Biophysics of the immune response
To formulate the mathematical model, we consider a part of the lymph node, i.e., the T cell zone, which contains various cell types, mainly the antigen presenting cells (APCs) and subsets of T lymphocytes. Naive T cells and some APCs (such as plasmocytoid Dendritic Cells, pDCs) enter the node with blood flow via the High Endothelial Venules (HEVs) whereas effector and/or memory T cells, and mainly DCs and macrophages home to lymph nodes via afferent lymphatic vessels [18, 19]. Following activation with pathogens, APCs acquire a motile state that allows their translocation to the T cell zone of draining lymph node with the afferent lymph flow [20, 21]. Therefore, we assume that the influx of APCs is proportional to the level of infection in the organism. Differentiation of naive T cells into CD4 + and CD8 + T cells occurs in the thymus from progenitor T cells [22]. We suppose that they enter lymph nodes already differentiated and that there is a given influx of each cell type.
The APCs bearing foreign antigens activate the clonal expansion of naive T lymphocytes. The activation of T cell division and death is regulated by a set of signals coming from the interactions of the antigen-specific T cell receptors (TCRs) with the MHC class I or class II presented peptides and IL-2 receptors binding IL-2. Naive T cells undergo asymmetric division [23] (Fig. 1). Some of the daughter cells continue to proliferate and differentiate. Mature CD4 + T cells produce IL-2 [22, 24, 25] which influences survival and differentiation of both CD4 + and CD8 + T cells. The proliferation of CD8 + T cells is stimulated by IL-2 [24]. They can expand their number many thousand-fold. In addition to IL-2 enhancing the proliferation of T cells, APCs start to secrete type I IFN which has an antiviral- and immunomodulatory function. In fact, the effect of IFN α depends on the temporal sequence of the signals obtained by naïve T cells [2]. It can change from a normal activation of T cells followed by their proliferation and differentiation to an already differentiated state followed by apoptosis as shown schematically in Fig. 2. Overall, the regulated death of T cells by apoptosis depends on the availability and the timing of TCR, IL-2 and IFN signalling.
Mature CD8 + T cells (effector cells) leave the lymph node and kill infected cells. Therefore, there is a negative feedback between production of mature CD8 + T cells and the influx of APCs.
In the model, an asymmetric T cell division is considered as shown in Fig. 3. Naive T cell entering the draining lymph node is recruited into the immune response after the contact interaction via the T cell receptor (TCR) with APC presenting the foreign antigen. The activation and prolonged contact with APC can results in polarity of the lymphocyte. The position of the contact with the APC determines the direction of cell division and the difference between the daughter cells in terms of their differentiation state. According to [23], the proximal daughter cell will undergo clonal proliferation and differentiation resulting in the generation of terminally differentiated effector cells (mature CD8 + T cells) that leave the lymph node for peripheral tissues to search and kill infected cells. The distal daughter cell becomes a memory cell. The memory cells are capable of self-renewal by slowly dividing symmetrically in the absence of recurrent infection.
Hybrid model of cell dynamics
In our model of cell dynamics, cells are considered as individual objects that can move, divide, differentiate and die. Their behavior is determined by the surrounding cells, by intracellular regulatory networks described by ordinary differential equations and by various substances in the extracellular matrix whose concentrations are described by partial differential equations. This approach was used to model hematopoiesis and blood diseases [11–17].
Cells and concentrations. Cells in the lymph node:
-
1.
n
APC
(x,t) - the density of APCs in T cell zone;
-
2.
n
CD4(x,t) - the density of CD4 + T cells in T cell zone (with different levels of maturity);
-
3.
n
CD8(x,t) - the density of CD8 + T cells in T cell zone (with different levels of maturity);
Extracellular variables:
-
4.
I
e
(x,t) - the concentration of IL-2 in T cell zone;
-
5.
C
e
(x,t) - the concentration of type I IFN in T cell zone;
Intracellular variables:
-
6.
I
i
(t) - the intracellular concentration of IL-2-induced signalling molecules in the ith cell;
-
7.
C
i
(t) - the intracellular concentration of type I IFN-induced signalling molecules in the ith cell;
The state variables at the level of the whole organism:
-
8.
N
ef
(t) - the total number of effector CD8 +T cells in the body;
-
9.
N
inf
(t) - the total number of infected cells in the body;
Cell displacement. In the model, cells are represented by individual elastic spheres. There are two mechanisms of motion of cells in the lymph node. First of all they move in a random way. This motion allows naive T cells to meet APCs which is necessary for their activation, division and differentiation. Second, each two cells, when they meet, they push each other due to a direct mechanical interaction. We consider this interaction as an elastic force acting on cells and influencing their motion. Let us describe it in more detail.
The cells divide and can increase their number which involves pushing each other leading to their displacement in the lymph node. We describe cells displacement by the following model. Let us denote the center of two cells by x
1 and x
2 and their radii by r
1 and r
2 respectively. Then, if the distance h
12 between the two cells is less than the sum of their radii (r
1+r
2), there will be a repulsive force f
12 between them. This force should depend on the difference between (r
1+r
2) and h
12. Let us consider the case of one cell interacting with different cells in the lymph node. The total force applied to this cell will be \(F_{i} = \sum _{j\ne i} f_{ij}\). We describe the motion of the particles as the motion of their centers which can be found by the applying Newton’s second law:
$$ m \ddot x_{i} + m \mu \dot x_{i} - \sum_{j \neq i} f_{ij} = 0, $$
(1)
where m is the mass of the particle, μ is the friction factor due to contact with the surrounding medium. The potential force between two cells is given explicitly by:
$$f_{ij} = \left\{ \begin{array}{ccc} K \frac{h_{0} - h_{ij}}{h_{ij} - (h_{0}-h_{1})} &,& h_{0}-h_{1} < h_{ij} < h_{0} \\ 0 &,& h_{ij} \geq h_{0} \end{array} \right., $$
where h
ij
is the distance between the centers of the two cells i and j, h
0 is the sum of their radii, K is a positive parameter and h
1 is the sum of the incompressible part of each cell. The force between the particles tends to infinity if h
ij
decreases to h
0−h
1.
Cell division and differentiation. APC and naive T cells enter the computational domain with a given frequency if there is available space. Naive T cells move in the computational domain randomly. If they contact APC, they divide asymmetrically (Fig. 3). The distant daughter cell is similar to the mother cell, the proximal daughter cell becomes differentiated.
When the cell reaches the half of its life cycle, it will increase its size. When it divides, two daughter cells appear, the direction of the axis connecting their centers is chosen randomly from 0 to 2π. The duration of the cell cycle is 18 hours with a random perturbation of −3 to 3 hours.
We consider two levels of maturity of CD4 + T cells and three levels of CD8 + T cells. If a differentiated cell has enough IL-2 (see the next paragraph), then it divides and gives two more mature cells. Finally differentiated cells leave the lymph node. In the simulations, this means that they are removed from the computational domain.
Intracellular regulation. The survival and differentiation of activated CD4 +- and CD8 + T lymphocytes depends on the amount of signalling via the IL-2 receptor and the type I IFN receptor. It is controlled primarily by the concentration of the above cytokines in the close proximity of the respective receptors. The signalling events lead to the up-regulation of the genes responsible for cell proliferation, differentiation and death. One can use similar type of equation to model qualitatively the accumulation of the respective intracellular signalling molecules linked to IL-2- and type I IFN receptors. The IL-2 dependent regulatory signal dynamics in individual cells can be described by the following equation:
$$ \frac{{dI}_{i}}{dt} = \frac{\alpha_{1}}{n_{T}} I_{e}({\mathbf{x_{i}}},t) - d_{1} I_{i}. $$
(2)
Here I
i
is the intracellular concentration of signalling molecules accumulated as a consequence of IL-2 signals transmitted through transmembrane receptor IL2R downstream the signaling pathway to control the gene expression in the ith cell. The concentrations inside two different cells are in general different from each other. The first term in the right-hand side of this equation shows the cumulative effect of IL-2 signalling. The extracellular concentration I
e
is taken at the coordinate x
i
of the center of the cell. The second term describes the degradation of IL-2-induced signalling molecules inside the cell. Furthermore, n
T
is the number of molecules internalized by T cell receptors.
In a similar way, the IFN-dependent regulatory signal dynamics in individual cells can be described by the following equation:
$$ \frac{{dC}_{i}}{dt} = \frac{\alpha_{2}}{n_{T}} C_{e}({\mathbf{x_{i}}},t) - d_{2} C_{i}. $$
(3)
Here C
i
is the intracellular concentration of signalling molecules accumulated as a consequence of IFN signals transmitted through transmembrane receptor IFNR downstream the signaling pathway to control the gene expression in the i-th cell. The concentrations inside two different cells are in general different from each other. The first term in the right-hand side of this equation shows the cumulative effect of IFN signalling. The extracellular concentration C
e
is taken at the coordinate x
i
of the center of the cell. The second term describes the degradation of IFN-induced signalling molecules inside the cell.
To model the fate regulation of growth versus differentiation of the activated cells in relation to the timing of the IL-2 and type I IFN signalling we implement the following decision mechanism.
-
C1
If the concentration of activation signals induced by type I IFN, C
i
, is greater than some critical level \(C_{i}^{*}\) at the beginning of the cell cycle and that of I
i
, is smaller than the critical level \(I_{i}^{*}\), then the cell will differentiate resulting in a mature cell.
-
C2
If the concentration of activation signals induced by IL-2, I
i
, is greater than some critical level \(I_{i}^{*}\) at the end of the cell cycle, then the cell will divide producing two more mature cells.
-
C3
If \(C_{i}<C_{i}^{*}\) at the beginning of cell cycle and \(I_{i}<I_{i}^{*}\) at the end of cell cycle, then the cell will die by apoptosis and will be removed from the computational domain.
Stochastic aspects of the model.
As it is discussed above, mechanical interaction of cells results in their displacement described by equation (1) for their centers. In order to describe random motion of cells we add random variables to the cell velocity in the horizontal and vertical directions.
Duration of cell cycle is given as a random variable in the interval [ T−τ,T+τ].
Extracellular dynamics of cytokines.
Proliferation and differentiation of T cells in the lymph node depends on the concentration of IL-2 and type I IFN. These cytokines are produced by mature CD4 + T cells and antigen-presenting cells, respectively. Their spatial distribution is described by a similar reaction-diffusion equation as follows
$$ \frac{\partial I_{e}}{\partial t} = D_{IL} \Delta I_{e} + W_{IL} - b_{1} I_{e}. $$
(4)
Here I
ex
is the extracellular concentration of IL-2, D is the diffusion coefficient, W
IL
is the rate of its production by CD4 + T cells, and the last term in the right-hand side of this equation describes its consumption and degradation. The production rate W
IL
is determined by mature CD4 + T cells. We consider each such cell as a source term with a constant production rate ρ
IL
at the area of the cell. Let us note that we do not take into account explicitly consumption of IL-2 by immature cells in order not to introduce an additional parameter. Implicitly this consumption is taken into account in the degradation term.
For type I IFN, the equation and the terms in it have a similar interpretation:
$$ \frac{\partial C_{e}}{\partial t} = D_{IFN} \Delta C_{e} + W_{IFN} - b_{2} C_{e}. $$
(5)
Initial and boundary conditions for both concentrations IL-2 and IFN are taken zero. As before, the production rate W
IFN
equals ρ
IFN
at the area filled by APC cells and zero otherwise.
Infection. Mature T cells leave the lymph node. The level of CD8 + T cells (effector cells) N
ef
in the body is determined by the equation
$$ \frac{d N_{ef}}{dt} = k_{1} T - k_{2} N_{ef}, $$
(6)
where T is their number in the lymph nodes. So the first term in the right-hand side of this equation describes production of effector cells in the lymph nodes and the second term their death in the body.
Denote by N
inf
the number virus-infected cells. We will describe it by the equation
$$ \frac{d N_{inf}}{dt} = f(N_{inf}) - k_{3} N_{ef} N_{inf}. $$
(7)
The first term in the right-hand side of this equation describes growth of the number of infected cells and the second term their elimination by the effector cells. The function f will be considered in the form:
$$ f(N_{inf}) = \frac{a N_{inf}}{1 + h N_{inf}} \;, $$
where a and h are some positive constants.
Finally, the influx of APC cells into the lymph nodes is proportional to the number of infected cells N
inf
.
This influx is limited by the place available in the lymph node. If there is a free place sufficient to put a cell, the new cells are added. Let us also note that the lymph nodes can increase due to infection in order to produce more effector cells.
