At equilibrium, with no inflammation signal, the amount of G is constant, and from Eq. From Eq.
High-resolution 3D imaging and topological mapping of the lymph node conduit system
The simulations reported in this paper have been carried out with free parameter values as listed in Table 1. Typically a simple time course is specified, with inflammation held at a fixed value for some period, then tapering to zero. We developed a general approach to incorporating a chemotactic influence into the probabilistic simulation of motion on a lattice. The obvious first step for simulating chemotaxis, then, is to simulate the chemokine concentration field. While the approach described below is directly applicable to situations where chemokine concentrations are simulated, including multiple chemokines, it is also amenable to a convenient simplification in the case of egress via discrete exit portals, employing the mechanism of control of cell egress by the gradient of some chemokine close to an exit portal  , .
In this case, instead of solving for chemokine concentrations it is reasonable to use a simplifying assumption to approximate the chemotactic influence, as a function of distance from the source of the chemokine — in the case simulated, the exit portal.
Consider a T cell that is experiencing a chemotactic attraction represented by C , where C is a 3-vector. In the absence of chemotaxis the jump probabilities associated with the N J possible jumps to neighbour sites e. For each possible jump direction, the probability is made proportional to the square of the cosine of the angle between the jump vector and the unit vector v , scaled by the inverse of the jump distance, and setting to zero the probabilities corresponding to jumps directed counter to the attracting influence, i.
Dividing by the jump distance is necessary because there are three possible values. In the example illustrated in Fig.
Table 2 shows the steps in computing the chemotaxis-only jump probabilities p c. The relative amounts of chemotaxis and normal motility depend on C , the strength of the chemotactic influence, which takes into account the susceptibility of the T cell to chemotaxis. This susceptibility may in general vary between cells, and between states of differentiation of a single cell. In the simplified 2D case there are eight possible jump directions. The assumption that the model uses to treat attraction to exit portals is to make the influence of an exit depend simply on the distance of the cell from the exit, r , through some function g r with a maximum value of 1.
A simple choice for g r , to convey how the chemotactic effect falls off with distance from the exit, is a bounded inverse square relation:.
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The fractional influence of chemotaxis is given by. When a cell is subject to more than one chemotactic influence, e. The net chemotactic attraction vector can be defined to be the vector sum of the M influences,. In the case of a cell that is within the range of attraction of more than one exit portal the combined effect of the multiple influences can be computed in the same way. The strength of each individual chemotactic attraction is given by , and the net chemotactic influence vector is. Transit times of individual cells can vary significantly about this mean.
The probability distribution of transit time is computed by simulating a steady-state condition, then recording both the entry and exit times of a large number of cells as they traffic through the blob. The transit time distribution has been determined for the cases in which all cells experience the same influence of exit chemotaxis, and for some cases in which a variable fraction of cells are subject to chemotaxis.
For a given value of egress probability P E the total rate of cell egress is expected to vary in proportion to the number of exit portals. Initial simulations were carried out with P E arbitrarily set to 0.
Through repeated simulation runs the value of N E required to maintain steady state was determined by iterative adjustment, for initial T cell population N ranging from 50 k to 1. A nearly linear power law was then found to provide a good fit for the relationship between N and N E Fig. Simulations were conducted with T cell population N ranging from 51 k to k and influx rates corresponding to a residence time of 12 h, to determine in each case the number of exit portals N E required for maintenance of steady state.
The product number of exit portals x residence time was plotted against N , and the best-fit power function determined. Further simulations confirmed that the number of exits predicted by this function also achieve steady-state system behaviour for cases with a residence time of 24 h. This formulation satisfies requirements 1 and 2 specified above. For any choice of T res consistent with observations i. Note that the egress probability P E of 0.
JCI - Transcytosis route mediates rapid delivery of intact antibodies to draining lymph nodes
It is to be expected that P E and N E will be in an inverse relationship to each other, since the total number of exit neighbour sites is proportional to N E , while the rate of egress is proportional to the product of P E and the number of exit sites. This led to the more general expression for the number of exits: 4. Hence extensive simulations showed that the functional relationship between the number of exits N E and the T cell population N was found to be almost linear. Effectively, the strategy of introducing a fixed number of exit portals, randomly distributed throughout the designated exit zone Fig.
Should exit portals change their permissiveness to T cell egress, the number of exits needed to support steady state flux falls inversely. The results reported above are for steady-state conditions, in which the influx rate is fixed and the T cell population remains approximately constant. In the case of an immune response, inflammation signals cause the T cell population to grow, and when the inflammation dies away the population falls. As the cell population changes the model automatically adjusts the number of exit portals, ensuring that Eq.
The locations of the exit portals are also adjusted automatically as the blob expands and contracts, in a way that keeps them near the boundary such that the 27 sites of the Moore neighbourhood of an exit portal are all within the blob, and at least one of these sites has a neighbour site that is outside and also maintains their separation. To focus on the effect of inflammation and check that the growth factor model performs correctly in conjunction with the way the number of exit portals is determined, simulations were run with specified time variation in the inflammation signal, but no corresponding antigen influx and no DCs.
The resulting variation in the T cell population over the day course of the simulation is therefore solely the result of trafficking changes. For these runs the initial population was k, the equilibrium residence time was either 12 or 24 hours, and the parameters of the trafficking sub-model were given the values listed in Table 1. In each case the inflammation signal A t was held at a constant level for 3. The five cases plotted in Fig. In each 10 day simulation run, with a starting population of k T cells, the external inflammation signal was held at fixed level 0.
The T cell population of the blob rises initially while inflammation-driven influx exceeds efflux, then falls back towards the initial level as steady-state balance of inflow and outflow is restored. These simulation results show that the trafficking model i. Although the chemotaxis sub-model is generally applicable to chemotactic influences within the LN, for convenience we simulated the effects of a chemotactic influence emanating from the exit portals.
This allows tracking of the effects of chemotaxis by tracking its effect on egress from the portals, which can be quantified through the residence time. We are not expressing acceptance of the reality of chemotactic egress, which is still undecided, but we do believe that the methods we have developed are appropriate for simulating this phenomenon. To provide a visual impression of how chemotaxis influences cell motion, a series of Supplementary Videos have been created and made available online as supporting material.
In each case a simplified scenario was simulated to make interpretation easier, using the following procedure. Steady-state cell trafficking was simulated in a spherical blob with a radius of There was a single exit portal located at the blob centre. In a simulation run, a specified number of cells that are initially located less than a specified distance D from the exit portal are randomly selected, made subject to a specified level of chemotaxis, and tagged.
As the simulation proceeds the locations of the tagged cells are recorded. The videos display the motion of these tagged cells for four hours. As the chemotaxis influence increases, the tagged cells have a greater tendency to move towards the exit portal, so the number of tagged cells remaining in the blob declines more rapidly. As the videos show, under this probabilistic method of simulating chemotaxis, the cells under chemotactic influence still maintain their random walk motility, so they have a tendency to move towards the exit portal, without moving directly up the chemotactic gradient.
Importantly, this chemotactic tendency is occurring inside a swarm of other untagged cells, which are continuing their random walk motility without any chemotactic responsiveness. Hence the method allows chemotaxis to operate on sub-populations of cells in crowded environments that are swarming with cells undergoing random walk motility. To explore the influence of chemotaxis in a simplified setting, simulations of steady-state trafficking were carried out for a blob of 10 k cells with a single exit portal at the centre.
The trajectories of these chemotactic cells were recorded over six hours of simulation. For five simulation runs with a range of values of the chemotaxis parameter K E , the number of tagged chemotactic cells remaining in the blob has been plotted as a function of time. To further test the methodology, we simulated chemotactic influences within larger blobs with a full complement of exit portals. Two scenarios were simulated: in the first, all T cells were equally susceptible to chemotaxis; in the second, only a subset of T cells was subject to chemotaxis.
This latter scenario is of interest because of abundant data showing that the expression of chemotactic receptors by lymphocytes changes dynamically during an immune response, so that at any stage of an immune response, different lymphocyte populations may differ substantially in their responses to a particular chemotactic influence. Under these circumstances, to maintain steady state, the number of exit portals must be reduced. The results, shown in Fig. For K E greater than about 4.
This expression provides the number of exit portals required in the model to maintain steady state, given the number of T cells, N , the residence time, T res , the egress probability, P E , and the chemotactic susceptibility, K E. Effectively these simulations show that chemotaxis acting on all cells is subject to rapid saturation, as spatial constraints reduce the ability of new cells to access the source of the chemotactic factors. Under the second scenario, only a subset of cells has enhanced chemotactic susceptibility, and the results are more interesting.
In this case the more chemotactic cells might be expected to gain preferential access to the exit portals, and as a result leave the paracortex more rapidly. Hence we simulated this scenario, and measured the dependence of residence time on the chemotactic sensitivity K E of the susceptible subset. Probability distributions of transit times with varying levels of chemotaxis have been computed by keeping track of T cell entry and exit times.
Transit times were recorded for these cells on exiting the paracortex. The transit time distribution for the base case, with no cells subject to exit chemotaxis is shown in Fig. Residence times for these cases are plotted in Fig. In these runs the probability of egress of a cell that is in the neighbourhood of an exit portal was fixed at 0. Note that in these simulations no time restriction was placed on the ability of cells to exit immediately after ingress. The tagged cells were tracked and their transit times determined.
A Dependence of residence time mean transit time on K E. For the purposes of comparison with experimentally observed chemotaxis, it is useful to have a measure of the chemotaxis generated numerically by this method. This measure is typically used in conjunction with an assay e.