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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.

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Acknowledgments

We would like to thank Yan Jun, Lu Zhang, Lindsey Muir, Geoff Patterson, Xin Guo, Anthony Bloch, an anonymous referee and especially Mike Shub for helpful discussions. We extend thanks to James Gimlett and Srikanta Kumar at Defense Advanced Research Projects Agency for support and encouragement.

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Corresponding author

Correspondence to Steve Smale.

Additional information

Communicated by Michael Shub.

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

$$\begin{aligned} \frac{{\hbox {d}}x}{{\hbox {d}}t}&=\frac{2}{1+y^{m}}-x\nonumber \\ \frac{{\hbox {d}}y}{{\hbox {d}}t}&=\frac{2}{1+x^{m}}-y, x,y\ge 0. \end{aligned}$$
(23)

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

$$\begin{aligned} \frac{{\hbox {d}}x}{{\hbox {d}}t}&=\frac{\alpha }{1+z^{m}}-x\nonumber \\ \frac{{\hbox {d}}y}{{\hbox {d}}t}&=\frac{\alpha }{1+x^{m}}-y\nonumber \\ \frac{{\hbox {d}}z}{{\hbox {d}}t}&=\frac{\alpha }{1+y^{m}}-z,\text { }x,y,z\ge 0. \end{aligned}$$
(24)

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

$$\begin{aligned} s+s^{m+1}-\alpha =0. \end{aligned}$$
(25)

The Jacobian matrix J at (sss) 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

$$\begin{aligned} \frac{{\hbox {d}}x}{{\hbox {d}}t}&=\frac{2}{1+z^{m}}-x\nonumber \\ \frac{{\hbox {d}}y}{{\hbox {d}}t}&=\frac{2x^{m}}{1+x^{m}}-y\nonumber \\ \frac{{\hbox {d}}z}{{\hbox {d}}t}&=\frac{2}{1+y^{m}}-z. \end{aligned}$$
(26)

The equilibria of the system C1 are

$$\begin{aligned} x=\frac{2}{1+z^{m}},\text { }y=\frac{2x^{m}}{1+x^{m}},\text { and }z=\frac{2}{1+y^{m}} \end{aligned}$$
(27)

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.

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Rajapakse, I., Smale, S. Mathematics of the Genome. Found Comput Math 17, 1195–1217 (2017). https://doi.org/10.1007/s10208-016-9316-x

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Keywords

  • Genome dynamics
  • Gene networks
  • Pitchfork bifurcation
  • Hopf bifurcation

Mathematics Subject Classification

  • 37G15
  • 92D10