A conceptual cellular interaction model of left ventricular remodelling post-MI: dynamic network with exit-entry competition strategy

Abstract

Background: Progressive remodelling of the left ventricle (LV) following myocardial infarction (MI) is an outcome of spatial-temporal cellular interactions among different cell types that leads to heart failure for a significant number of patients. Cellular populations demonstrate temporal profiles of flux post-MI. However, little is known about the relationship between cell populations and the interaction strength among cells post-MI. The objective of this study was to establish a conceptual cellular interaction model based on a recently established graph network to describe the interaction between two types of cells.

Results: We performed stability analysis to investigate the effects of the interaction strengths, the initial status, and the number of links between cells on the cellular population in the dynamic network. Our analysis generated a set of conditions on interaction strength, structure of the network, and initial status of the network to predict the evolutionary profiles of the network. Computer simulations of our conceptual model verified our analysis.

Conclusions: Our study introduces a dynamic network to model cellular interactions between two different cell types which can be used to model the cellular population changes post-MI. The results on stability analysis can be used as a tool to predict the responses of particular cell populations.

Figure 1

The structure of the dynamic network in the exit-entry updating process During the exit-entry updating process, a vacated vertex is replaced by a new macrophage according to the fitness function determined by its neighbouring cells. An original network is shown in the left part of Figure 1 (Left). If a cell, marked in gray, exits the network, the possibility of replacing this cell with either a type C (classically activated macrophage) or type A (alternatively activated macrophage) cell is determined by the fitness function of the neighbouring cells. Cost function of each cell linked to the vacated cell is shown in the middle of the figure (Middle). In this case, the fitness function of all type C cells is calculated as F_A = 4(1 − ω)+ ω(10a + 2b). Similarly, the fitness function determined by the connected type C cells can be calculated as F_C = 4(1 − ω)+ ω(3c + 5d). The gray vertex may be replaced by a type A cell with probability , or a type C cell with probability of according to the exit-entry evolutionary strategy, which is shown in the right part of Figure 1 (Right).

Figure 2

Effects of interaction strengths on the stability of the dynamic network with different interaction matrix (P_A → 1) Figure 2 demonstrates the evolutionary dynamics of a network with interaction matrix [1 0; 0 1], k=4, and initial populations of type A and type C cells are set as 9900 and 100 in a total population of 10000 cells, based on previously published experimental results. The intensity of selection ω equals to 0.01. In the left part of Figure 2 (Left), X axis represents the variable P_A, the ratio of type A cellular population over the total population. The Y axis represents the evolutionary rate of type A cellular population denoted as Ṗ_A. The red star is the initial status of P_A. At the initial status, Ṗ_A is positive, making P_A increase until P_A reaches 1 where Ṗ_A decreases to 0. In the right part of Figure 2 (Right), the simulation results demonstrate that population of type A cells increases until it dominates the whole population within 300 generations.

Figure 3

Effects of interaction strengths on the stability of the dynamic network with different interaction matrix (P_A → 1) Figure 3 demonstrates the evolutionary dynamics with interaction matrix as [0 -0.5; 1 0], k=4, and initial populations of type A and C cells are set as 9900 and 100 in a total population of 10000 cells. The intensity of selection ω equals to 0.01. As shown in the left part of figure (Left), Ṗ_A is negative in the region of P_A ∈ (0, 1). It means the population of type A cells decreases in the interval of P_A ∈ (0, 1) until all type A cells exit the system, and Ṗ_A = 0. In the right part of the figure (Right), the simulation results demonstrate that population of type A cells decreases until all type A cells exit the system within 600 generations.

Figure 4

Effects of interaction strength on the stability of the dynamic network with different interaction matrix (P_A ∈ (0, 1)) Figure 4 shows the evolutionary dynamics with interaction matrix as [0 1; 1 0], k=4, and initial populations of type A and C cells are set as 9900 and 100 in a total population of 10000, respectively. The intensity of selection ω equals to 0.01. Variable Ṗ_A is negative when the initial status of P_A stays in the region between the marked root (red square) and 1 in the left part of the figure(Left). When P_A goes to the marked root, Ṗ_A reaches 0. Accordingly, the population of type A cells decreases in the interval of [root, 1] until P_A goes to the root denoted at 0.5 in the simulation. The simulation results shown in the right part of the figure (Right) demonstrate that the population of type A cells approached 5000 within 1000 generations.

Figure 5

Effects of initial status on the stability of the dynamic network with different interaction matrix (P_A ∈ (0, 1)) Figure 5 shows the evolutionary dynamics with interaction matrix as [0 1; 1 0], k=4, and initial populations of A and B are as 4000 and 6000 in a total population of 10000 cells. The intensity of selection equals to 0.01. Ṗ_A is positive when the initial status of P_A stays between 0 and the marked root (red square) in the left part of figure (Left). Accordingly, the population of type A cells increases from the initial status in the interval of [0, root] until P_A goes to the root denoted at 0.5 in this simulation. The simulation results shown in the right part of Figure 5 (Right) demonstrate that the population of type A cells approaches 5000 within 800 generations.

Figure 6

Effects of initial status on the stability of the dynamic network Figure 6 shows the evolutionary dynamics with interaction matrix as [0 -1; -2 0], k=4, and initial populations of A and B are set as 9900 and100, respectively, in a total population of 10000. The intensity of selection ω equals to 0.01. As shown in the left part of Figure 6, in the case that initial status of P_A stays between the root and 1, Ṗ_A is positive. Therefore, P_A increases from the initial status until it goes to 1 and Ṗ_A goes to 0. Simulation results demonstrate that population of type A cells reach 10000 within 300 generations as shown the right part of Figure 6 (Right).

Figure 7

Effects of initial status on the stability of the dynamic network Figure 7 shows the evolutionary dynamics with interaction matrix as [0 -1; -2 0], k=4, and initial populations of A and B are set as 2000 and 8000, respectively, in a total population of 10000. The intensity of selection ω equals to 0.01. As shown in the left part of Figure 7, in the case that initial status of P_A stays between 0 and the root (red square), Ṗ_A is negative. Thus, P_A decreases from the initial status until it goes to 0 and Ṗ_A goes to 0. Simulation results demonstrate the type A cells totally exit the system within 400 generations as shown the right part of Figure 7 (Right).

Figure 8

Effects of number of links on the stability of the dynamic network Figure 8 shows evolutionary profiles of two dynamic networks with exit-entry interaction matrix [0 b; 1 0] (Left) and [1 0; 0 d] (Right) were simulated with different number of links (k=4, Green stars, k=5 Yellow square, and k=6 Red circles. Population of initial type C cells was set as 100 with a total population of 10,000 cells in the system. The intensity of selection ω was set as 0.01. The markers are the average populations of type C cells after five generations over 10⁴ runs. In the left part of the figure (Left), the horizontal coordinate represents the changes of b value and the vertical coordinate represents the population of type C cells. In the right figure, the horizontal coordinate represents the changes of d value and the vertical coordinate represents the population of type C cells.