Periodic Reporting for period 1 - AlgSignSen (The Algebraic Geometry of Chemical Reaction Networks: Structural conditions for uniquely determined Sign-sensitivities.)
Periodo di rendicontazione: 2019-03-01 al 2021-02-28
Many biological processes of communication between cells, activation and deactivation of cellular processes and some regulatory processes can be modelled by means of dynamical systems, determined by the interaction of the species involved in each process. These species can be, for instance, different molecules or proteins. The interactions between them can be represented by directed graphs which are known in the biochemical context as Reaction Networks (RNs) [F]. The project AlgSignSen was designed to introduce some new ideas from Algebraic Geometry to the study of RNs, and in particular to the sensitivity analysis at steady state.
As an example, the following RN represents a possible way of interaction between two species, A and B: A+B→2B B→A
Equations that describe the evolution of such a process can be given, in terms of the concentrations of the species involved and some constants called reaction rate parameters. These rates carry information about the speed at which the system evolves. Interesting information about this evolution can be obtained by looking at the states (concentrations) in which the system is in equilibrium, the so-called steady states. Under certain assumptions, these states are the common solutions of a finite collection of polynomial equations, where the variables encode the concentrations of species and the coefficients depend on the reaction rates.
For the former network, we can obtain the polynomial equations -k_1 ab+k_2 b=0, k_1 ab-k_2 b=0.
The set of common solutions to these equations can be seen as a geometrical object, and geometrical objects which are defined by polynomial equations are the subject of study of Algebraic Geometry. The subset of those solutions which correspond to positive concentrations is called the positive steady state variety.
The steady state (or states) that can be reached by a system depends on the initial concentrations of the involved species: the initial amounts of the species evolve according to the specific dynamics of the system, and eventually reach a steady state. Different initial concentrations lead, under the same environmental conditions, to different steady states. An interesting question is: how do small changes in the initial concentrations affect the steady state reached in each case? This is what we refer to as the sensitivity of the system to (small) perturbations of the initial concentrations. A species X can see its concentration at steady state increased, decreased or untouched by the effect of a perturbation. This is the kind of sensitivities that we have investigated in the AlgSignSen project. When X is not affected (at steady state) by any perturbation of initial concentrations, we say that the system has zero sensitivity for X.
[F] M. Feinberg. Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, 202. Springer International Publishing, 2019
As an example, the following RN represents a possible way of interaction between two species, A and B: A+B→2B B→A
Equations that describe the evolution of such a process can be given, in terms of the concentrations of the species involved and some constants called reaction rate parameters. These rates carry information about the speed at which the system evolves. Interesting information about this evolution can be obtained by looking at the states (concentrations) in which the system is in equilibrium, the so-called steady states. Under certain assumptions, these states are the common solutions of a finite collection of polynomial equations, where the variables encode the concentrations of species and the coefficients depend on the reaction rates.
For the former network, we can obtain the polynomial equations -k_1 ab+k_2 b=0, k_1 ab-k_2 b=0.
The set of common solutions to these equations can be seen as a geometrical object, and geometrical objects which are defined by polynomial equations are the subject of study of Algebraic Geometry. The subset of those solutions which correspond to positive concentrations is called the positive steady state variety.
The steady state (or states) that can be reached by a system depends on the initial concentrations of the involved species: the initial amounts of the species evolve according to the specific dynamics of the system, and eventually reach a steady state. Different initial concentrations lead, under the same environmental conditions, to different steady states. An interesting question is: how do small changes in the initial concentrations affect the steady state reached in each case? This is what we refer to as the sensitivity of the system to (small) perturbations of the initial concentrations. A species X can see its concentration at steady state increased, decreased or untouched by the effect of a perturbation. This is the kind of sensitivities that we have investigated in the AlgSignSen project. When X is not affected (at steady state) by any perturbation of initial concentrations, we say that the system has zero sensitivity for X.
[F] M. Feinberg. Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, 202. Springer International Publishing, 2019
The project begun with the use of the approach in [Fel] to study sensitivities, in the search of algebraic conditions that guarantee positive/negative/zero sensitivities for a network, depending on its structure. Our research led soon to a connection with other robustness measurements, and we explored how this way of computing sensitivities could be helpful in the determination of the property ACR (Absolute Concentration Robustness) [SF]: a system has ACR in a species X if it has the same concentration of the species X at any possible steady state. For a system with ACR in X, the concentration of X when the system reaches a steady state is independent of the initial concentration of all species.
The property of ACR is of high interest and has been studied in the literature from an experimental and from a theoretical point of view, for specific biological mechanisms and in general ([A]), but no general procedure allows to detect it, outside of some very restrictive conditions in the network ([SF],[K]). Different attempts are currently being made to understand ACR ([CGK]).
We formalized the connection between zero sensitivity and ACR: ACR implies a very high degree of robustness against any perturbations of the initial concentrations. However, zero sensitivity does not imply ACR: even if small perturbations do not affect X at steady state, there still can be steady states for which X has different concentrations. This brought the definition of an intermediate property, local ACR, which is in many real examples equivalent to ACR.
We analyzed the geometric properties of the positive steady state variety and formulated the conditions under which zero sensitivity and local ACR coincide. We developed a practical criterion to check the existence of local ACR for a given dynamical system. Moreover, in the application to RNs, this criterion gave us a method to decide on the existence of local ACR for the network based on its structure, independently of the reaction rates.
The results can be found in [PF] and [PF2], and have been exposed in several events for mathematicians (seminars, SIAM conferences), biologists (SMB2021), and for the RN community in different seminars and conferences.
[A] U. Alon, M. G. Surette, N. Barkai, and S. Leibler. Robustness in bacterial chemotaxis. Nature, 397(6715):168-171, 1999
[CGK] D. Cappelletti, A. Gupta, M. Khammash. A hidden integral structure endows absolute concentration robust systems with resilience to dynamical concentration disturbances. J. R. Soc. Interface, 17(171):20200437, 2020
[Fel] E. Feliu. Sign-sensitivities for reaction networks: an algebraic approach. Math. Biosci. Eng., 16(6):8195–8213, 2019
[K] R. L. Karp, M. Pérez Millán, T. Dasgupta, A. Dickenstein, and J. Gunawardena. Complex-linear invariants of biochemical networks. J. Theoret. Biol., 311:130-138, 2012
[SF] G. Shinar and M. Feinberg. Structural sources of robustness in biochemical reaction networks. Science, 327(5971):1389–1391, 2010
The property of ACR is of high interest and has been studied in the literature from an experimental and from a theoretical point of view, for specific biological mechanisms and in general ([A]), but no general procedure allows to detect it, outside of some very restrictive conditions in the network ([SF],[K]). Different attempts are currently being made to understand ACR ([CGK]).
We formalized the connection between zero sensitivity and ACR: ACR implies a very high degree of robustness against any perturbations of the initial concentrations. However, zero sensitivity does not imply ACR: even if small perturbations do not affect X at steady state, there still can be steady states for which X has different concentrations. This brought the definition of an intermediate property, local ACR, which is in many real examples equivalent to ACR.
We analyzed the geometric properties of the positive steady state variety and formulated the conditions under which zero sensitivity and local ACR coincide. We developed a practical criterion to check the existence of local ACR for a given dynamical system. Moreover, in the application to RNs, this criterion gave us a method to decide on the existence of local ACR for the network based on its structure, independently of the reaction rates.
The results can be found in [PF] and [PF2], and have been exposed in several events for mathematicians (seminars, SIAM conferences), biologists (SMB2021), and for the RN community in different seminars and conferences.
[A] U. Alon, M. G. Surette, N. Barkai, and S. Leibler. Robustness in bacterial chemotaxis. Nature, 397(6715):168-171, 1999
[CGK] D. Cappelletti, A. Gupta, M. Khammash. A hidden integral structure endows absolute concentration robust systems with resilience to dynamical concentration disturbances. J. R. Soc. Interface, 17(171):20200437, 2020
[Fel] E. Feliu. Sign-sensitivities for reaction networks: an algebraic approach. Math. Biosci. Eng., 16(6):8195–8213, 2019
[K] R. L. Karp, M. Pérez Millán, T. Dasgupta, A. Dickenstein, and J. Gunawardena. Complex-linear invariants of biochemical networks. J. Theoret. Biol., 311:130-138, 2012
[SF] G. Shinar and M. Feinberg. Structural sources of robustness in biochemical reaction networks. Science, 327(5971):1389–1391, 2010
The theoretical development as well as the practical criterion and some illustrative examples can be found in [PF]. We formulated our results for a wider class of dynamical systems than those coming from Reaction Networks. This method provides a tool for biologists and chemists in order to easily test a network for local ACR. It makes it possible to understand which structures of a network allow this property with very basic knowledge on the network.
We explored the whole BIOMODELS database in the search for networks whose structure allows ACR. We found very few networks with an interesting structure for which ACR is possible, confirming that the property is uncommon. We further analysed theoretical models, considering all possible networks with bimolecular complexes up to n species and r reactions. We measured the abundance of local ACR for different choices of n and r. The results of this analysis, together with the practical algorithm that can be used by non-mathematicians, will soon be available in [PF2].
On a different use of algebraic techniques to the study of RNs, [PH] provides conditions for a desired behaviour of the probability distribution of the states, when the network is studied from a stochastic point of view.
[PF] Local and global robustness at steady state, with E. Feliu. Math Meth Appl Sci. 2021; 1- 24
[PF2] Local ACR in small networks. B. Pascual-Escudero, E. Feliu (in preparation)
[PH] An algebraic approach to product-form stationary distributions for some reaction networks, B. Pascual-Escudero, L. Hoessly. (arXiv:2012.03227)
We explored the whole BIOMODELS database in the search for networks whose structure allows ACR. We found very few networks with an interesting structure for which ACR is possible, confirming that the property is uncommon. We further analysed theoretical models, considering all possible networks with bimolecular complexes up to n species and r reactions. We measured the abundance of local ACR for different choices of n and r. The results of this analysis, together with the practical algorithm that can be used by non-mathematicians, will soon be available in [PF2].
On a different use of algebraic techniques to the study of RNs, [PH] provides conditions for a desired behaviour of the probability distribution of the states, when the network is studied from a stochastic point of view.
[PF] Local and global robustness at steady state, with E. Feliu. Math Meth Appl Sci. 2021; 1- 24
[PF2] Local ACR in small networks. B. Pascual-Escudero, E. Feliu (in preparation)
[PH] An algebraic approach to product-form stationary distributions for some reaction networks, B. Pascual-Escudero, L. Hoessly. (arXiv:2012.03227)