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35
GENE REGULATORY NETWORKS WITH DISTRIBUTED DELAY 1
© A. Ponossov, A. Shindiapin, Yu. Nepomnyashchikh
A method to study asymptotic properties of solutions to systems of differential equations with distributed time-delays and Boolean-type nonlinearities (step functions) is offered. Such systems arise in many applications, but this particular contribution deals with specific examples of such systems coming from gene regulatory networks. A challenge is to analyze stable stationary points which belong to the discontinuity set of the system (thresholds). We will describe an algorithm of localizing stationary points in the presence of delays as well as stability analysis around such points. The basic technical tool consists in replacing step functions by moderately steep sigmoids (called «logoids») and investigating the smooth systems thus obtained. In addition, we make use of the so-called «linear chain trick» which we, however, apply to the system in question in a slightly modified form.
Let us consider a system of delay differential equations with switch-like nonlinearities
xi = Fi(Z\,Zn) — Gi(Z\,Zn)xi,
Zi = Zi(yi),
yi(t) = (№iXi)(t) (t ^ 0;i = l,..,n),
describing gene regulatory networks with autoregulation.
Assumpt ion 1.1: Fi,Gi are affine functions in each variable, satisfying
Fi(Z\,Zn) ^ 0, Gi(Z1,Zn) ^ 5 > 0 for 0 ^ Z ^ 1 and i = 1, ...,n.
Assumption 1.2: For some Qi > 0
( 0 if yi <Qi
Zi(yi) = Zi(yi,Qi):= < 0.5 if yi = Qi
1 if yi > Qi
Assumption 1.3: The integral operator is given by
(№iXi)(t) = C0Xi(t) + f Ki(t — s)xi(s)ds, t ^ 0,
J — (X
where Ki(u) = c\Kl(u) + ci2Kf(u), civ ^ 0 (v = 0,1, 2), c0 + c\ + ci2 = 1, and Ki(u) = aie—aiU, ai > 0 (the weak generic delay kernel),
K2(u) = afue—aiW, ai > 0 (the strong generic delay kernel).
Below we assume that a natural number j (1 ^ j ^ n) is fixed. For the sake of simplicity we assume in what follows that there is no delay in the variables xr for r = j.
1The work is partially supported by CIGENE — Centre for Integrative Genetics and by NUFU — Norwegian
Programme for Development, Research and Education.
We are investigating singular stationary points, i. e. points P0 belonging to a set where exactly one of the variables yj assumes the value yj = Qj. Then, it can be shown that in a sufficiently small neighborhood of such a point the other functions Zi are identically equal to 0 or 1: Zr(yr) = Br for any r = j, where Br is a Boolean variable. Put Br = (Br)r=j and
d d J = W3 Fj(Zj,BR) — Gj(Zj,BR)Qj
and observe that J is, in fact, independent of Zj, as Fj and Gj are affine with respect to Zj (and the other variables as well).
The following result is a particular case of the existence theorem proved in [1]: Theorem1. Assume that J is not zero and the system,
Fj (Zj ,Br) — Gj (Zj ,BR)Qj =0
Fr (Zj ,Br) — Gr (Zj, Br)yr = 0 (r = j)
with the constraints
0 <Zj < 1
Zr (yr ) = Br (r = j)
has a solution Zj, yr (r = j). Then for any delay operator №j, described above, there exists a singular stationary point P0 for the main system. This point is independent of the choice of the operator №j.
Stability analysis around singular stationary points is more involved. A typical result is formulated below.
Theorem 2. Assume that c30 = 0. Let also the assumptions of Theorem 1 be fulfilled. Then the following statements are valid:
1. If J > 0, then P0 is unstable.
2. If J < 0, c{ = 0, then P0 is unstable.
3. If J < 0, cj = 0 and Gj (Zj) < aj (cii)—l(1 — 2c{), then P0 is unstable.
4. If J < 0, cj = 0 and Gj (Zj) > aj (cJ]) — l(1 — 2cJl), then P0 is asymptotically stable spiral
point.
Analysis of higher order delay kernels has been performed with the help of MATHEMATICA.
REFERENCES
1. Ponossov A. Gene regulatory networks and delay differential equations // Special issue of Electronic J. Diff. Eq. 2004. V. 12. P. 117-141.
Ponossov Arcady
Norwegian University of Life Sciences Norway, As
e-mail: [email protected]
Shindiapin Andrei
Eduardo Mondlane University
Maputo, Mozambique
e-mail: [email protected]
Nepomnyashchikh Yuri Eduardo Mondlane University Maputo, Mozambique e-mail: [email protected]
Поступила в редакцию 5 мая 2007 г.
