Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, USA

Department of Mechanical Engineering, University of Texas at San Antonio, San Antonio, USA

Center for Research in Biological Systems, University of California at San Diego, La Jolla, California 92093-0043, USA

Department of Medicine, University of Texas Health Science Center at San Antonio, San Antonio, USA

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.

Background

Progressive remodelling of the left ventricle (LV) following myocardial infarction (MI) involves spatial-temporal cellular interactions among different cell types

Macrophages are believed to first undergo classical activation, and then proceed through the alternative activation pathway

The evolution of a dynamic network has been carried out in game theory, social networks, and other biological systems

Results

We developed a dynamic network including two types of macrophages based on a previous graphic model published by Nowak and colleagues

Mathematical model of exit-entry updating law in a dynamic network

A total of N well-mixed cells are distributed over the network. Each cell occupies a vertex of the structured network and links to k other adjacent cells. A linkage between two cells is the edge of the network, denoting the interaction strength between cells. A general interaction matrix can be written as

where

where

The structure of the dynamic network in the exit-entry updating process

**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 _{A}_{C}

In this study, an exit-entry strategy was chosen for the conceptual model, since exit-entry is a fundamental cellular migration scheme for cellular interaction post-MI. In the exit-entry strategy, each iterative step in the exit-entry evolutionary process is called a generation. During the evolutionary process, a cell is chosen randomly to exit in each generation. Assuming a vacated vertex caused by cellular exiting will be only replaced with either a new type _{A}_{A}_{C}_{A}_{C}_{A}

For phenotypes with weak selection, the primary differential equations were set up as

where

We define _{A}_{C}_{AA}_{AC}_{CA}_{CC}_{X|Y}

Since _{AA}_{A|A}P_{A}_{A} and q_{A|A}

Theoretical analysis

In the case of weak selection, _{A}

Case 1: Stable equilibrium at _{A}

A stable equilibrium, _{A}

where _{A0} is the initial position of _{A}

Case 2: Stable equilibrium at _{A}

Case 3: Stable equilibrium of _{A}

Computational simulations

Based on the theoretical analysis, we predicted three types of evolutionary profiles: 1) population of type

Effects of the interaction strengths on the evolutionary profiles

We have run three sets of computer simulations with interaction matrix in the form of _{A}_{1} in Figure _{A}_{2} in Figure _{A}_{3} in Figure

Effects of interaction strengths on the stability of the dynamic network with different interaction matrix (_{A}

**Effects of interaction strengths on the stability of the dynamic network with different interaction matrix ( P** Figure

Effects of interaction strengths on the stability of the dynamic network with different interaction matrix (_{A}

_{A}_{A}_{A}_{A}_{A}

Effects of interaction strength on the stability of the dynamic network with different interaction matrix (_{A}

**Effects of interaction strength on the stability of the dynamic network with different interaction matrix ( P** Figure

Effects of initial status on the evolutionary profiles of a dynamic network

To investigate the effects of the initial status on the evolutionary profiles, we run three more computer simulations. While sharing the same interaction matrix, the number of links, initial status of _{A}

Effects of initial status on the stability of the dynamic network with different interaction matrix (_{A}

**Effects of initial status on the stability of the dynamic network with different interaction matrix ( P** Figure

In simulation pairs shown in Figures _{A}_{A}

Effects of initial status on the stability of the dynamic network

**Effects of initial status on the stability of the dynamic network** Figure _{A}_{A}_{A}_{A}

Effects of initial status on the stability of the dynamic network

**Effects of initial status on the stability of the dynamic network** Figure _{A}_{A}_{A}_{A}

Effects of the number of links on the evolutionary profiles of a dynamic network

We also designed computer simulations as shown in Figure

Effects of number of links on the stability of the dynamic network

**Effects of number of links on the stability of the dynamic network** Figure ^{4} 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

All the initial conditions, interaction strength, and the number of links listed in the simulations satisfied the condition associated with the specified equilibrium. The simulations verified predictions on the evolutionary profiles of the network based on our theoretical analysis.

Discussion

We have used a dynamic network model to study the cellular interactions with an exit-entry strategy. Our results demonstrated that evolutionary profiles of a dynamic network could be stabilized at different states by perturbing the interaction strength matrix, the number of links, and initial status of the network. We have quantified conditions for stable states in terms of interaction strengths, the initial status, and the number of links in the network. Our computer simulations verified predictions of our analytic results. While we used an exit-entry strategy presented by game theory

Here we have two remarks of our methods. First, we only considered an exit-entry strategy in a structured dynamic network. The exit-entry strategy was chosen because it was the most fundamental and logical cellular function for an initial investigation of the interactions between populations of classically and alternatively activated macrophages post-MI. There exist other evolutionary strategies such as entry-exit, mutation, and imitation. These strategies will need to be considered and potentially incorporated in future models. Secondly, the structure of the dynamic network is fixed by assuming a weak selection,

We provide here the first application of a dynamic network model to describe macrophage interactions. We have obtained explicit conditions that determine interaction strength and have established a structure of the network that allows us to predict the stability and equilibrium of the post-MI dynamic network. Our simulation results confirmed the prediction of the stability and the equilibriums of the network.

Conclusions

We used a new approach to model the cellular interactions between macrophage activation types in the post-MI setting. The results on stability analysis can be used as a useful tool to predict the responses of specific cellular populations.

Methods

Stability analysis of the exit-entry dynamic network

The established mathematical model in equation (2) and (5) is a high order nonlinear system. In a weak selection, _{A|A}

The conditional probabilities can then be rewritten as

In a weak selection, equations (2-3) can be simplified as

Stability of _{A}_{A}

Define the parameters _{0} = (^{2} – ^{2} − 1)

_{1} = (^{2} − 2^{2} –

and _{2} = −(_{A}_{A}

Stability of the system can be checked with 3 cases based on the position of the third root and sign of the parameter _{2}. The relations of the three equilibriums have been shown in Figures (

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Y.J and M.L.L designed the research; Y.J and Y.W performed computational analysis and simulation. J.Y.Y involved in the analysis and provided useful insights in the application to cellular functions. Y.J, Y.W, H.C.H, and M.L.L analyzed the results and wrote the manuscript.

Acknowledgements

The authors acknowledge grant support from NIH R01 HL75360, AHA Grant-in-Aid 0855119F, and the Morrison Fund (to MLL), grant support from NSF 0644646 and 0602834 (to HCH), and grant support from NSF 0649172 and NIH 1SC2 HL101430 (to YJ).

This article has been published as part of