Appendix A: 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 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 xdim be the dimensional version of dimensionless x such that xdim=ax. Then,
dfdxdim=adfdx.
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 x0 is
f(x)=f(x0)+dfdx|x=x0(x−x0)+higher order terms.
For a bivariate function the Taylor series about point (x0,y0) is
f(x,y)=f(x0,y0)+∂f∂x|x=x0,y=y0(x−x0)+∂f∂y|x=x0,y=y0(y−y0)+higher order terms.
Hill functions¶
Activating Hill function¶
f(x)=xn1+xn,dfdx=nxn−1(1+xn)2.
Repressive Hill function¶
f(x)=11+xn,dfdx=−nxn−1(1+xn)2.
Two activators affecting a single gene¶
AND logic¶
f(x,y)=xnxyny1+xnxyny,∂f∂x=nxxnx−1yny(1+xnxyny)2,∂f∂y=nyxnxyny−1(1+xnxyny)2.
OR logic¶
f(x,y)=xnx+yny1+xnx+yny,∂f∂x=nxxnx−1(1+xnx+yny)2,∂f∂y=nyyny−1(1+xnx+yny)2.
Two repressors affecting a single gene¶
AND logic¶
f(x,y)=1(1+xnx)(1+yny),∂f∂x=−nxxnx−1(1+xnx)2(1+yny),∂f∂y=−nyyny−1(1+xnx)(1+yny)2.
OR logic¶
f(x,y)=1+xnx+yny(1+xnx)(1+yny),∂f∂x=−nxxnx−1yny(1+xnx)2(1+yny),∂f∂y=−nyxnxyny−1(1+xnx)(1+yny)2.
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¶
f(x,y)=xnx1+xnx+yny,∂f∂x=nxxnx−1(1+yny)(1+xnx+yny)2,∂f∂y=−nyxnxyny−1(1+xnx+yny)2.
OR logic¶
f(x,y)=1+xnx1+xnx+yny,∂f∂x=nxxnx−1yny(1+xnx+yny)2,∂f∂y=−ny(1+xnx)yny−1(1+xnx+yny)2.