A stochastic differential model for cell migration and mutual cell-cell interactions

Motivated by two biological questions concerning the way radiation treatment affects cell behaviour and the way interactions between cells control cell segregation and cluster formation, we constructed a mathematical model for cell migration and interactions. Starting from first principles and basic biological assumptions, we arrived at a stochastic differential equation where the drift term accounts for short term interactions between cells and the random term accounts for independent cell motion. Likelihoods of different values for model parameters given particle paths were obtained using Girsanov theorem, and then used to estimate actual parameter values by maximising these likelihoods (MLE). Accuracy of this method was tested by comparing estimated parameter values given computer generated paths to the actual values used to generate these paths, for a few different drift functions. Application of this technique to a data set containing real cell paths which were observed in laboratory experiments studying the effect of radiation treatment on cell migration and interaction unveiled a clear trend in cells’ response: radiation dosage of 10Gy was found to increase cell motility by 50% and diminish cell adhesion effectively to zero. An extended version of our model which further accounts for cell births and interactions between different population types was designed to help understand cell segregation and cluster formation regulated by cell membrane proteins called Eph and ephrin. First this helped identify the significant components in controlling the behaviour and dynamics displayed by the biological system, which has countlessly more components over many time and distance scales. Second, when compared against experimental results it was able to replicate both the dynamics and range of cell segregation that was observed in the laboratory by our collaborators. We thus present here a powerful yet simple model which is both generic and versatile. With only a small number of parameters that can be estimated from data containing cell paths, it holds information regarding independent cell motion and mutual cell-cell interactions, and can reproduce, predict and help analyse dynamics and behaviours observed in laboratory experiments