Appendix D: Regulatory functions and their derivatives

Because we use certain functions to describe regulation of gene expression, it is useful to have a reference for the functions and their first derivatives (and also some second derivatives).

Dimensional forms

In all cases, the functions are dimensionless, and the arguments of the function are given in dimensionless form. To convert to a dimensional derivative, use the chain rule. Let \(x_\mathrm{dim}\) be the dimensional version of dimensionless \(x\) such that \(x_\mathrm{dim} = a x\). Then,

\begin{align} \frac{df}{\mathrm{d}x_\mathrm{dim}} = a \,\frac{\mathrm{d}f}{\mathrm{d}x}. \end{align}

Taylor series

In performing linear stability analysis, we need to express functions as Taylor series. For a univariate function (such a simple Hill functions), the Taylor series about the point \(x_0\) is

\begin{align} f(x) = f(x_0) + \left.\frac{\mathrm{d}f}{\mathrm{d}x}\right|_{x=x_0}\,(x-x_0) + \text{higher order terms.} \end{align}

For a bivariate function the Taylor series about point \((x_0, y_0)\) is

\begin{align} f(x, y) = f(x_0, y_0) + \left.\frac{\partial f}{\partial x}\right|_{x=x_0, y=y_0}\,(x-x_0) + \left.\frac{\partial f}{\partial y}\right|_{x=x_0, y=y_0}\,(y-y_0)+ \text{higher order terms.} \end{align}

Hill functions

Activating Hill function

\begin{align} &f(x) = \frac{x^n}{1 + x^n}, \\[1em] &\frac{\mathrm{d}f}{\mathrm{d}x} = \frac{n x^{n-1}}{(1 + x^n)^2},\\[1em] &\frac{\mathrm{d}^2f}{\mathrm{d}x^2} = -\frac{n x^{n-2}\left(1 - n + (1+n)x^n\right)}{(1 + x^n)^3}. \end{align}

Repressive Hill function

\begin{align} &f(x) = \frac{1}{1 + x^n}, \\[1em] &\frac{\mathrm{d}f}{\mathrm{d}x} = -\frac{n x^{n-1}}{(1 + x^n)^2},\\[1em] &\frac{\mathrm{d}^2f}{\mathrm{d}x^2} = \frac{n x^{n-2}\left(1 - n + (1+n)x^n\right)}{(1 + x^n)^3}. \end{align}

Two activators affecting the same gene

AND logic

\begin{align} &f(x, y) = \frac{x^{n_x} y^{n_y}}{(1 + x^{n_x})\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial x} = \frac{n_x\,x^{n_x-1} y^{n_y}}{(1 + x^{n_x})^2\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial y} = \frac{n_y\,x^{n_x} y^{n_y-1}}{(1 + x^{n_x})\,(1 + y^{n_y})^2}. \end{align}

OR logic

\begin{align} &f(x,y) = \frac{x^{n_x} + y^{n_y} + x^{n_x} y^{n_y}}{(1 + x^{n_x})\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial x} = \frac{n_x\,x^{n_x-1}}{(1 + x^{n_x})^2\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial y} = \frac{n_y\,y^{n_y-1}}{(1 + x^{n_x})\,(1 + y^{n_y})^2}. \end{align}

OR logic, single occupancy

\begin{align} &f(x,y) = \frac{x^{n_x} + y^{n_y}}{1 + x^{n_x}+ y^{n_y}},\\[1em] &\frac{\partial f}{\partial x} = \frac{n_x\,x^{n_x-1}}{(1 + x^{n_x} + y^{n_y})^2},\\[1em] &\frac{\partial f}{\partial y} = \frac{n_y\,y^{n_y-1}}{(1 + x^{n_x}+ y^{n_y})^2}. \end{align}

Two repressors affecting the same gene

AND logic

\begin{align} &f(x, y) = \frac{1}{(1 + x^{n_x})\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial x} = -\frac{n_x\,x^{n_x-1}}{(1 + x^{n_x})^2\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial y} = -\frac{n_y\,y^{n_y-1}}{(1 + x^{n_x})\,(1 + y^{n_y})^2}. \end{align}

OR logic

\begin{align} &f(x, y) = \frac{1 + x^{n_x} + y^{n_y}}{(1 + x^{n_x})\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial x} = -\frac{n_x\,x^{n_x-1}\,y^{n_y}}{(1 + x^{n_x})^2\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial y} = -\frac{n_y\,x^{n_x}\,y^{n_y-1}}{(1 + x^{n_x})\,(1 + y^{n_y})^2}. \end{align}

AND logic, single occupancy

\begin{align} &f(x, y) = \frac{1}{(1 + x^{n_x} + y^{n_y})},\\[1em] &\frac{\partial f}{\partial x} = -\frac{n_x\,x^{n_x-1}}{(1 + x^{n_x} + y^{n_y})^2},\\[1em] &\frac{\partial f}{\partial y} = -\frac{n_y\,y^{n_y-1}}{(1 + x^{n_x} + y^{n_y})^2}. \end{align}

One activator and one repressor affecting the same gene

Here, we assign \(x\) to be the concentration of the activator and \(y\) to be the concentration of the repressor.

AND logic

\begin{align} &f(x, y) = \frac{x^{n_x}}{(1 + x^{n_x})\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial x} = \frac{n_x\,x^{n_x-1}}{(1 + x^{n_x})^2\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial y} = -\frac{n_y\,x^{n_x}\,y^{n_y-1}}{(1 + x^{n_x})\,(1 + y^{n_y})^2}. \end{align}

OR logic

\begin{align} &f(x, y) = \frac{1 + x^{n_x} + x^{n_x}y^{n_y}}{(1 + x^{n_x})\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial x} = \frac{n_x\,x^{n_x-1}\,y^{n_y}}{(1 + x^{n_x})^2\,(1 + y^{n_y})},\\[1em] &\frac{\partial f}{\partial y} = -\frac{n_y\,y^{n_y-1}}{(1 + x^{n_x})\,(1 + y^{n_y})^2}. \end{align}

AND logic, single occupancy

\begin{align} &f(x, y) = \frac{x^{n_x}}{1 + x^{n_x} + y^{n_y}},\\[1em] &\frac{\partial f}{\partial x} = \frac{n_x\,x^{n_x-1}(1+y^{n_y})}{(1 + x^{n_x} + y^{n_y})^2},\\[1em] &\frac{\partial f}{\partial y} = -\frac{n_y\,x^{n_x}\,y^{n_y-1}}{(1 + x^{n_x} + y^{n_y})^2}. \end{align}

OR logic, single occupancy

\begin{align} &f(x, y) = \frac{1 + x^{n_x}}{1 + x^{n_x} + y^{n_y}},\\[1em] &\frac{\partial f}{\partial x} = \frac{n_x\,x^{n_x-1}\,y^{n_y}}{(1 + x^{n_x} + y^{n_y})^2},\\[1em] &\frac{\partial f}{\partial y} = -\frac{n_y\,(1+x^{n_x})\,y^{n_y-1}}{(1 + x^{n_x} + y^{n_y})^2}. \end{align}