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Complexity of gene circuits, Pfaffian functions and the morphogenesis problem
C. R. Acad. Sci. Paris, Ser. I 337 (2003) 721–724
Systèmes dynamiques/Informatique théorique
Complexity of gene circuits, Pfaffian functions and the morphogenesis problem Sergey Vakulenko a , Dmitry Grigoriev b a Institute of Mechanical Engineering Problems, St Petersburg, Russia b IRMAR, Université de Rennes, Beaulieu, 35042 Rennes, France
Received 13 September 2003; accepted 14 October 2003 Presented by Christophe Soulé
Abstract We consider a model of gene circuits. We show that these circuits are capable to generate any spatio–temporal patterns. We give lower bounds on the number of genes required to create a given pattern. To cite this article: S. Vakulenko, D. Grigoriev, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Complexité de réseaux de gènes, fonctions de Pfaff et le problème de morphogenèse. On considère un modèle de réseaux de gènes. Nous démontrons que ces réseaux peuvent engendrer toutes les structures spatio–temporelles et nous obtenons des bornes inférieures du nombre de gènes du réseau qui engendrent une structure prescrite. Pour citer cet article : S. Vakulenko, D. Grigoriev, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Version française abrégée Le problème de morphogenèse biologique a été étudié dans de nombreux travaux, à partir du papier célèbre de Turing [9]. Deux questions fondamentales ouvertes, qui se posent naturellement, sont les suivantes. Premièrement, existent-ils des modèles mathématiques engendrant toutes les structures spatio–temporelles possibles ? De plus, peut-on trouver des algorithmes qui résolvent le problème suivant : étant donnée une structure, déterminer les paramètres du modèle qui engendre cette structure ? Cet article traite un système dynamique important en biologie [4,7,8]. C’est un circuit de gènes qui est une version simplifiée, en temps discret, du modèle [7,8]. Nous démontrons que : A Toutes les structures spatio–temporelles peuvent être obtenues par de tels circuits ; E-mail address: [email protected] (D. Grigoriev). 1631-073X/$ – see front matter 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crma.2003.10.021
722
S. Vakulenko, D. Grigoriev / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 721–724
B Il existe des algorithmes universels qui nous permettent de définir les paramètres du circuit qui produit une structure donnée ; C Etant donnée une structure, on peut majorer le nombre de gènes du circuit qui engendre cette structure. La formulation stricte et la démonstration du dernier résultat sont basées sur les méthodes de Khovanski [5]. Les algorithmes B utilisent les résultats et les algorithmes classiques de la théorie des réseaux en couche (voir [1]) et une idée, empruntée de biologie : les circuits génétiques réels ont une structure modulaire [3].
1. Introduction Mathematically, the biological morphogenesis problem (how complicated patterns emerge from homogeneous or almost homogeneous structures) was posed first by Turing [9]. Turing’s approach was developed in many works (for example, [6]). However, the following fundamental question was left open: whether there exist mathematical models generating any spatio–temporal patterns? In the framework of these models, whether there are algorithms ‘programming’ these patterns? This article deals with special circuits of the neural type playing a key role in biology [4,7,8]. These circuits are dynamical systems with discrete time and can be considered as a simplified version of the gene networks [7,8]. We show that A Any patterns can be obtained by these gene circuits; B Parameters of a circuit generating given pattern, can be found by an algorithm; C Given a pattern, one can evaluate the minimal number of genes in a network that generates this pattern. There exists a connection between ‘complexity’ of the circuits and pattern complexity. Below we give a strict formulation and proofs of these results. The time discrete model has an important advantage: in this case we can use the methods of [5]. It allows us to obtain C.
2. Model We consider the following chain of functions uti (x) defined iteratively by m t +1 t ui (x) = σ Kij uj (x) + µi θi (x) − ηi , u0j (x) = 0,
(1)
j =1
where t = 0, 1, 2, . . ., T , i = 1, 2, . . . , m and x ∈ Ω, Ω ∈ Rd is a bounded domain, d = 1, 2, 3. Here m is the number of genes involved in the circuit, ui (x, t) the concentration of the i-th gene, the matrix Kij defines a gene interaction, ηi are activation thresholds, µi are coefficients. Assume σ is a strictly monotone increasing function satisfying limz→−∞ σ (z) = 0, limz→∞ σ (z) = 1 and a differential equation σ = P (σ ), where P a polynomial. The well known example is σ (z) = (1 + tanh(z))/2 (here P = σ (1 − σ )/2). Functions θi (x) define concentrations of so-called maternal genes [10,8].
3. Main results A. Consider the following problem:
S. Vakulenko, D. Grigoriev / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 721–724
723
Pattern generation problem on [T0 , T ]. Let T0 , T be integers such that 0 T0 < T . Given a sequence of continuous functions zt (x) ∈ [0, 1], x ∈ Ω, t = 0, 1, . . . , T , and a positive ε, to find m, Kij , ηi , µi such that the solution of problem (1) satisfies supzt (x) − ut1 (x) < ε, x = (x1 , . . . , xd ) ∈ Ω, t = T0 , . . . , T . (2) x,t
Notice that σ and θi are fixed. Theorem 3.1. Suppose T0 2 and there exist continuous functions φl (θ ), l = 1, . . . , d, defined on Rm such that xl = φl (θ1 (x), . . . , θm (x)) for each x ∈ Ω ⊂ Rd . Then the pattern generation problem has a solution. B. Let us introduce a measure of the complexity of the pattern z(x). The complexity C1 (z(·), c) is the number of connected components of the set Dc,t = {x: z(x) = c}. Theorem 3.2. Consider chain (1). The number C1 of the connected components of the pattern uT1 (x) can be bounded from above by C1 < 2(rθ +T m) (dθ + T deg P )O(rθ +T m+n) . 2
(3)
Thus, given C1 , we can bound from below R = rθ + T m roughly as (log2 C1 )1/2 , provided that log(deg P ), log(dθ ), n1/2 are less than rθ + T m. The quantity R can be interpreted as a ‘complexity’ of the gene circuit (1), where rθ is the sum of the lengths of Pfaffian chains (see [5] for all necessary definitions and bounds on Pfaffian chains) for θi , dθ is the maximum of the degrees of Pfaffian chains determining θi , deg P is the degree of the polynomial that defines σ . Analogous estimates can be obtained for other complexity measures (see [12]).
4. Outline of proofs A brief proof of Theorem 3.1 can be obtained by the following lemma. Lemma 4.1. Superposition lemma. Consider a family consisting of p circuits (1) generating functions uti,s , where t = 0, . . . , T1 , s = 1, . . . , p and i = 1, . . . , ms (where the index s marks the s-th circuit). Denote by ut a vector function composed of all the functions uti,s , i.e., ut = (ut1,1 , ut2,1 , . . . , utm1 ,1 , . . . , utmp ,p ). t t N Suppose p z (x) = F (u (x)), where F is a continuous function defined on N -dimensional cube Q = [0, 1] and N = s=1 ms is the number of the functions involved in the circuits. (In other words, the target pattern can be expressed through the functions generated by the family.) Then for any ε > 0 there exists a circuit (1) satisfying (3) with T0 = 2 and T = T1 + 2. The main idea of the proof is based on the well known biological fact: networks have a modular structure [10,3]. Moreover, we use the following approximation result [1,11]: for any κ > 0 there exist M and Akj s , bk , ηk such that M −1 σ (z) − bk σ Akj s uj,s − ηk < κ, uj,s ∈ QN . (4) k=1
j,s
Let us construct now a large circuit by given networks and including additional variables vk , w, k = 1, . . . , M. Their time evolution is defined as M vkt +1 = σ Akj s utj,s − ηk , bk vkt . wt +1 = σ (5) j,s
k=1
724
S. Vakulenko, D. Grigoriev / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 721–724
Relations (4) and (5) imply approximation (2) (we set u1 = w) if κ = κ(ε) is a sufficiently small and M is large enough. Theorem 3.1 follows from Lemma 4.1. To show this, we define the circuit by utm+1 = σ (um+1 ), uti = σ (θi ), i = 1, 2, . . . , m. This proof gives an effective algorithm solving the pattern generation problem. Indeed, the key step of the proof (approximation (4)) can be performed by constructive procedures [1,11]. To prove Theorem 3.2, we notice that, under our hypothesis on σ (the derivative of σ is a polynomial of σ ), iterations (1) defines a Pfaffian chain. Thus, Khovanskii estimates [5] can be applied and we obtain (3) by an induction (see [2,12]).
References [1] A. Barron, Universal approximation bounds for superpositions of a sigmoidal functions, IEEE Trans. Inform. Theory 39 (1993) 930–945. [2] D. Grigoriev, N. Vorobjov, Complexity lower bounds for computation trees with elementary transcendental functions gates, Theoret. Comput. Sci. 157 (1996) 185–214. [3] L.H. Hartwell, J.J. Hopfield, S. Leibler, A.W. Murray, From molecular to modular cell biology, Nature 402 (1999) C47–C52. [4] J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. USA 79 (1982) 2554–2558. [5] A. Khovanskii, Fewnomials, in: Transl. Math. Monographs, Vol. 88, American Mathematical Society, 1991. [6] H. Meinhardt, in: P.K. Maini, H.G. Othmer (Eds.), Mathematical Models for Biological Pattern Formation, in: IMA Vol. Math. Appl., Vol. 121, Springer, 2000. [7] E. Mjolness, D.H. Sharp, J. Reinitz, A connectionist model of development, J. Theor. Biol. 152 (1991) 429–453. [8] J. Reinitz, D.H. Sharp, Mechanism of formation of eve stripes, Mechanisms of Developments 49 (1995) 133–158. [9] A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B 237 (1952) 37–72. [10] L. Wolpert, R. Beddington, T. Jessell, P. Lawrence, E. Meyerowitz, J. Smith, Principles of Development, Oxford University Press, 2002. [11] S. Vakulenko, Dissipative systems generating any structurally stable chaos, Adv. Differential Equations 5 (2000) 1139–1178. [12] S. Vakulenko, D. Grigoriev, Complexity of patterns generated by genetic circuits and Pfaffian functions, Preprint IHES, 2003.
Systèmes dynamiques/Informatique théorique
Complexity of gene circuits, Pfaffian functions and the morphogenesis problem Sergey Vakulenko a , Dmitry Grigoriev b a Institute of Mechanical Engineering Problems, St Petersburg, Russia b IRMAR, Université de Rennes, Beaulieu, 35042 Rennes, France
Received 13 September 2003; accepted 14 October 2003 Presented by Christophe Soulé
Abstract We consider a model of gene circuits. We show that these circuits are capable to generate any spatio–temporal patterns. We give lower bounds on the number of genes required to create a given pattern. To cite this article: S. Vakulenko, D. Grigoriev, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Complexité de réseaux de gènes, fonctions de Pfaff et le problème de morphogenèse. On considère un modèle de réseaux de gènes. Nous démontrons que ces réseaux peuvent engendrer toutes les structures spatio–temporelles et nous obtenons des bornes inférieures du nombre de gènes du réseau qui engendrent une structure prescrite. Pour citer cet article : S. Vakulenko, D. Grigoriev, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Version française abrégée Le problème de morphogenèse biologique a été étudié dans de nombreux travaux, à partir du papier célèbre de Turing [9]. Deux questions fondamentales ouvertes, qui se posent naturellement, sont les suivantes. Premièrement, existent-ils des modèles mathématiques engendrant toutes les structures spatio–temporelles possibles ? De plus, peut-on trouver des algorithmes qui résolvent le problème suivant : étant donnée une structure, déterminer les paramètres du modèle qui engendre cette structure ? Cet article traite un système dynamique important en biologie [4,7,8]. C’est un circuit de gènes qui est une version simplifiée, en temps discret, du modèle [7,8]. Nous démontrons que : A Toutes les structures spatio–temporelles peuvent être obtenues par de tels circuits ; E-mail address: [email protected] (D. Grigoriev). 1631-073X/$ – see front matter 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crma.2003.10.021
722
S. Vakulenko, D. Grigoriev / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 721–724
B Il existe des algorithmes universels qui nous permettent de définir les paramètres du circuit qui produit une structure donnée ; C Etant donnée une structure, on peut majorer le nombre de gènes du circuit qui engendre cette structure. La formulation stricte et la démonstration du dernier résultat sont basées sur les méthodes de Khovanski [5]. Les algorithmes B utilisent les résultats et les algorithmes classiques de la théorie des réseaux en couche (voir [1]) et une idée, empruntée de biologie : les circuits génétiques réels ont une structure modulaire [3].
1. Introduction Mathematically, the biological morphogenesis problem (how complicated patterns emerge from homogeneous or almost homogeneous structures) was posed first by Turing [9]. Turing’s approach was developed in many works (for example, [6]). However, the following fundamental question was left open: whether there exist mathematical models generating any spatio–temporal patterns? In the framework of these models, whether there are algorithms ‘programming’ these patterns? This article deals with special circuits of the neural type playing a key role in biology [4,7,8]. These circuits are dynamical systems with discrete time and can be considered as a simplified version of the gene networks [7,8]. We show that A Any patterns can be obtained by these gene circuits; B Parameters of a circuit generating given pattern, can be found by an algorithm; C Given a pattern, one can evaluate the minimal number of genes in a network that generates this pattern. There exists a connection between ‘complexity’ of the circuits and pattern complexity. Below we give a strict formulation and proofs of these results. The time discrete model has an important advantage: in this case we can use the methods of [5]. It allows us to obtain C.
2. Model We consider the following chain of functions uti (x) defined iteratively by m t +1 t ui (x) = σ Kij uj (x) + µi θi (x) − ηi , u0j (x) = 0,
(1)
j =1
where t = 0, 1, 2, . . ., T , i = 1, 2, . . . , m and x ∈ Ω, Ω ∈ Rd is a bounded domain, d = 1, 2, 3. Here m is the number of genes involved in the circuit, ui (x, t) the concentration of the i-th gene, the matrix Kij defines a gene interaction, ηi are activation thresholds, µi are coefficients. Assume σ is a strictly monotone increasing function satisfying limz→−∞ σ (z) = 0, limz→∞ σ (z) = 1 and a differential equation σ = P (σ ), where P a polynomial. The well known example is σ (z) = (1 + tanh(z))/2 (here P = σ (1 − σ )/2). Functions θi (x) define concentrations of so-called maternal genes [10,8].
3. Main results A. Consider the following problem:
S. Vakulenko, D. Grigoriev / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 721–724
723
Pattern generation problem on [T0 , T ]. Let T0 , T be integers such that 0 T0 < T . Given a sequence of continuous functions zt (x) ∈ [0, 1], x ∈ Ω, t = 0, 1, . . . , T , and a positive ε, to find m, Kij , ηi , µi such that the solution of problem (1) satisfies supzt (x) − ut1 (x) < ε, x = (x1 , . . . , xd ) ∈ Ω, t = T0 , . . . , T . (2) x,t
Notice that σ and θi are fixed. Theorem 3.1. Suppose T0 2 and there exist continuous functions φl (θ ), l = 1, . . . , d, defined on Rm such that xl = φl (θ1 (x), . . . , θm (x)) for each x ∈ Ω ⊂ Rd . Then the pattern generation problem has a solution. B. Let us introduce a measure of the complexity of the pattern z(x). The complexity C1 (z(·), c) is the number of connected components of the set Dc,t = {x: z(x) = c}. Theorem 3.2. Consider chain (1). The number C1 of the connected components of the pattern uT1 (x) can be bounded from above by C1 < 2(rθ +T m) (dθ + T deg P )O(rθ +T m+n) . 2
(3)
Thus, given C1 , we can bound from below R = rθ + T m roughly as (log2 C1 )1/2 , provided that log(deg P ), log(dθ ), n1/2 are less than rθ + T m. The quantity R can be interpreted as a ‘complexity’ of the gene circuit (1), where rθ is the sum of the lengths of Pfaffian chains (see [5] for all necessary definitions and bounds on Pfaffian chains) for θi , dθ is the maximum of the degrees of Pfaffian chains determining θi , deg P is the degree of the polynomial that defines σ . Analogous estimates can be obtained for other complexity measures (see [12]).
4. Outline of proofs A brief proof of Theorem 3.1 can be obtained by the following lemma. Lemma 4.1. Superposition lemma. Consider a family consisting of p circuits (1) generating functions uti,s , where t = 0, . . . , T1 , s = 1, . . . , p and i = 1, . . . , ms (where the index s marks the s-th circuit). Denote by ut a vector function composed of all the functions uti,s , i.e., ut = (ut1,1 , ut2,1 , . . . , utm1 ,1 , . . . , utmp ,p ). t t N Suppose p z (x) = F (u (x)), where F is a continuous function defined on N -dimensional cube Q = [0, 1] and N = s=1 ms is the number of the functions involved in the circuits. (In other words, the target pattern can be expressed through the functions generated by the family.) Then for any ε > 0 there exists a circuit (1) satisfying (3) with T0 = 2 and T = T1 + 2. The main idea of the proof is based on the well known biological fact: networks have a modular structure [10,3]. Moreover, we use the following approximation result [1,11]: for any κ > 0 there exist M and Akj s , bk , ηk such that M −1 σ (z) − bk σ Akj s uj,s − ηk < κ, uj,s ∈ QN . (4) k=1
j,s
Let us construct now a large circuit by given networks and including additional variables vk , w, k = 1, . . . , M. Their time evolution is defined as M vkt +1 = σ Akj s utj,s − ηk , bk vkt . wt +1 = σ (5) j,s
k=1
724
S. Vakulenko, D. Grigoriev / C. R. Acad. Sci. Paris, Ser. I 337 (2003) 721–724
Relations (4) and (5) imply approximation (2) (we set u1 = w) if κ = κ(ε) is a sufficiently small and M is large enough. Theorem 3.1 follows from Lemma 4.1. To show this, we define the circuit by utm+1 = σ (um+1 ), uti = σ (θi ), i = 1, 2, . . . , m. This proof gives an effective algorithm solving the pattern generation problem. Indeed, the key step of the proof (approximation (4)) can be performed by constructive procedures [1,11]. To prove Theorem 3.2, we notice that, under our hypothesis on σ (the derivative of σ is a polynomial of σ ), iterations (1) defines a Pfaffian chain. Thus, Khovanskii estimates [5] can be applied and we obtain (3) by an induction (see [2,12]).
References [1] A. Barron, Universal approximation bounds for superpositions of a sigmoidal functions, IEEE Trans. Inform. Theory 39 (1993) 930–945. [2] D. Grigoriev, N. Vorobjov, Complexity lower bounds for computation trees with elementary transcendental functions gates, Theoret. Comput. Sci. 157 (1996) 185–214. [3] L.H. Hartwell, J.J. Hopfield, S. Leibler, A.W. Murray, From molecular to modular cell biology, Nature 402 (1999) C47–C52. [4] J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. USA 79 (1982) 2554–2558. [5] A. Khovanskii, Fewnomials, in: Transl. Math. Monographs, Vol. 88, American Mathematical Society, 1991. [6] H. Meinhardt, in: P.K. Maini, H.G. Othmer (Eds.), Mathematical Models for Biological Pattern Formation, in: IMA Vol. Math. Appl., Vol. 121, Springer, 2000. [7] E. Mjolness, D.H. Sharp, J. Reinitz, A connectionist model of development, J. Theor. Biol. 152 (1991) 429–453. [8] J. Reinitz, D.H. Sharp, Mechanism of formation of eve stripes, Mechanisms of Developments 49 (1995) 133–158. [9] A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B 237 (1952) 37–72. [10] L. Wolpert, R. Beddington, T. Jessell, P. Lawrence, E. Meyerowitz, J. Smith, Principles of Development, Oxford University Press, 2002. [11] S. Vakulenko, Dissipative systems generating any structurally stable chaos, Adv. Differential Equations 5 (2000) 1139–1178. [12] S. Vakulenko, D. Grigoriev, Complexity of patterns generated by genetic circuits and Pfaffian functions, Preprint IHES, 2003.