IL-2 Stimulation of Regulatory T Cells: A Stochastic and Algorithmic Approach

Abstract

Regulatory T cells express IL-2 receptor (IL-2R) complexes on their surface, but do not produce IL-2 molecules. Survival of a population of regulatory T cells depends on the production of IL-2 by other cells, such as effector T cells. We formulate a stochastic version of the model of Busse (Dynamics of the IL-2 cytokine network and T-cell proliferation, Logos, Berlin, 2010, [1]), for the synthesis of IL-2R by a regulatory T cell in constitutive (ligand-independent) and in ligand-induced conditions, with the assumption that synthesis is a function of the number of IL-2/IL-2R bound complexes present on the cell surface. Exact analysis of the stochastic Markov process, by considering its master equation, is usually not possible. Here, we develop an algorithmic approach, which leads to the analysis of suitable random variables. In particular, we focus on the time to reach a threshold number of IL-2/IL-2R bound complexes on the cell surface, and the number of receptors synthesised in this time. These descriptors provide a way to quantify the rates at which IL-2/IL-2R bound complexes and IL-2R free receptors are formed in the cell, and how these rates relate to each other. By following first-step arguments, the different order moments of these random variables are obtained. We illustrate our approach with numerical realisations. The contributions of the constitutive and the ligand-induced synthesis pathways are quantified under different signalling hypotheses.

L. de la Higuera and M. López-García have contributed equally to this work.

Appendix

In order to efficiently analyse the first descriptor studied in Sect. 3.1, we express the system of equations given by Eq. (2), in matrix form as

$$\begin{aligned} {\varvec{\varphi }}(z) \ = \ \mathbf{A}(z) \; {\varvec{\varphi }}(z) + \mathbf{b}(z)\; , \end{aligned}$$

(7)

where the constant B has been omitted to simplify the notation. The vector of unknowns, ${\varvec{\varphi }}(z)$, is structured, due to the organisation of ${\mathscr {S}}$, in levels and sub-levels, by blocks as follows

$$\begin{aligned} {\varvec{\varphi }}(z)= & {} \left( \begin{array}{c} {\varvec{\varphi }}_0(z)\\ {\varvec{\varphi }}_1(z)\\ \vdots \\ {\varvec{\varphi }}_{B-1}(z) \end{array}\right) ,\quad {\varvec{\varphi }}_{k}(z) \ = \ \left( \begin{array}{c} {\varvec{\varphi }}^k_0(z)\\ {\varvec{\varphi }}^k_1(z)\\ \vdots \\ {\varvec{\varphi }}^k_{n_R^{max}-k}(z) \end{array}\right) ,\quad 0\le k\le B-1, \end{aligned}$$

with ${\varvec{\varphi }}^k_r(z)=(\varphi _{(r,k,0)}(z),\ldots ,\varphi _{(r,k,n_R^{max}-r-k)}(z))^T$, and where $^T$ represents the transpose operator. In a similar way, the organisation of states within ${\mathscr {S}}$ by levels and sub-levels, and the consideration of the transition rates given in Eq. (1), lead to

$$\begin{aligned} \mathbf{A}(z)= & {} \left( \begin{array}{cccccc} \mathbf{A}_{0,0}(z) &{} \mathbf{A}_{0,1}(z) &{} \mathbf{0} &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{A}_{1,0}(z) &{} \mathbf{A}_{1,1}(z) &{} \mathbf{A}_{1,2}(z) &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{A}_{2,1}(z) &{} \mathbf{A}_{2,2}(z) &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{A}_{B-2,B-2}(z) &{} \mathbf{A}_{B-2,B-1}(z)\\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{A}_{B-1,B-2}(z) &{} \mathbf{A}_{B-1,B-1}(z) \end{array}\right) , \end{aligned}$$

where the sub-matrix $\mathbf{A}_{k,k'}(z)$ contains in an ordered fashion, those coefficients in the system, Eq. (2), related to transitions from states in level ${\mathscr {S}}(k)$ to states in level ${\mathscr {S}}(k')$. The specific structure by sub-levels allows us to write the following expressions

$$\begin{aligned} \mathbf{A}_{k,k}(z)= & {} \left( \begin{array}{cccccc} \mathbf{B}^{k,k}_{0,0}(z) &{} \mathbf{B}^{k,k}_{0,1}(z) &{} \mathbf{0} &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{B}^{k,k}_{1,0}(z) &{} \mathbf{B}^{k,k}_{1,1}(z) &{} \mathbf{B}^{k,k}_{1,2}(z) &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{B}^{k,k}_{2,1}(z) &{} \mathbf{B}^{k,k}_{2,2}(z) &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{B}^{k,k}_{n_R^{max}-k-1,n_R^{max}-k-1}(z) &{} \mathbf{B}^{k,k}_{n_R^{max}-k-1,n_R^{max}-k}(z)\\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{B}^{k,k}_{n_R^{max}-k,n_R^{max}-k-1}(z) &{} \mathbf{B}^{k,k}_{n_R^{max}-k,n_R^{max}-k}(z) \end{array}\right) ,\\ \mathbf{A}_{k,k-1}(z)= & {} \left( \begin{array}{cccccc} \mathbf{B}^{k,k-1}_{0,0}(z) &{} \mathbf{B}^{k,k-1}_{0,1}(z) &{} \mathbf{0} &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{B}^{k,k-1}_{1,1}(z) &{} \mathbf{B}^{k,k-1}_{1,2}(z) &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{B}^{k,k-1}_{2,2}(z) &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{B}^{k,k-1}_{n_R^{max}-k-1,n_R^{max}-k}(z) &{} \mathbf{0}\\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{B}^{k,k-1}_{n_R^{max}-k,n_R^{max}-k}(z) &{} \mathbf{B}^{k,k-1}_{n_R^{max}-k,n_R^{max}-k+1}(z) \end{array}\right) ,\\ \mathbf{A}_{k,k+1}(z)= & {} \left( \begin{array}{cccccc} \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{B}_{1,0}^{k,k+1}(z) &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{B}_{2,1}^{k,k+1}(z) &{} \mathbf{0} &{} \dots &{} \mathbf{0} &{} \mathbf{0} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{B}^{k,k+1}_{n_R^{max}-k-1,n_R^{max}-k-2}(z) &{} \mathbf{0}\\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \dots &{} \mathbf{0} &{} \mathbf{B}^{k,k+1}_{n_R^{max}-k,n_R^{max}-k-1}(z) \end{array}\right) , \end{aligned}$$

where the dimensions of the sub-blocks $\mathbf{0}$ in the previous expressions have been omitted. We note that, in fact, the dimensions of a sub-block $\mathbf{0}$ corresponding to the group of rows ${\mathscr {S}}(k;r)$ and the group of columns ${\mathscr {S}}(k';r')$ are $J(k;r)\times J(k';r')$. Expressions for the sub-matrices $\mathbf{B}^{k,k'}_{r,r'}(z)$ can be obtained from Eq. (2) as

$$\begin{aligned} (\mathbf{B}_{r,r}^{k,k-1}(z))_{ij}= & {} \left\{ \begin{array}{ll} \gamma \; k \; (z+\varDelta _{(r,k,i)})^{-1}, &{} {\mathrm{if}\,j=i+1},\\ 0, &{} \mathrm{otherwise}, \end{array}\right. \end{aligned}$$

where $1\le i\le J(k;r)$, $1\le j\le J(k-1;r)$, $1\le k\le B-1$ and $0\le r\le n_R^{max}-k$;

$$\begin{aligned} (\mathbf{B}_{r,r+1}^{k,k-1}(z))_{ij}= & {} \left\{ \begin{array}{ll} k_\mathrm{off} \; k \; (z+\varDelta _{(r,k,i)})^{-1}, &{} {\mathrm{if}\,j=i},\\ 0, &{} \mathrm{otherwise}, \end{array}\right. \end{aligned}$$

where $1\le i\le J(k;r)$, $1\le j\le J(k-1;r{+}1)$, $1\le k\le B{-}1$ and $0\le r\le n_R^{max}-k{-}1$;

$$\begin{aligned} (\mathbf{B}_{r,r-1}^{k,k}(z))_{ij}= & {} \left\{ \begin{array}{ll} k_s \; r \; (z+\varDelta _{(r,k,i)})^{-1}, &{} {\mathrm{if}\,j=i},\\ 0, &{} \mathrm{otherwise}, \end{array}\right. \end{aligned}$$

where $1\le i\le J(k;r)$, $1\le j\le J(k;r-1)$, $0\le k\le B-1$ and $1\le r\le n_R^{max}-k$;

$$\begin{aligned} (\mathbf{B}_{r,r}^{k,k}(z))_{ij}= & {} \left\{ \begin{array}{ll} k_e \; i \; (z+\varDelta _{(r,k,i)})^{-1}, &{} {\mathrm{if}\,j=i-1},\\ 0, &{} \mathrm{otherwise}, \end{array}\right. \end{aligned}$$

where $1\le i\le J(k;r)$, $1\le j\le J(k;r)$, $0\le k\le B-1$ and $0\le r\le n_R^{max}-k$;

$$\begin{aligned} (\mathbf{B}_{r,r+1}^{k,k}(z))_{ij}= & {} \left\{ \begin{array}{ll} (\nu _0+\nu _1 \; \sigma (k)) \; (z+\varDelta _{(r,k,i)})^{-1}, &{} {\mathrm{if}\,j=i},\\ \delta \; i \; (z+\varDelta _{(r,k,i)})^{-1}, &{} {\mathrm{if}\,j=i-1},\\ 0, &{} \mathrm{otherwise}, \end{array}\right. \end{aligned}$$

where $1\le i\le J(k;r)$, $1\le j\le J(k;r+1)$, $0\le k\le B-1$ and $0\le r\le n_R^{max}-k-1$; and

$$\begin{aligned} (\mathbf{B}_{r,r-1}^{k,k+1}(z))_{ij}= & {} \left\{ \begin{array}{ll} k_\mathrm{on}\; r \; n_L \; (z+\varDelta _{(r,k,i)})^{-1}, &{} {\mathrm{if}\,j=i},\\ 0, &{} \mathrm{otherwise}, \end{array}\right. \end{aligned}$$

where $1\le i\le J(k;r)$, $1\le j\le J(k{+}1;r{-}1)$, $0\le k\le B-2$ and $1\le r\le n_R^{max}-k$. Finally, the expression for the vector $\mathbf{b}(z)$ in Eq. (7) is given by

$$\begin{aligned} \mathbf{b}(z)= & {} \left( \begin{array}{c} \mathbf{0}\\ \mathbf{0}\\ \vdots \\ \mathbf{0}\\ \mathbf{A}_{B-1,B}(z) \; \mathbf{e}_{J(B)} \end{array}\right) , \end{aligned}$$

where $\mathbf{e}_j$ represents a column vector of ones with dimension j. Then, following a forward-elimination backward-substitution method suggested by Ciarlet [32, p. 144], Algorithm 1 is obtained. This Algorithm allows us to compute all the Laplace–Stieltjes transforms in Eq. (2) in an efficient and recursive manner.

Algorithm 1 [to obtain the Laplace–Stieltjes transforms ${\varphi }^B_{(n_1,n_2,n_3)}(z)$]

Finally, the order moments, $m_{(n_1,n_2,n_3)}^{B,(l)}$, of the random variable $T^B_{(n_1,n_2,n_3)}$ can be obtained by means of a matrix formalism similar to that described for Eq. (3). In particular, the system given by Eq. (3) can be expressed in matrix form as

$$\begin{aligned} \mathbf{m}^{(l)}= & {} \mathbf{A}(0) \; \mathbf{m}^{(l)}+{\tilde{\mathbf{b}}}^{l}, \end{aligned}$$

with

$$\begin{aligned} ({\tilde{\mathbf{b}}}^{l})_i= & {} l \; \frac{1}{\varDelta _i} \; (\mathbf{m}^{(l-1)})_i,\quad 0\le i\le \sum \limits _{k=0}^{B-1} \; \#{\mathscr {S}}(k), \end{aligned}$$

where $\varDelta _i$ represents the value $\varDelta _{(n_1,n_2,n_3)}$ for the state $(n_1,n_2,n_3)$ corresponding to row i. The vector ${\tilde{\mathbf{b}}}^l$ can be structured by blocks as follows

$$\begin{aligned} {\tilde{\mathbf{b}}}^l= & {} \left( \begin{array}{c} {\tilde{\mathbf{b}}}^l_0\\ {\tilde{\mathbf{b}}}^l_1\\ \vdots \\ {\tilde{\mathbf{b}}}^l_{B-2}\\ {\tilde{\mathbf{b}}}^l_{B-1} \end{array}\right) . \end{aligned}$$

Similar arguments to those used to derive Algorithm 1 lead to Algorithm 1 (continuation), which allows us to compute the moments in vector $\mathbf{m}^{(p)}$ from those previously computed in vector $\mathbf{m}^{(p-1)}$, starting at $\mathbf{m}^{(0)}={\varvec{\varphi }}(0)$ and until the desired order, $p=l$, is reached.

Algorithm 1 (Continuation) [to obtain the l-th order moments $m_{(n_1,n_2,n_3)}^{B,(l)}$]

We now turn to the second descriptor analysed in Sect. 3.2. Equation (6) can be expressed in matrix form as

$$\begin{aligned} {\varvec{\phi }}(s) \ = \ \mathbf{\overline{A}}(s) \; {\varvec{\phi }}(s)+\mathbf{\overline{b}}, \end{aligned}$$

(8)

where we are omitting again B in the notation, and where the probability generating functions $\phi _{(n_1,n_2,n_3)}^B(s)$ for $(n_1,n_2,n_3)\in \cup _{k=0}^{B-1}{\mathscr {S}}(k)$ are stored in a column vector ${\varvec{\phi }}(s)$, which is organised in sub-vectors following the structure of levels and sub-levels of ${\mathscr {S}}$. This follows similar arguments to those used for the vector ${\varvec{\varphi }}(z)$. A direct comparison between Eqs. (2) and (6) allows us to write $\mathbf{\overline{A}}(s)=\mathbf{A}(z=0)$, except for sub-blocks $\mathbf{B}_{r,r+1}^{k,k}(z)$, which should be replaced by $\mathbf{\overline{B}}_{r,r+1}^{k,k}(s)$ given by

$$\begin{aligned} (\mathbf{\overline{B}}_{r,r+1}^{k,k}(s))_{ij}= & {} \left\{ \begin{array}{ll} (\nu _0 + \nu _1 \; \sigma (k)) \; s \; \varDelta _{(r,k,i)}^{-1}, &{} {\mathrm{if}\,j=i},\\ \delta \; i \; \varDelta _{(r,k,i)}^{-1}, &{} {\mathrm{if}\,j=i-1},\\ 0, &{} \mathrm{otherwise}, \end{array}\right. \end{aligned}$$

for $1\le i\le J(k;r)$, $1\le j\le J(k;r+1)$, $0\le k\le B-1$ and $0\le r\le n_R^{max}-k-1$. Finally, the vector $\mathbf{\overline{b}}=\mathbf{b}(z=0)$ and Algorithm 1 leads to the computation of the vector ${\varvec{\phi }}(s)$ from Eq. (8).

The factorial moments $n_{(n_1,n_2,n_3)}^{B,(p)}$ and the probabilities $\alpha _{(n_1,n_2,n_3)}^B(a)$ of the random variable $N^B_{(n_1,n_2,n_3)}$ can be computed following similar arguments to those provided above, and are not included here.