Mathematics of the Genome

Abstract

This work gives a mathematical foundation for bifurcation from a stable equilibrium in the genome. We construct idealized dynamics associated with the genome. For this dynamics, we investigate the two main bifurcations from a stable equilibrium. Finally, we give mathematical proofs of existence and points of bifurcation for the repressilator and the toggle gene circuits.

Appendices

Appendix 1

The existence of a pitchfork bifurcation in the two-gene case

When $\alpha _{1}=\alpha _{2}=2,$ and $m=n>0,$ the system in Eq.  (11) can be written as

$$\begin{aligned} \frac{{\hbox {d}}x}{{\hbox {d}}t}&=\frac{2}{1+y^{m}}-x\nonumber \\ \frac{{\hbox {d}}y}{{\hbox {d}}t}&=\frac{2}{1+x^{m}}-y, x,y\ge 0. \end{aligned}$$

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Proposition 2

For our system given by A1, for $0\le m\le 2,$ every equilibria must be $\left( 1,1\right) .$ This can be proved by numerics.

The Jacobian matrix J at (1, 1)

$$\begin{aligned} J=\left( \begin{array} [c]{cc} -1 &{} -m\frac{\alpha y^{m-1}}{\left( y^{m}+1\right) ^{2}}\\ -m\frac{\alpha x^{m-1}}{\left( x^{m}+1\right) ^{2}} &{} -1 \end{array} \right) =\left( \begin{array} [c]{cc} -1 &{} -\frac{1}{2}m\\ -\frac{1}{2}m &{} -1 \end{array} \right) \end{aligned}$$

and the $det(J)=1-\frac{1}{4}m^{2}.$ Therefore, $0\le m<2,$ $\ det(J)>0,$ and $m=2$, $\ det(J)=0;$ this is a condition for a pitchfork bifurcation.

Appendix 2

The existence of a Hopf bifurcation in the three-gene case

A simple case of the repressilator can be written as

$$\begin{aligned} \frac{{\hbox {d}}x}{{\hbox {d}}t}&=\frac{\alpha }{1+z^{m}}-x\nonumber \\ \frac{{\hbox {d}}y}{{\hbox {d}}t}&=\frac{\alpha }{1+x^{m}}-y\nonumber \\ \frac{{\hbox {d}}z}{{\hbox {d}}t}&=\frac{\alpha }{1+y^{m}}-z,\text { }x,y,z\ge 0. \end{aligned}$$

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The diagonal $x=y=z=s,$ is parameterized by s. Then, the coordinates of an equilibrium state are (s, s, s),  and the equilibrium path is described by

$$\begin{aligned} s+s^{m+1}-\alpha =0. \end{aligned}$$

(25)

The Jacobian matrix J at (s, s, s) is

$$\begin{aligned} J=\left( \begin{array} [c]{ccc} -1 &{} 0 &{} -\frac{ms^{m}}{\left( s^{m}+1\right) }\\ -\frac{ms^{m}}{\left( s^{m}+1\right) } &{} -1 &{} 0\\ 0 &{} -\frac{ms^{m}}{\left( s^{m}+1\right) } &{} -1 \end{array} \right) . \end{aligned}$$

The real part of the complex eigenvalues is, Re$(\lambda )=$ $-\frac{1}{2}\left( \frac{-ms^{m}}{\left( s^{m}+1\right) }\right) -1.$ The condition for a Hopf bifurcation is thus

$$\begin{aligned} -\frac{1}{2}\left( \frac{-ms^{m}}{\left( s^{m}+1\right) }\right) -1=0. \end{aligned}$$

This further simplifies to $s^{m}(2-m)=-2$ and for any $m>2,$ the following equation determines a bifurcation value for s.

$$\begin{aligned} s=\root m \of {\frac{2}{m-2}} \end{aligned}$$

Therefore, this satisfies the conditions for a generic Hopf bifurcation. See [25] for a related treatment.

Appendix 3

The existence of a pitchfork bifurcation in the three-gene case

As an example in Case 2 (Sect. 5), for the system described in Eq. (12), we take

$$\begin{aligned} \frac{{\hbox {d}}x}{{\hbox {d}}t}&=\frac{2}{1+z^{m}}-x\nonumber \\ \frac{{\hbox {d}}y}{{\hbox {d}}t}&=\frac{2x^{m}}{1+x^{m}}-y\nonumber \\ \frac{{\hbox {d}}z}{{\hbox {d}}t}&=\frac{2}{1+y^{m}}-z. \end{aligned}$$

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The equilibria of the system C1 are

$$\begin{aligned} x=\frac{2}{1+z^{m}},\text { }y=\frac{2x^{m}}{1+x^{m}},\text { and }z=\frac{2}{1+y^{m}} \end{aligned}$$

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Proposition 3

For our system given by C1, $0\le m\le 2,$ all equilibria $\left( x,y,z\right)$ must be $\left( 1,1,1\right) .$ This can be proved by numerics ( Stephen Lindsly).

The Jacobian matrix is

$$\begin{aligned} J=\left( \begin{array} [c]{ccc} -1 &{} 0 &{} -2m\frac{z^{m-1}}{\left( z^{m}+1\right) ^{2}}\\ 2m\frac{x^{m-1}}{\left( x^{m}+1\right) ^{2}} &{} -1 &{} 0\\ 0 &{} -2m\frac{y^{m-1}}{\left( y^{m}+1\right) ^{2}} &{} -1 \end{array} \right) =\left( \begin{array} [c]{ccc} -1 &{} 0 &{} -\frac{1}{2}m\\ \frac{1}{2}m &{} -1 &{} 0\\ 0 &{} -\frac{1}{2}m &{} -1 \end{array} \right) . \end{aligned}$$

The $det(J)=$ $\frac{1}{8}m^{3}-1,$ which is zero exactly when $m=2$. This satisfies the conditions for the existence of a Pitchfork bifurcation.