Fastest Time to Cancer by Loss of Tumor Suppressor Genes

Abstract

Genetic instability promotes cancer progression (by increasing the probability of cancerous mutations) as well as hinders it (by imposing a higher cell death rate for cells susceptible to cancerous mutation). With the loss of tumor suppressor gene function known to be responsible for a high percentage of breast and colorectal cancer (and a good fraction of lung and other types as well), it is important to understand how genetic instability can be orchestrated toward carcinogenesis. In this context, this paper gives a complete characterization of the optimal (time-varying) cell mutation rate for the fastest time to a target cancerous cell population through the loss of both copies of a tumor suppressor gene (TSG). Similar to the (1-step) oncogene activation model previously analyzed, the optimal mutation rate of the present 2-step model changes qualitatively with the convexity of the (mutation rate dependent) cell death rate. However, the structure of the Hamiltonian for the new model differs significantly and intrinsically from that of the 1-step model and a completely new approach is needed for the solution of the present 2-step problem. Considerable insight on the biology of optimal switching (between corner controls) is extracted from numerical results for cases with nonconvex death rates.

Optimal Proliferation and Differentiation of Chlamydia Trachomatis

“…In addition to being ubiquitous in engineering science, it has also been found to occur in models for the life sciences. These include models on cancers chemotherapy, intestinal crypt development, genetic instability, and other linear constrained optimal control models (see [10], [12], [9], [14], and [16] and references therein). For a frequently encountered performance objective such as the time optimal problem (TOP), the solution process leading to a bang-bang control can be found in most texts on optimal control such as [4], [12], and [15].…”

“…Evidently, u op (t) = u max as long as λ g (t) < 1; hence, t s is the instant nearest to T for which λ g (t) < 1 does not hold. It follows that the switch condition λ g (t s ) = 1 (14) determines t s . We have then the following partial result for the optimal control u op (t).…”

“…PROPOSITION 1. The optimal control u op (t) that maximizes H [u] must consist of the upper corner control u max for an interval (t s , T ] adjacent to the terminal time with t s determined uniquely by (14) to be…”

Evolution of genetic instability in heterogeneous tumors

“…All these patterns of instability are realistic; the time-dependent stable – to unstable – back to stable pattern is especially interesting, given biological evidence supporting it (see the Discussion section for more details). The ideas presented in [29] were further developed mathematically and given analytical rigor in [55, 47]. …”

“…In the approach of [30, 29, 47], the instability-induced mutation rate, p , was treated as a function of time, and the optimization procedure sought the function p ( t ) that maximized tumor growth. The colony of cells was considered homogeneous with respect to the time-dependent mutation rate, p ( t ), that is, all cells were assumed to have the same mutation rate.…”