Control of Intracellular Molecular Networks Using Algebraic Methods

Abstract

Many problems in biology and medicine have a control component. Often, the goal might be to modify intracellular networks, such as gene regulatory networks or signaling networks, in order for cells to achieve a certain phenotype, what happens in cancer. If the network is represented by a mathematical model for which mathematical control approaches are available, such as systems of ordinary differential equations, then this problem might be solved systematically. Such approaches are available for some other model types, such as Boolean networks, where structure-based approaches have been developed, as well as stable motif techniques. However, increasingly many published discrete models are mixed-state or multistate, that is, some or all variables have more than two states, and thus the development of control strategies for multistate networks is needed. This paper presents a control approach broadly applicable to general multistate models based on encoding them as polynomial dynamical systems over a finite algebraic state set, and using computational algebra for finding appropriate intervention strategies. To demonstrate the feasibility and applicability of this method, we apply it to a recently developed multistate intracellular model of E2F-mediated bladder cancerous growth and to a model linking intracellular iron metabolism and oncogenic pathways. The control strategies identified for these published models are novel in some cases and represent new hypotheses, or are supported by the literature in others as potential drug targets. Our Macaulay2 scripts to find control strategies are publicly available through GitHub at https://github.com/luissv7/multistatepdscontrol.

Appendix

In this Appendix, we denote finite fields with either ${\mathbb {F}}_q$ or ${\mathbb {F}}_p$, where p is assumed to be a prime number while q is assumed to be a power of a prime number.

1.1 Converting Mixed-State Models into Polynomial Dynamical Systems

Let q be the smallest number which is a power of a prime number such that $q\ge |X_i|$ for all i. Consider the finite field ${\mathbb {F}}={\mathbb {F}}_q.$

We can identify $X_i\hookrightarrow {\mathbb {F}}$ by an injective map $\iota _i$ for i from 1 to n. Let $\iota =(\iota _1,\ldots , \iota _n)$. We can now consider the dynamical system ${\mathbf {F}}$ as a subsystem of a dynamical system $\hat{{\mathbf {F}}}:{\mathbb {F}}^n\rightarrow {\mathbb {F}}^n$ as follows.

Define the map $\alpha _i:{\mathbb {F}}\rightarrow X_i$ as $\alpha _i(x)=\iota _i^{-1}(x)$ if $\iota _i^{-1}(x)\ne \emptyset$ and $\alpha _i(x)=c_i$ where $c_i\in X_i$ otherwise. Let $\alpha =(\alpha _1,\ldots , \alpha _n)$. Now, consider the map $\hat{{\mathbf {F}}}=\iota \circ {\mathbf {F}} \circ \alpha$.

$$\begin{aligned} {\mathbb {F}}^n\overset{\alpha }{\rightarrow }X\overset{{\mathbf {F}}}{\rightarrow }X \overset{\iota }{\rightarrow }{\mathbb {F}}^n. \end{aligned}$$

Notice that $\alpha _i$ essentially “crushes” the points in ${\mathbb {F}}-\iota _i(X_i)$ into a constant in $\iota _i(X_i)$.

Example 1

We use a slight abuse of notation for convenience: When we write an integer m here, we mean the representative in the particular finite field. Consider the sets $X_1=\{0,1,2,\ldots ,5\},X_2=\{0,\ldots ,4\}.$

We embed $X_1\overset{\iota _1}{\hookrightarrow }{\mathbb {F}}_7,X_2\overset{\iota _2}{\hookrightarrow }{\mathbb {F}}_7$ by inclusion. Define $\alpha _1, \alpha _2$ as follows.

$$\begin{aligned} \alpha _1(x)= & {} {\left\{ \begin{array}{ll} x &{} {\text { if }} x\in \{0,1,\ldots ,5\} \\ 5 &{} {\text {if }} x =6 \end{array}\right. }\\ \alpha _2(x)= & {} {\left\{ \begin{array}{ll} x &{} {\text { if }} x\in \{0,1,\ldots ,4\} \\ 4 &{} {\text {if }} x =5,6. \end{array}\right. } \end{aligned}$$

Here $\iota =(\iota _1, \iota _2)$, $\alpha =(\alpha _1, \alpha _2)$, and ${\mathbb {F}}_7^2\overset{\alpha }{\rightarrow }X\overset{{\mathbf {F}}}{\rightarrow }X \overset{\iota }{\rightarrow }{\mathbb {F}}_7^2.$

Veliz-Cuba et al. (2010) previously used a similar transformation for a finite field of prime order, ${\mathbb {F}}_p$, where the elements outside of ${\mathbb {F}}_p-\iota (X)$ were sent into the “largest” element $(p-1)$. However, in a general finite field ${\mathbb {F}}_q$, there is no adequate concept of the “largest element”. Notice that $\hat{{\mathbf {F}}}(x_1,\ldots ,x_n)=(x_1,\ldots ,x_n)$ if and only if $(x_1,\ldots ,x_n)$ is in the image of $\iota$ and $(\iota _1^{-1}(x_1),\ldots ,\iota _n^{-1}(x_n))$ is a fixed point of ${\mathbf {F}}.$ In particular, we can now “extend” the discrete dynamical system ${\mathbf {F}}$ to a discrete dynamical system $\hat{{\mathbf {F}}}:{\mathbb {F}}^n \rightarrow {\mathbb {F}}^n$ without changing the dynamics of the original system.

1.2 An Approach for Deriving a Polynomial Dynamical System from a Mixed-State Dynamical System

A common approach to representing mixed-state dynamical systems is to give Boolean expressions for when a certain node will attain a given value based on the state of the other nodes (Zañudo et al. 2017; Remy et al. 2015). For example, in the signaling network model presented in Remy et al. (2015), the rule for representing how E2F3 attains values 1 or 2 are shown in Table 4.

In the case that some of the variables are Boolean (can only take one of two values), and the other variables are in a set of the same prime cardinality q, we can convert to a polynomial dynamical system over ${\mathbb {F}}_q$. If a variable $x_i$ was Boolean to start with, we replace $x_i$ with $x_i^{q-1}$. For a variable, $x_i$ that was not Boolean, we can write the polynomial representation by taking advantage of indicators functions $q_j(x)=(\Pi _{i\in {\mathbb {F}}_q, i\ne j} (x-i))^{q-1}$ for $j\in {\mathbb {F}}_q.$ For example, if a variable appears in a Boolean expression as $x_i=j$, then we substitute that variable with $(\Pi _{i\in {\mathbb {F}}_q, i\ne j} (x-i))^{q-1}.$ Recall that the operator AND is equivalent to the product over ${\mathbb {F}}_2,$ the operator OR is equivalent to the operator $(x,y)\rightarrow x+y-(x+y)$ and NOT is equivalent to $x\rightarrow 1+x$. Over ${\mathbb {F}}_q$, we define x AND y to be $(x,y)\rightarrow (x\cdot y)^{q-1}$, NOT x to be $x\rightarrow 1-x^{q-1}$ and x OR y to be $(x,y)\rightarrow -(x\cdot y)^{q-1}+x^{q-1}+y^{q-1}.$

Example 2

Consider the update rule for the transcription factor E2F3 from Remy et al. (2015), which takes values in the set ${\mathbb {F}}_3,$ and whose value depends on the nodes RB1, $\text {CHECK1\_2}$, and RAS (Table 4). Here, the variables RB1 and RAS were Boolean variables, so we first substitute them with RB1$^2$ and RAS$^2.$ We then apply indicator functions for variables that were not Boolean. For example, $\text {CHECK1\_2}$=2 now becomes $q_2(\text {CHECK1\_2})$ where $q_2(x)=x+2\cdot x^2.$

The final polynomial equation can now be formed by adding the individual functions together, times their respective value (Table 5).

$$\begin{aligned} {\hbox {E2F3}}^{*}= & {} 1\cdot (1-{\hbox {RB1}}^{2}) \cdot (1- q_2({\hbox {CHEK1}}\_2))\cdot {\hbox {RAS}}^{2}\\&+\,2\cdot (1-{\hbox {RB1}}^2)\cdot q_2 ({\hbox {CHECK1}}\_2) \cdot {\hbox {RAS}}^{2} \end{aligned}$$

Table 4 Original equations for E2F3 from Remy et al. (2015)

Table 5 The results of applying the conversion rules to the rules in Table 4

1.3 Continuity Condition and Steady States

The continuity condition is a restriction that the state of each variable does not change by more than one unit at each time step (see, e.g., Chifman et al. 2017 for details). Intuitively, the continuity condition represents that a biological quantity cannot suddenly go from high to low (or low to high) without reaching an intermediate step. Here we show that the continuity condition on polynomial dynamical systems used in Chifman et al. (2017) does not change steady states.

Fix a prime p and consider the finite field $k={\mathbb {F}}_p$. Fix the notation

$$\begin{aligned} {\mathbf {x}}=(x_1,\ldots ,x_{i-1},x_i,x_{i+1},\ldots ,x_n), \end{aligned}$$

and let ${\mathbf {F}}_i:=f_i({\mathbf {x}})$. We will always assume that the representative for $x_i$ is in the set $\{0,1,\ldots ,p-1\}$.

We will say that $f:k[x_1,\ldots ,x_n]\rightarrow k^n$ is continuous if $|x_i-f_i({\mathbf {x}})|_{{\mathbb {R}}}\in \{0,1\} \text { for } 0\le x_i\le p-1, 1\le i \le n$. Let

$$\begin{aligned} h(x,y)={\left\{ \begin{array}{ll} x+1 &{} y>x \\ x &{} x=y \\ x-1 &{} y<x \end{array}\right. } \end{aligned}$$

Any PDS ${\mathbf {F}}:k^n\rightarrow k^n$ can be made continuous by considering $\hat{{\mathbf {F}}}:k^n\rightarrow k^n$ where $\hat{{\mathbf {F}}}_i=h\circ ({\mathbf {F}}_i\times \pi _i)$ where $\pi _i$ is the projection onto the ith coordinate.

Theorem 1

Let ${\mathbf {F}}:k^n\rightarrow k^n$ be a polynomial dynamical system over a finite field k and let $\hat{{\mathbf {F}}}:k^n\rightarrow k^n$ be the polynomial dynamical system where the continuity condition has been applied to ${\mathbf {F}}$. Then the set of fixed points of ${\mathbf {F}}$, FIX(${\mathbf {F}})$ is equal to FIX($\hat{{\mathbf {F}}})$

Proof

Let $x \in$FIX(${\mathbf {F}}$), $\pi _i:k^n\rightarrow k$ be the projection onto the ith coordinate.

Notice $\hat{{\mathbf {F}}}_i=h\circ ({\mathbf {F}}_i\times \pi _i)$. Then $\hat{{\mathbf {F}}}_i(x)=h\circ ({\mathbf {F}}_i\times \pi _i)(x)=h({\mathbf {F}}_i(x),x_i)=h(x_i,x_i)=x_i$.

Now, if $x\in \text {FIX}(\hat{{\mathbf {F}}})$, we have $h({\mathbf {F}}_i(x),x_i)=x_i$ for all i. This can only happen if $x_i={\mathbf {F}}_i(x)$ for all i.

As a result, we have FIX$({\mathbf {F}})=\,$FIX($\hat{{\mathbf {F}}})$.

$\square$