CC BY
72
arg(z)
possible values of variables, including their branch cut values.
For example, the value ^—z2 can be correctly presented as ize‘'t n J or \/—z(where \x\ means the maximal integer less than or equal to x, it is Floor function), the formula \/1 — z2 = y/1 — zy/1 + z is correct formula for all complex z-plane and formula V—1 + z2 = V—1 + zV1 + z is correct not everywhere at the complex z-plane (it is wrong for z < —1), the correct formula for logarithm from product has special additional term, including floor-arg construction: log(wz) = log(w) + log(z) +
+ 2in
arg(w)
2n
arg(^)
2n
The author of the talk works as developer of calculus related functionality of Mathematica for over 18 years.He derived thousands of new formulas in different areas of elementary and special functions, which are included into the functions site (http : // functions.wolfram.com/) and describe mentioned above Mathematica's world of mathematical formulas. This talk shows how developing mathematical formulas for functions site makes effect on developing system Mathematica and back.
Аннотация: В докладе показывается, как разработка математических формул для сайта математических функций влияет на развитие системы "MathematicanH наоборот.
Ключевые слова: математический формулы; сайт функций; система "Mathematica".
Маричев Олег Игоревич ведущий разработчик Wolfram Research Inc. США, Урбана e-mail: oleg@wolfram.com
Oleg Marichev leading former Wolfram Research Inc. USA, Urbana, IL e-mail: oleg@wolfram.com
УДК 517.958
THE FILIPPOV THEORY AND ITS APPLICATION TO GENE REGULATORY
NETWORKS 1
© A. Machina
Keywords: piecewise-linear differential system; singular stationary point; differential inclusion; Filippov solution.
Abstract: We study some properties of piecewise-linear differential systems describing gene regulatory networks, where the dynamics is governed by sigmoid-type nonlinearities which are close to or coincide with the step functions. To overcome the difficulty of describing the dynamics of the system near singular stationary points (i.e. belonging to the discontinuity set of the system) we use the concept of Filippov solutions. It consists in replacing differential equations with discontinuous right-hand sides with differential inclusions with multi-valued functions. The global existence and some basic properties of Filippov solutions such as continuous dependence on parameters are studied. We also study uniqueness and non-uniqueness of the Fillipov solutions in singular
1The work is supported by Norwegian State Educational Loan Fund.
domains. The concept of Filippov stationary point is extensively exploited in the paper. We compare two approaches in defining the singular stationary points: based on Fillipov theory and based on replacing step functions by steep sigmoids and investigating the smooth systems thus obtained.
An important feature of genetic networks as well as of many other complex biological systems is the presence of thresholds causing switch-like interactions between genes. Such interactions can be described by the so-called "sigmoids", smooth monotone functions assuming the values between 0 and 1 and rapidly increasing around the threshold. The resulting nonlinear system can be however too complicated to be studied theoretically and even numerically, as the number of the system’s variables may be huge. To simplify the analysis it is common to replace sigmoids with step functions, which converts the original smooth system into a switching system with discontinuous right-hand sides. Such a replacement can only be considered admissible if the dynamics of the "idealized"(i.e. switching) systems do mimic the dynamics of the original smooth system. In [lj and [2] it is observed that in many cases the qualitative behavior of the solutions does not changed under such replacements. This analysis was continued in the papers [3-6], where a special emphasis was put on the behavior of solutions around steady states (equilibrium concentrations) lying close to one or more threshold values. Such states are of interest as they represent homeostatic states in the model. However, if we use the idealized model based on discontinuous right-hand sides, we should first be able to define the very notion of the homeostatic states (which in the limit may end up in the discontinuity set) and to describe an efficient way to identify such states without any additional information from the smooth system.
Basically, there are two ways of solving this problem. The first one is based on the implicit function theorem and goes back to the papers [4-6]. Another approach utilizes the concept of differential inclusions and the so-called "Filippov solutions"[7, 8]. Both approaches have their advantages and disadvantages. For instance, in the second approach we may obtain steady states that are not limits of the proper steady states coming from the smooth model. On the other hand, the results obtained in the framework of the first approach can be too restrictive. However, as far as we know in the available literature there is no attempt to compare these two approaches from the mathematical point of view.
The present work is aimed to fill partly the gap between the two approaches. However, the main objective of the work is different. Although the very idea to use the Filippov framework to the analysis of gene regulatory networks seems to be suggested in [7], no detailed justification of this approach was offered in this paper. Nor was it compared mathematically to the alternative approach from [4-6].
We wanted to look at the Filippov approach a little bit more systematically starting with the very concept of the Filippov solution which can be defined in three different ways (in [7] only one of the definitions is used). We proved that two of the definitions (where the right-hand sides are convex though constructed in different ways) are in fact equivalent in the case of the gene regulatory networks, while the third gives a different inclusion with a non-convex right-hand side, thus giving a different set of stationary solutions. However, it is the latter definition that covers the homeostatic states in the model, while the first two may produce stationary solutions of quite a different nature.
We studied some basic properties of Filippov solutions of the systems in question putting emphasis on global existence and continuous dependence on parameters. In particular, these results can be used to justify similarities between the "real-world"model based on smooth interactions (sigmoids) and the idealized model based on step-like interactions.
We studied uniqueness and non-uniqueness of Filippov solutions in the singular domains (i. e. in the set of discontinuity points of the right-hand side). For instance, we showed that the solution is unique in the black and transparent walls (see e.g. [4]), while the white walls usually give rise to infinitely many Filippov solutions.
Stationary solutions were also discussed. We compared the two approaches mentioned above. Roughly speaking we showed that the main difference between them amounts to the difference between non-equivalent definitions of the Filippov solutions. In the case of the non-convex right-hand side we get stationary points in the sense of [4], while in the case of the convex right-hand side we obtain stationary
points in the sense of [4]. Although the second approach gives more stationary points than the first one, we showed that the Filippov stationary points that are limits of convergent (as steepness parameter q ^ 0) sequences of stationary points of smooth systems are indeed the Fillipov solutions in the sense of definition with non-convex right-hand sides. We also introduced some examples of stationary points in the sense of [7], which at the same time are not stationary points in the sense of [4].
References
1. Glass L., Kaufmann S.A. Co-operative components, spatial localization and oscillatory cellular dynamics // J. Theor. Biol. 1972. V. 34. P. 219-237.
2. Glass L., Kaufmann S.A. The logical analysis of continuous, non-linear biochemical control networks j j J. Theor. Biol. 1973. V. 39. P. 103-129.
3. Mestl Т., Plahte E., and Omholt S.W. A mathematical framework for describing and analysing gene regulatory networks // J. Theor. Biol. 1995. V. 176. P. 291-300.
4. Plahte E., Kjoglum S. Analysis and generic properties of gene regulatory networks with graded response functions 11 J. Physica D. 2005. V. 201. P. 150-176.
5. Plahte E., Mestl Т., Omholt S.W. Global analysis of steady points for systems of differential equations with sigmoid interactions j j J. Dynamics and Stability of Systems. 1994. V. 9. P. 275-291.
6. Plahte E., Mestl Т., Omholt S. W. A methodological basis for the description and analysis of systems with complex switch-like interactions j j J. Math. Biol. 1998. V. 36. P. 321-348.
7. Gouze J.-L. and Sari T. A class of piecewise linear differential equations arising in biological models // Dynamical Systems: An International Journal. 2002. V. 17. P. 299-316.
8. Jong H. de, Gouze J.-L., Hernandez C., Page М., Sari Т., and Geiselmann J. Qualitative simulations of genetic regulatory networks using piecewise linear models j j Bulletin of mathematical biology. 2004. V. 66 (2). P. 301-340.
Аннотация: Рассматриваются свойства кусочно-линейных дифференциальных систем, описывающих генные регуляторные сети, в которых динамика управляется сигмоидальными нелинейными функциями, близкими по поведению к ступенчатым функциям. Для того чтобы преодолеть трудность описания динамики вблизи сингулярных стационарных точек (т.е. принадлежащих множеству разрыва системы), используется понятие решений Филиппова. Оно состоит в замене дифференциальных уравнений с разрывными правыми частями дифференциальными включениями с многозначными функциями. Изучается вопрос о существовании глобального решения Филиппова, а также его некоторые основные свойства, как, например, непрерывная зависимость от параметров. Рассматривается единственность и неединственность решений Филиппова в сингулярных областях. Вводится понятие стационарной точки по Филиппову. Сравниваются два подхода к определению стационарной точки: основанное на теории Филиппова и основанное на замене ступенчатых функций сигмоидами.
Ключевые слова: кусочно-линейные дифференциальные системы; сингулярные стационарные точки; дифференциальные включения; решения Филиппова.
Machina Anna Мачина Анна Николаевна
post-graduate student аспирант
Norwegian University of Life Sciences Норвежский университет
Norway, Aas естественных наук
e-mail: annama@umb.no Норвегия, Ос
e-mail: annama@umb.no
