Dynamics of actively regulated gene networks

Sensitive dependence on initial conditions in gene networks

“…12 and developed particularly by Glass over many years (e.g., Ref. 11), then by Plahte,19 Mestl, 20 Gouz e, 4,6,13 Edwards,7,8 Farcot, 9 Ironi, 14 and their coauthors.…”

“…The simplest such example is damped oscillations via negative feedback between a pair of variables, which can be hard to identify. Ironi and coauthors 14 pvdd@math.uvic.ca 1054-1500/2013/23(2)/025101/9/$30.00 V C 2013 AIP Publishing LLC 23, 025101-1 an improved software system that can handle some types of networks not handled properly by GNA. Additional challenges remain, however, as the current analysis 14 has restrictive assumptions.…”

“…This kind of "homeostatic" situation has been called "active regulation." 14 For a differential equation model with step function interaction terms, there are discontinuity hyperplanes in the vector field at threshold values in which the variables corresponding to concentrations of protein products of such actively regulating genes can become stuck. Then the standard theory of smooth ordinary differential equations (ODEs) does not hold and some special analytic tools are required to determine behaviour.…”

Linear extensions of a partial order subject to algebraic constraints

“…Because the pattern η 1 = η 2 = η 3 = {0} occurs twice and η 1 = η 2 = {0}, η 3 = {0, 1} four times, there are at most 2 · 1 + 4 · 2 = 10 realizable linear extensions of < consistent with (13). Identical considerations are valid when the inequality in (13) is reversed, and therefore there exist at most 2 · 10 = 20 realizable linear extensions.…”

“…It is typically assumed that the interactions have switch-like behavior and thus the nonlinearities in these ODEs are often assumed to have a sigmoidal shape. One common approximation [3,4,6,7,8,9,10,11,12,13] yields nonlinearities that take the form of piecewise constant functions. Definition 2.2.…”