Lessons and perspectives for applications of stochastic models in biological and cancer research

Abstract

The effects of randomness, an unavoidable feature of intracellular environments, are observed at higher hierarchical levels of living matter organization, such as cells, tissues, and organisms. Additionally, the many compounds interacting as a well-orchestrated network of reactions increase the difficulties of assessing these systems using only experiments. This limitation indicates that elucidation of the dynamics of biological systems is a complex task that will benefit from the establishment of principles to help describe, categorize, and predict the behavior of these systems. The theoretical machinery already available, or ones to be discovered to help solve biological problems, might play an important role in these processes. Here, we demonstrate the application of theoretical tools by discussing some biological problems that we have approached mathematically: fluctuations in gene expression and cell proliferation in the context of loss of contact inhibition. We discuss the methods that have been employed to provide the reader with a biologically motivated phenomenological perspective of the use of theoretical methods. Finally, we end this review with a discussion of new research perspectives motivated by our results.

Figure 1

(A) is a representation of a self-repressing gene, and (B) represents an externally regulated gene. The protein number is denoted by n, while its synthesis (degradation) rate is denoted by k(ρ). The ON to OFF and the OFF to ON state switching rates are indicated by h, and f, respectively.

Figure 2

(A) Fano factor versus average protein number for the self-repressing gene. The value of a is fixed as 500. The values of b^s are indicated within the graph, while we varied the value of z₀. (B) Spatial profile of mRNA average amounts along the AP axis of a D. melanogaster embryo. We also included the fluctuation in the positions of the borders of the peak expression along with the standard deviation of n at each nucleus along the AP axis. The positions of the borders are computed at the point where <n> is half of its maximal value at the position 41.5% of the embryo length.

Figure 3

(A) Representative configuration of the co-culture experiment at the confluence regime. (B) Evolution of the individual populations until confluence is reached and their fitting by a sigmoidal curve. (C) Experimental ratio of melanoma to normal cells over time. (D) Simulation of the ratio of melanoma to normal cells over time.

Figure 4

(A) Cartoon of our model for proliferation under different allelophylic degrees. (B) Spatial configuration achieved in our simulations at the co-culture confluence regime. (C) Experimental cell-to-cell distance distribution (the blue/red curve indicates the cancer/normal cells histogram). (D) Normal-to-normal (red) and melanoma-to-melanoma (blue) cell count in a 50 × 50 µm² region as they are further away from the interface with the melanoma clusters.