) + (9). There is now no distinction between external and internal species, although mother, daughter, memory and naïve T cells remain distinct. The resulting single compartment model is expressed by:

$$ \frac{{{\text{d}}T^{*} }}{{{\text{d}}t}} = H_{1} (N - 1)p_{\text{T}}^{*} T_{\text{m}}^{*} \left( {1 - \frac{{T_{\text{d}}^{*} }}{{T_{\text{dL}} }}} \right) - \delta_{\text{T}}^{*} T^{*} - H_{2} \beta T_{\text{d}}^{*} $$

(26)

$$ \frac{{{\text{d}}A}}{{{\text{d}}t}} = a_{\text{A}} + p_{\text{A}} A\left( {1 - \frac{A}{{A_{\text{L}} }}} \right) - \delta_{\text{A}} A - \alpha_{\text{A}} T_{\text{i}}^{*} A $$

(27)

$$ \frac{{{\text{d}}A^{*} }}{{{\text{d}}t}} = - \delta_{\text{A}}^{*} A^{*} + \alpha_{\text{A}} T_{\text{i}}^{*} A $$

(28)

together with Eqs. (1), (2), (3) and (10) which are unchanged.

Stability of the resting state

The resting state is one of several fixed points in the system. The stability of the system with respect to this state is determined from the eigenvalues of the Jacobian of the time evolution operator. Analysis of the eigenvalues shows that the system is unstable with respect to the resting state when activated species are present. This differs from some existing single-compartment models, including those closely related to the single-compartment reduction of our model, whose resting state can demonstrate stability for sufficiently small perturbations [47].