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 cancer 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. Similar to the (one-step) oncogene activation model previously analyzed, the optimal mutation rate of the present two-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 one-step model, and a completely new approach is needed for the solution of the present two-step problem. Considerable insight into the biology of optimal switching (between corner controls) is extracted from numerical results for cases with nonconvex death rates.

Appendix

1.1 Cell Populations for Upper Corner Control

With the two corner controls playing a key role in the optimal solution, we note that the state equations admit an exact solution for these controls. For the upper corner control $u_{1}(t)=1$, the state equations simplify to

$$\begin{aligned} \frac{\hbox {d}x_{0}^{(1)}}{\hbox {d}t}=-2\mu x_{0}^{(1)},\qquad \frac{\hbox {d}x_{1}^{(1)}}{\hbox {d}t}=2\mu x_{0}^{(1)}-\left( \mu +u_{m}\right) x_{1}^{(1)},\quad \frac{ \hbox {d}x_{2}^{(1)}}{\hbox {d}t}=\frac{u_{m}+\mu }{\sigma }x_{1}^{(1)}.\nonumber \\ \end{aligned}$$

(82)

The following exact solutions for these uncoupled first-order state equations are immediate.

Lemma 49

For $u(t)=u_{1}(t)\equiv 1$ and a general set of initial conditions $x_{k}^{(1)}(T_{i})=x_{k}^{(i)}\ge 0,\,k=0,1,2$, the exact solution of the three uncoupled first-order state Eq. (82) is

$$\begin{aligned} x_{0}^{(1)}&= x_{0}^{(i)}e^{-2\mu \tau },\qquad x_{1}^{(1)}=\frac{2\mu x_{0}^{(i)}}{u_{m}-\mu }\left[ e^{-2\mu \tau }-e^{-(\mu +u_{m})\tau } \right] +x_{1}^{(i)}e^{-(\mu +u_{m})\tau },\qquad \quad \end{aligned}$$

(83)

$$\begin{aligned} x_{2}^{(1)}&= x_{2}^{(i)}+\frac{x_{1}^{(i)}}{\sigma }\left\{ 1-e^{-(u_{m}+\mu )\tau }\right\} \\&\quad +\frac{x_{0}^{(i)}}{\sigma \left( u_{m}-\mu \right) }\left[ (\mu +u_{m})(1-e^{-2\mu \tau })-2\mu (1-e^{-(\mu +u_{m})\tau })\right] \nonumber \end{aligned}$$

(84)

with $\tau =t-T_{i}$ and a superscript “$(1)$” for upper corner control. The normalized mutated cell populations $x_{1}^{(1)}$ and $x_{2}^{(1)}$ are positive function of time for $\tau =t-T_{i}>0$.

Remark 50

The concavity of $x_{2}^{(1)}(t)$ also follows from

$$\begin{aligned} \frac{\hbox {d}^{2}x_{2}^{(1)}}{\hbox {d}t^{2}}=\frac{u_{m}+\mu }{\sigma }\frac{\hbox {d}x_{1}^{(1)}}{\hbox {d}t}=-\frac{\left( u_{m}+\mu \right) ^{2}}{\sigma }x_{1}^{(1)}<0. \end{aligned}$$

Remark 51

For the interval adjacent to the initial time $t=T_{i}=0$, the exact solutions (83) and (84) simplify by the known initial conditions to

$$\begin{aligned} x_{0}^{(1)}&= e^{-2\mu t},\quad x_{1}^{(1)}=\frac{2\mu }{u_{m}-\mu } \left[ e^{-2\mu t}-e^{-(\mu +u_{m})t}\right] , \end{aligned}$$

(85)

$$\begin{aligned} x_{2}^{(1)}&= \frac{1}{\sigma }\left[ 1-\frac{1}{\left( u_{m}-\mu \right) } \left\{ \left( u_{m}+\mu \right) e^{-2\mu t}-2\mu e^{-(\mu +u_{m})t}\right\} \right] . \end{aligned}$$

(86)

1.2 Cell Populations for Lower Corner Control

For the lower corner control $u_{0}(t)=0$, the first two state Eqs. (1) and (2) simplify to

$$\begin{aligned} \frac{\hbox {d}x_{0}^{(0)}}{\hbox {d}t}&= x_{0}^{(0)}\left( 1-2\mu -x_{0}^{(0)}-x_{1}^{(0)}\right) , \end{aligned}$$

(87)

$$\begin{aligned} \frac{\hbox {d}x_{1}^{(0)}}{\hbox {d}t}&= 2\mu x_{0}^{(0)}+x_{1}^{(0)}\left( 1-\mu -x_{0}^{(0)}-x_{1}^{(0)}\right) ,\qquad \end{aligned}$$

(88)

They may be solved exactly (by Mathematica or Maple) allowing for the satisfaction of two initial conditions

$$\begin{aligned} x_{0}^{(0)}(T_{\mathrm{s}})=x_{0}^{(s)},\ x_{1}^{(0)}(T_{\mathrm{s}})=x_{1}^{(s)}. \end{aligned}$$

(89)

Instead of writing down the exact solution, we need only the following simple bounds on $x_{0}^{(0)}$ and $x_{1}^{(0)}$ for our purposes:

$$\begin{aligned} -2\mu x_{0}^{(0)}&\le \frac{\hbox {d}x_{0}^{(0)}}{\hbox {d}t}\le x_{0}^{(0)}\left( 1-2\mu -x_{0}^{(0)}\right) ,\qquad \end{aligned}$$

(90)

$$\begin{aligned} 2\mu x_{0}^{(0)}-\mu x_{1}^{(0)}&\le \frac{\hbox {d}x_{1}^{(0)}}{\hbox {d}t}\le 2\mu x_{0}^{(0)}+x_{1}^{(0)}\left( 1-\mu -x_{1}^{(0)}\right) , \end{aligned}$$

(91)

given the nonnegativity of the three cell population and $0\le x_{0}^{(0)}+x_{1}^{(0)}\le 1$ by Lemma 4. It is possible to simplify the upper bound for $dx_{1}^{(0)}/dt$ further by replacing the right-hand side of (91) with $2\mu x_{0}^{(0)}+x_{1}^{(0)}\left( 1-\mu \right)$. We refrain from doing so to get sharper results.

Lemma 52

$$\begin{aligned} x_{0}^{(s)}e^{-2\mu \tau }&\le x_{0}^{(0)}\le 1-2\mu \end{aligned}$$

(92)

$$\begin{aligned} x_{1}^{(s)}+2x_{0}^{(s)}\left\{ e^{-\mu \tau }-e^{-2\mu \tau }\right\}&\le x_{1}^{(0)}\le x_{1}^{(s)} \end{aligned}$$

(93)

where

$$\begin{aligned} \tau =t-T_{\mathrm{s}},\ \ x_{k}^{(s)}=x_{k}^{(0)}(t=T_{\mathrm{s}})\ \end{aligned}$$

(94)

Proof

The various inequalities are straightforward consequences of the inequalities (90) and (91) along with the switch conditions (89): In particular, we have

$$\begin{aligned} x_{0}^{(0)}\!\le \! \frac{(1-2\mu )C_{0}e^{(1-2\mu )\tau }}{1+C_{0}e^{(1-2\mu )\tau }}\!\le \! 1-2\mu ,\quad x_{1}^{(0)}\!\le \! x_{1}^{(p)}\!-\!\frac{C_{1}\gamma e^{-\gamma \tau }}{1+C_{1}e^{-\gamma )\tau }}\!\le \! x_{1}^{(p)}\lesssim 1\!+\!\mu ,\nonumber \\ \end{aligned}$$

(95)

where

$$\begin{aligned} C_{0}\!=\!\frac{x_{0}^{(s)}}{1\!-\!2\mu \!-\!x_{0}^{(s)}}>0,\quad C_{1}\!=\!\frac{x_{1}^{(p)}-x_{1}^{(s)}}{x_{1}^{(s)}+x_{1}^{(p)}\!+\!\mu -1}>0,\quad \gamma =2x_{1}^{(p)}+\mu -1>0, \end{aligned}$$

and

$$\begin{aligned} 2x_{1}^{(p)}=(1-\mu )+\sqrt{1+6\mu -15\mu ^{2}}=2\left\{ 1+\mu +O(\mu ^{2})\right\} \lesssim 2(1+\mu ), \qquad \end{aligned}$$

(96)

keeping in mind $x_{1}^{(s)}<1<x_{1}^{(p)}$. $\square$

Remark 53

We note that the upper bound for $x_{1}^{(0)}$ is unrealistic since $x_{0}^{(0)}+x_{1}^{(0)}<1$ (by Lemma 1) and $x_{0}^{(0)}\ge 0$. However, $x_{1}^{(p)}\lesssim 1+\mu$ is only an (overly conservative) upper bound and does not contradict other more realistic results for cell populations (such as Lemma 1).

For $u=0$, the third state Eq. (3) takes the form

$$\begin{aligned} \frac{\hbox {d}x_{2}^{(0)}}{\hbox {d}t}=\frac{\mu }{\sigma }x_{1}^{(0)}+x_{2}^{(0)}(a-x_{0}^{(0)}-x_{1}^{(0)}). \end{aligned}$$

(97)

which is a linear first-order ODE for the only unknown $x_{2}^{(0)}$ and can be solved with the help of an integrating factor. Even without the explicit solution, we see from (97) that $x_{2}(t)$ increases without bound as $t \rightarrow \infty$ since $a\gg 1$.

More useful for our analysis is the following upper and lower bound for $x_{2}^{(0)}$:

Lemma 54

The following inequalities hold for $u=0$ and $x_{2}^{(0)}(t=T_{\mathrm{s}})=x_{2}^{(s)}$:

$$\begin{aligned} x_{2}^{(0)}(t)\ge x_{2}^{(s)}e^{(a-1)\tau },\quad \tau =t-T_{\mathrm{s}}. \end{aligned}$$

(98)

Proof

With $0\le x_{0}^{(0)}+x_{1}^{(0)}\le 1$ by Lemma 1, the ODE (97) implies

$$\begin{aligned} \frac{\hbox {d}x_{2}^{(0)}}{\hbox {d}t}\ge \frac{\mu }{\sigma }x_{1}^{(0)}+x_{2}^{(0)}(a-1)\ge x_{2}^{(0)}(a-1). \end{aligned}$$

from which we get

$$\begin{aligned} x_{2}^{(0)}(\tau )\ge x_{2}^{(s)}e^{(a-1)\tau }. \end{aligned}$$

$\square$