Law of localization in chemical reaction networks

In living cells, chemical reactions are connected by sharing their products and substrates, and form complex networks, e.g. metabolic pathways. Here we developed a theory to predict the sensitivity, i.e. the responses of concentrations and fluxes to perturbations of enzymes, from network structure alone. Responses turn out to exhibit two characteristic patterns, $localization$ and $hierarchy$. We present a general theorem connecting sensitivity with network topology that explains these characteristic patterns. Our results imply that network topology is an origin of biological robustness. Finally, we suggest a strategy to determine real networks from experimental measurements.


I. INTRODUCTION
Cells have many chemical reactions, each of which is mediated by organic catalysts, enzymes. Reactions are not independent but connected and form complex networks.
The dynamics of chemical concentrations are considered as the origin of physiological functions. However, dynamical behavior based on the network is not understood well.
One experimental approach to study such network systems is sensitivity analysis where the amount or activity of enzyme is perturbed and responses (concentrations of chemicals in the system) are measured [1] (see Fig. 1). However, the results of such experiments are very difficult to interpret, because theoretical criteria to evaluate the results of perturbations from network structures are not established. There are other difficulties in understanding dynamical behaviors of reaction systems in biology. First, although a huge amount of information of reaction networks is available on databases [2][3][4], they provide no more than knowledge of identified reactions in biochemistry. It is possible that the information is incomplete, including many unidentified reactions or regulations. Second, in spite of the recent progress in biosciences, it is still difficult or almost impossible to determine quantitative details of the dynamics, such as functions for reaction rates, parameter values, or initial states.
In order to circumvent these difficulties, we introduce a mathematical method, named structural sensitivity analysis [13,14], to determine responses of chemical reaction systems to the perturbation of the enzyme amount or activity based only on network structure.
From analyses we found that qualitative responses at a steady state are determined from information of network structure only. We also found that response patterns, e.g. distribution of nonzero responses of chemical concentrations in the network, exhibit two characteristic features, localization and hierarchy depending on the structure of networks and position of perturbed reactions. Finally we found a general theorem connecting the network topology and the response patterns directly, and governing the characteristic patterns of responses. This theorem, which we call the law of localization, is not only theoretically important, but also practically useful for examining real biological systems.
In the context of adaptation, there were some previous studies, which reported confined nonzero responses in specific systems [9][10][11][12]. However, they did not find general laws of such response patterns, nor any topological conditions.

II. STRUCTURAL SENSITIVITY ANALYSIS
We study concentration changes in a reaction system under perturbation of reaction rate parameters, assuming that the system is in a steady state [13,14]. We label chemicals by m (m = 1, . . . , M ) and reactions by j (j = 1, . . . , R). A state of the system is specified by concentrations x m (t) and obeys the following differential equations [15,16] Here, ν is called a stoichiometric matrix. W i is called a flux, which depends on metabolite concentrations and also on a reaction rate k i . We do not assume specific forms for W i , but assume that each W i is an increasing function of its substrate concentration; Below, we abbreviate and emphasize nonzero ∂W i ∂xm as r im . In this framework, enzyme knockdown of the j-th reaction corresponds to changing the rate as k j → k j + δk j (triangles in FIG. 2). By assuming steady state [5][6][7][8], the flux is expressed, in terms of a basis { c n } of ker ν, as W = N k n=1 µ n c n , where N k is the dimension of the kernel and µ n are N k coefficients depending on reaction rates. Under the j-th knockdown, we have The i-th component of δ j W is also expanded as From (3) (4), the response of steady state concentration δ j x ≡ d x dk j δk j (circles in FIG. 2) and flux δ j W (arrows in FIG. 2) to each perturbation k j → k j + δk j is determined from network structure only [13,14]. The result for each perturbation is simultaneously obtained through the following matrix computation: where the matrix A is given as In (5) and (6), the horizontal and vertical lines are the partitions of the matrices. We then obtain the flux change δ j W from (3), or in a matrix notation. We call the inverse of A as the sensitivity matrix S. Note that δ j x, δ j µ, δ j W , c n are column vectors with M, N k , R, and R components respectively, and ∂W i ∂xm is an R-by-M matrix. We assume networks with dim ker ν T = 0 throughout this paper, which guarantees the matrix A is square, i.e. R = M + N k .
Comments are in order. First, our theory depends only on the structure of reaction networks. The network structure is reflected in the distribution of nonzero entries in the A-matrix, which determines the qualitative responses. Second, as a generalization of our method, we account for regulations such as allosteric effects by relaxing (2) as Then, regulations add additional r im in the A-matrix, and the response is still determined through (5).

III. LOCALIZATION AND HIERARCHY
Let us see some results of structural sensitivity analysis.
The flux changes only when we perturb the top reaction 1 (the 1st column of S). The perturbation to reactions 2 or 3 changes only its substrate concentration (the 2nd, 3rd column of S).
Again, only the perturbation to the input rate, corresponding to the 1st column in   Proof.-The theorem is proved from the distribution of nonzero entries of the A-matrix.
(i) Suppose a subnetwork Γ is a buffering structure. Then by appropriately choosing a basis of the kernel of ν and the orderings of the indices of the A-matrix, we can always rewrite the A-matrix as The lower left block vanishes because A is output-complete. (ii) As explained already, the concentration change δ j x m is proportional to A −1 mj ∝ DetÂ (j;m) , whereÂ (j;m) is the minor matrix associated with the row of the j-th reaction and the column of the m-th metabolite. Then, DetÂ (j;m) = 0 for i ∈ r, m / ∈ m follows because the upper left block in the minorÂ (j;m) , which was originally square in (12), is horizontally long.
We illustrate the law of localization in the example networks in FIG. 2 and FIG. 3 (Left). should be limited within Γ 4 , which is observed in the 4th column in (10). Similarly, the response to perturbation of reaction 6 is explained by Γ 5 . from network topology, surprisingly overlap biologically identified sub-circuits, the pentose phosphate pathway (yellow in FIG. 4, 5), the tricarboxylic acid cycle (blue) and the glycolysis (green). This correspondence may be understood from an evolutional point of view by considering the advantage of buffering structures.

V. DISCUSSIONS AND CONCLUSIONS
Here we discuss the biological significances of buffering structures (and nest of them) in two different levels. The first discussion is on the physiological importance. A buffering structure prohibits influence of given perturbation from expanding to the outside, like a "firewall." In other words, it is a substructure with robustness emerging from the network topology. The carbon metabolism network of E. coli possesses multiple nested firewalls (FIG. 5), and are expected to be robust to fluctuations of enzymes in it. We expect that such a topological characteristic of reaction networks could be the evolutionary origin of homeostasis in biological systems. A set of chemical reactions satisfying the condition of buffering structure by chance in evolutionarily early time would be positively selected as an advantageous circuit. We then expect that buffering structures in existing biological networks today might be generated and selected in such ways.