Network topology creates independent control of multiple checkpoints in the cell cycle system

Significance

Cells possess various biological functions, which are appropriately regulated in response to external signals and environment. Functions arise from dynamics of a single system composed of interconnected chemical reactions. Intuitively, controlling one function may affect others. We theoretically demonstrate that when a subnetwork of a reaction network satisfies a certain topological condition, it becomes a controllable module independent of others. This study reveals that such modules actually exist in yeast cell cycle system, enabling independent control of critical timing in the cell cycle, separate from other functions. Furthermore, by comparing theory with experiments, we predict existence of an unknown reaction and confirm its existence. Our method, based solely on topology, enables us to update network knowledge through prediction and verification.

Abstract

In living cells, numerous chemical reactions are interconnected by sharing substrates and products, forming a huge reaction network. The various functions of cells emerge from the dynamics of such interconnected system. Cells regulate the concentrations of key biochemicals by controlling the amount or activity of enzymes that catalyze each reaction, thereby achieving control of cellular functions. However, in such an interconnected system, can different chemicals responsible for different biological functions be controlled independently? If so, by what mechanism? This paper mathematically demonstrates that “modularity,” where parts of a system are controlled independently of others, arises solely from network topology. Furthermore, using the cell cycle system as an example, we show through a combination of theory and experiments that such “regulatory modules” actually exist in living organisms, performing important roles. In the cell cycle, the G₁-S and G₂-M transitions are strictly controlled by distinct protein complexes, requiring the specific activation of different complexes at different phases. This suggests that different transitions should be independently controlled. However, two cell-cycle-control complexes share a common protein component, raising the question of how phase-specific control is achieved. Analysis of a known cell cycle network using a topology-based theory revealed that the two complexes belong to different regulatory modules. Experimental verification confirms the existence of a module. Moreover, by comparing theoretical predictions with experimental verification, we theoretically predict the necessity of an unknown reaction and experimentally confirm it. This prediction and verification approach using model-free theory enables the updating of the network information.

Within living cells, various types of chemical reactions occur. It is known that these reactions do not work in isolation; instead, multiple reactions operate in a chain-like manner by sharing substrates and products. Information on reactions identified through experimental methods is compiled in multiple databases (1–6). According to these databases, the numerous types of reactions operating within living organisms form a single, interconnected large network. Fig. 1A is a schematic map illustrating the outline of such a chemical reaction network.

Fig. 1.

A buffering structure acts as a regulatory module in chemical reaction systems. (A) Schematic representation of a metabolic pathway from the KEGG database (1). Multiple pathways are connected to form a single connected network. (B) Schematic of a buffering structure. A subnetwork that satisfies certain topological conditions—most importantly, the index λ:= − (number of chemicals) + (number of reactions) − (number of cycles) + (number of conserved quantities), is equal to 0—is called a buffering structure. The effects of a perturbation within a buffering structure are confined to inside the structure at steady state. (C) Graphical representation of a chemical reaction system comprising of two chemicals (P,Q) and three reactions (1,2,3). Solid lines indicate chemical reactions. The red triangle indicates the reaction that is activated. The signs (increase or decrease) of responses are represented by + and − for chemicals, while chemicals without signs remain unchanged. The subnetwork enclosed by a red box is a buffering structure, with the index λ being equal to 0. (D) A numerical simulation of the network in (C). The concentrations of the chemicals (P, Q) and reaction rates of reaction 2 ($r_2$) are plotted. Reaction rate functions and parameters are set as described in SI Appendix, section 1.2. The dynamics are in the steady state at $t=12$. At $t=12$, $k_2$ is increased by a factor of 2. (E) Intuitive understanding of buffering structures using the water tank analogy.

The dynamics of chemical substances arising from such reaction networks are considered the origin of cellular functions. For example, at the center of Fig. 1A, a subsystem called the central metabolism exists. Through a series of reactions contained within this system, glucose taken up into the cell is decomposed into smaller molecules, and chemical energy is extracted during this process. Furthermore, each reaction within living cells is catalyzed by enzymes. By regulating the amount or activity of each enzyme, organisms control the dynamics of chemical reactions and, consequently, regulate cellular physiological functions. In addition to the central metabolic pathway, the chemical reaction network within living organisms includes numerous other subsystems with distinct biological functions, such as the lipid metabolism pathway and the amino acid synthesis pathway (1).

The above constitutes fundamental knowledge in modern biology, yet these facts harbor the following mystery. Cells must individually control biological functions in response to their environment. For example, under nutrient-poor conditions, cells activate the central metabolic pathway to obtain chemical energy more efficiently. However, a change in the behavior of one part of an interconnected system could affect other parts because of the single interconnected network. For instance, activating the central metabolic system might alter the functioning of lipid metabolism or amino acid synthesis. For cells possessing diverse functions, could using a single interconnected system be disadvantageous? In other words, are the different functions arising from one system inevitably coupled when controlled, or can they be controlled independently? If independent control is possible, what mechanism enables it?

We have addressed this question using a mathematical approach. In a previous study, we developed a theoretical framework—structural sensitivity analysis (SSA)— which determines the qualitative response of steady-state chemical concentrations to changes in enzyme amount or activity solely from the network topology, without requiring any assumptions about parameters or reaction rate function forms (7). Based on SSA, we derived a general mathematical law describing the emergence of independent control in chemical reaction systems: the “law of localization” (8, 9). Under certain topological conditions, the effects of a change in an enzymatic parameter are localized to a limited part of the network and do not propagate beyond it. These conditions are given solely by network topology, independent of the choices of reaction rate functions (Fig. 1B). More precisely, the subnetwork $\mathrm{\Gamma }=({\mathfrak{m}}_{\mathrm{\Gamma }},{\mathfrak{r}}_{\mathrm{\Gamma }})$ (${\mathfrak{m}}_{\mathrm{\Gamma }}$ is a subset of chemical species and ${\mathfrak{r}}_{\mathrm{\Gamma }}$ is a subset of reactions) is defined as a buffering structure if it satisfies the following two conditions:

1. All emanating reaction arrows from ${\mathfrak{m}}_{\mathrm{\Gamma }}$ are included within ${\mathfrak{r}}_{\mathrm{\Gamma }}$.

2. The index

   $$\lambda (\mathrm{\Gamma }):=-|{\mathfrak{m}}_{\mathrm{\Gamma }}|+|{\mathfrak{r}}_{\mathrm{\Gamma }}|-N({\mathfrak{r}}_{\mathrm{\Gamma }})+N_c({\mathfrak{m}}_{\mathrm{\Gamma }})$$

   equals zero.

Here, $N({\mathfrak{r}}_{\mathrm{\Gamma }})$ is the number of cycles consisting of reactions in ${\mathfrak{r}}_{\mathrm{\Gamma }}$, and $N_c({\mathfrak{m}}_{\mathrm{\Gamma }})$ is the number of independent conserved quantities that include at least one element in ${\mathfrak{m}}_{\mathrm{\Gamma }}$. The law of localization states that any perturbation to its internal parameters affect only the concentrations of its internal chemicals ${\mathfrak{m}}_{\mathrm{\Gamma }}$ and reaction rates of internal reactions ${\mathfrak{r}}_{\mathrm{\Gamma }}$ at steady state, exerting no influence externally (8, 9). Conversely, when the effect of parameter changes is confined to a limited part of a network, a corresponding buffer structure always exists (10, 11).

For example, in the network shown in Fig. 1C, where reaction 2 uses P as the substrate and produces Q as the product, SSA shows that the steady-state concentration of chemical P decreases as a result of an increase in the enzyme activity catalyzing reaction 2 (Fig. 1 C and D and SI Appendix, Fig. S1 A and B). The steady-state concentration of chemical Q is not affected by this perturbation. These responses are understood by the buffering structure that encompasses reaction 2 and chemical P but not Q.

The function of a buffering structure can be understood intuitively as the behavior of a water tank with an inflow and an outflow. In Fig. 1E, increasing the enzyme activity of reaction 2 corresponds to enlarging the outlet. Then the outflow of water naturally increases, reducing the water level. The decrease in the water level induces a decrease in the water pressure at the bottom of the tank, which in turn decreases the outflow. This process stops when the inflow and outflow are in balance, realizing a new steady state, where the only net change is a decrease in the water level. In other words, the increase in the outlet size (enzyme activity) is compensated by the decrease in the water level (chemical concentration), and there is no change in the downward flow at steady state.

In chemical reaction systems, the phenomenon where the effect of parameter changes is confined to specific chemical concentrations at steady state is known as perfect adaptation(12–18). Our law of localization demonstrates that this phenomenon arises purely from the network topology and provides a general theoretical principle for such behavior.

The significance of the law of localization becomes clearer when viewed from an evolutionary perspective. If a subnetwork that happens to satisfy topological conditions arises during evolution, it naturally acquires modularity, enabling it to behave independently of other parts of the network. Furthermore, the only origin of this behavioral modularity is the buffer structure. If a subnetwork whose independent control enhances cellular fitness exists, that subnetwork will likely evolve into a buffering structure. Such buffering structures, once established, are also expected to be preserved during evolution.

In this study, we address the following questions: Do buffering structures exist within the chemical reaction systems of living cells, and do they perform biological functions?

We choose the cell cycle system as a good example where multiple functions arise from a single connected reaction system. Progression through the cell cycle (${\mathrm{G}}_1$/S/${\mathrm{G}}_2$/M) is strictly controlled to ensure accurate genome replication and its subsequent distribution to the two daughter cells (19). The ${\mathrm{G}}_1$-S and ${\mathrm{G}}_2$-M boundaries are called ${\mathrm{G}}_1$-S and ${\mathrm{G}}_2$-M checkpoints, respectively. Transitions through these checkpoints are driven by different cyclin-dependent kinase (CDK)-cyclin complexes. In the fission yeast Schizosaccharomyces pombe, increases in the concentration of CDK-Cig2 and CDK-Cdc13 trigger the transitions through the ${\mathrm{G}}_1$-S and ${\mathrm{G}}_2$-M checkpoints, respectively (20) (Fig. 2A).

Fig. 2.

The cell cycle system. (A) Schematic representation of the cell cycle system in fission yeast. (B) As CDK-Cig2 and CDK-Cdc13 share the common CDK, it was intuitively hypothesized that these two complexes compete for CDK binding.

Notably, these two CDK–cyclin complexes are controlled by a same chemical reaction system (Fig. 3A). They differ in their cyclin components (Cig2 and Cdc13) but share the same CDK (Cdc2). In addition, the total amount of CDK remains almost constant throughout the cell cycle (21). Based on these facts, it has been hypothesized that Cig2 and Cdc13 may compete for CDK binding (22). Namely, when the expression level of Cig2 increases, not only CDK-Cig2 concentration will increase, but CDK-Cdc13 concentration might decrease. Conversely, if the expression level of Cdc13 increases, not only CDK-Cdc13 concentration will increase, but CDK-Cig2 concentration might decrease (Fig. 2B). In addition, as shown by time-lapse observations of cyclins (21), the expression of Cig2 is not completely phase-specific, and there is a temporal overlap between the expressions of the two cyclins in the cell cycle. If the hypothetical competition should exist, the increased synthesis rate of Cdc13 during the ${\mathrm{G}}_1$-S transition phase would result in a decrease in CDK-Cig2 concentration, potentially inhibiting the ${\mathrm{G}}_1$-S transition. However, the responses of the CDK–cyclin complexes to the changes in cyclin synthesis rates have not yet been confirmed from either theoretical or experimental viewpoints.

Fig. 3.

Analysis based on network structure demonstrates that CDK-Cig2 and CDK-Cdc13 are independently regulated. (A) A graphical representation of the cell cycle network. Each node indicates a chemical and each edge indicates a reaction (solid arc). Arrow-headed and line-headed dashed arc represents activating or inhibitory regulation, respectively. A reaction rate parameter (enzyme activity) for each reaction ($k_n$) is depicted above or below the edge. Green arrows indicate reactions whose enzyme modulations affect the steady-state concentration of CDK-Cig2. Blue arrows indicate reactions whose enzyme modulations affect the steady-state concentration of CDK-Cdc13. The subnetwork enclosed by the green curve is a buffering structure containing CDK-Cig2 but not CDK-Cdc13 (BS1). The subnetwork enclosed by the blue curve is a buffering structure containing CDK-Cdc13 but not CDK-Cig2 (BS2). The intersection of BS1 and BS2 is also a buffering structure. The union of the intersection and the gray-enclosed region is another buffering structure, and the purple-enclosed regions are also buffering structures. (B and C) Numerical calculation of the network dynamics are performed to illustrate the results in (A). Reaction rate functions are assumed to be combinations of mass action and Michaelis–Menten kinetics. Initial values are set to a steady state of the system. At $t=150$, the Cig2 synthesis rate [$k_{19}$, (B)] or the Cdc13 synthesis rate [$k_7$, (C)] is increased by a factor of 2. The units of time are arbitrary units. (D) Numerical calculation of the network dynamics is conducted to recapitulate the cell cycle dynamics. We assume that the value of $k_{19}$ or $k_7$ change with time as shown Below. (E) A graphical representation of a simplified cell cycle network, consisting of cyclin synthesis reactions, CDK–cyclin complex formation and dissociation reactions, and cyclin degradation reactions. (F) Summary of theoretical predictions based on the network structure.

We analyze the cell cycle network from a literature (23) using SSA and reveal that two cyclin–CDK complexes are contained within distinct buffering structures. This suggests that the concentrations of each CDK–cyclin complex are regulated independently from each other.

To experimentally verify this prediction, we generate fission yeast mutants in which the synthesis rate of either Cig2 or Cdc13 is increased, and measure the resulting changes in the concentrations of CDK-Cig2 and CDK-Cdc13 using a recently developed quantitative technique (24). The results show that the concentration of CDK-Cdc13 is regulated by the Cdc13 synthesis rate, independently from CDK-Cig2. This demonstrates that independent control of the essential cellular function is achieved via buffering structures within the cell cycle system; the law of localization is not merely theoretical hypothesis but operates in living systems.

Conversely, the change in the Cig2 synthesis rate not only affects CDK-Cig2 but also CDK-Cdc13, resulting in deviations from the predictions. However, the discrepancy between the theoretical predictions derived solely from the network topology and the experimental results indicates the need to revise the network structure. We theoretically predict the existence of a Cdc13 degradation pathway that is not present in the original network data. We subsequently confirmed its existence through experiments.

These results, obtained through theoretical prediction and experimental verification, have not only deepened our understanding of cell cycle function. The prediction-verification research using model-free theory proposes an approach to systems biology that advances by comparing network structures—the blueprints—with experimental results, confirming our understanding step by step.

Results

Structural Sensitivity Analysis (SSA) of the Cell Cycle Network Predicts Independent Control of CDK-Cig2 and CDK-Cdc13.

Here, we study the network structure of the cell cycle system proposed in a previous study (23) (Fig. 3A). The reaction network provides the fundamental understanding of the cell cycle system in fission yeast, and essentially the same network is used in the latest modeling research (25). This system consists of 20 chemicals and 24 reactions:

$$\begin{array}{cc}\hfill R_1& :\mathrm{CDK}-\mathrm{Cdc}13\stackrel{k_1\text{; Wee1}}{\to }\mathrm{CDK}-\mathrm{Cdc}13-\mathrm{p},\hfill \\ \hfill R_2& :\mathrm{CDK}-\mathrm{Cdc}13-\mathrm{p}\stackrel{k_2\text{; Cdc25-p}}{\to }\mathrm{CDK}-\mathrm{Cdc}13,\hfill \\ \hfill R_3& :\mathrm{Wee}1-\mathrm{p}\stackrel{k_3}{\to }\mathrm{Wee}1,\hfill \\ \hfill R_4& :\mathrm{Wee}1\stackrel{k_4\text{; CDK-Cdc13}}{\to }\mathrm{Wee}1-\mathrm{p},\hfill \\ \hfill R_5& :\mathrm{Cdc}25-\mathrm{p}\stackrel{k_5}{\to }\mathrm{Cdc}25,\hfill \\ \hfill R_6& :\mathrm{Cdc}25\stackrel{k_6\text{; CDK-Cdc13}}{\to }\mathrm{Cdc}25-\mathrm{p},\hfill \\ \hfill R_7& :0\stackrel{k_7}{\to }\mathrm{Cdc}13,\hfill \\ \hfill R_8& :\mathrm{CDK}+\mathrm{Cdc}13\stackrel{k_8}{\to }\mathrm{CDK}-\mathrm{Cdc}13,\hfill \\ \hfill R_9& :\mathrm{CDK}-\mathrm{Cdc}13\stackrel{k_9;{\mathrm{APC}}^{\mathrm{Slp}1},{\mathrm{APC}}^{\mathrm{Ste}9}}{\to }\mathrm{CDK}+\mathrm{Cdc}13-\mathrm{ub},\hfill \\ \hfill R_{10}& :\mathrm{Cdc}13-\mathrm{ub}\stackrel{k_{10}}{\to }0,\hfill \\ \hfill R_{11}& :\mathrm{APC}+\mathrm{Slp}1\stackrel{k_{11}\text{; CDK-Cdc13}}{\to }{\mathrm{APC}}^{\mathrm{Slp}1},\hfill \\ \hfill R_{12}& :{\mathrm{APC}}^{\mathrm{Slp}1}\stackrel{k_{12}}{\to }\mathrm{APC}+\mathrm{Slp}1,\hfill \\ \hfill R_{13}& :0\stackrel{k_{13}}{\to }\mathrm{Rum}1,\hfill \\ \hfill \end{array}$$

$$\begin{array}{cc}\hfill R_{14}& :\mathrm{CDK}-\mathrm{Cdc}13+\mathrm{Rum}1\stackrel{k_{14}}{\to }\mathrm{CDK}-\mathrm{Cdc}13-\mathrm{Rum}1,\hfill \\ \hfill R_{15}& :\mathrm{CDK}-\mathrm{Cdc}13-\mathrm{Rum}1\stackrel{k_{15}}{\to }\mathrm{CDK}-\mathrm{Cdc}13+\mathrm{Rum}1,\hfill \\ \hfill R_{16}& :\mathrm{Rum}1\stackrel{k_{16};\text{CDK-Cdc13, CDK-Cig2}}{\to }0,\hfill \\ \hfill R_{17}& :{\mathrm{APC}}^{\mathrm{Ste}9}\stackrel{k_{17}\text{; CDK-Cdc13}}{\to }\mathrm{APC}+\mathrm{Ste}9,\hfill \\ \hfill R_{18}& :\mathrm{APC}+\mathrm{Ste}9\stackrel{k_{18}}{\to }{\mathrm{APC}}^{\mathrm{Ste}9},\hfill \\ \hfill R_{19}& :0\stackrel{k_{19};(\mathrm{CDK}-\mathrm{Cig}2)}{\to }\mathrm{Cig}2,\hfill \\ \hfill R_{20}& :\mathrm{CDK}+\mathrm{Cig}2\stackrel{k_{20}}{\to }\mathrm{CDK}-\mathrm{Cig}2,\hfill \\ \hfill R_{21}& :\mathrm{CDK}-\mathrm{Cig}2\stackrel{k_{21}}{\to }\mathrm{CDK}+\mathrm{Cig}2-\mathrm{ub},\hfill \\ \hfill R_{22}& :\mathrm{CDK}-\mathrm{Cig}2+\mathrm{Rum}1\stackrel{k_{22}}{\to }\mathrm{CDK}-\mathrm{Cig}2-\mathrm{Rum}1,\hfill \\ \hfill R_{23}& :\mathrm{CDK}-\mathrm{Cig}2-\mathrm{Rum}1\stackrel{k_{23}}{\to }\mathrm{CDK}-\mathrm{Cig}2+\mathrm{Rum}1,\hfill \\ \hfill R_{24}& :\mathrm{Cig}2-\mathrm{ub}\stackrel{k_{24}}{\to }0.\hfill \end{array}$$

Above the arrow of each reaction $R_n$, we display $k_n$, $M_n^{+}$ and $M_n^{-}$. Here, $k_n$ is the reaction rate parameter for $R_n$. $M_n^{+}$ and $M_n^{-}$ represent the sets of chemicals that are not substrates of the reaction $R_n$ but regulate the reaction rate positively and negatively, respectively. $M_n^{-}$ is indicated in parentheses and is present only for $R_{19}$ (CDK-Cig2).

The values of each $k_n$ change over the course of the cell cycle as a consequence of various biological factors, including enzyme synthesis and degradation rates, changes in cell size (26), and alterations in enzyme spatial distribution during cell growth (27). In particular, changes in the transcription and translation rates of cyclins (Cig2 and Cdc13) are considered the major driving force behind cell cycle progression (23, 28–30). In other words, the reaction parameters for Cig2 synthesis ($k_{19}$) and Cdc13 synthesis ($k_7$) are thought to change in a stage-dependent manner. While the yeast cell cycle takes around two hours, key reactions in the network, including phosphorylation and protein–protein binding reactions, are known to occur on substantially faster timescales (31–34). Therefore, under given parameter values, it can be assumed that the system quickly reaches a steady state (quasi-steady state): Temporal changes in the reaction parameters of cyclin synthesis ($k_7$ or $k_{19}$) are slow and thus are thought to cause shifts in this steady state.

Using SSA, we systematically identify the reaction parameters that affect the concentration of CDK-Cig2 or CDK-Cdc13 (Table 1 and Fig. 3A). We find that only a small number of reaction parameters can have an impact on these concentrations (Table 1 and Fig. 3A). Even more remarkably, there is no overlap between the reaction parameters affecting CDK-Cig2 and those affecting CDK-Cdc13 (Table 1 and Fig. 3A). In particular, increasing the synthesis rate of Cig2 ($k_{19}$) raises the steady-state concentration of CDK-Cig2 but has no effect on that of CDK-Cdc13. Similarly, increasing the synthesis rate of Cdc13 ($k_7$) raises the steady-state concentration of CDK-Cdc13 but does not affect that of CDK-Cig2. These results are different from the naïve prediction based on the competition model.

Table 1.

Structural sensitivity analysis (SSA) of the fission yeast cell cycle network

A sensitivity matrix indicates qualitative changes in steady-state concentrations of each chemical (row) in response to the increase in each parameter (column). The symbol “0” denotes no change, while the symbols “+” and “−” represent an increase and decrease, respectively. The symbol “±” indicates that the steady-state concentration changes, but the direction of the change cannot be determined solely from the network structure and depends on reaction rate functions or parameter values. Reaction parameters shown in green or blue represent parameters whose modulations alter the steady-state concentration of CDK-Cig2 or CDK-Cdc13, respectively. The sensitivity matrix including perturbations of conserved quantities is shown in SI Appendix.

To illustrate the SSA prediction, we perform numerical simulations of the cell cycle system dynamics by assigning mass action or Michaelis–Menten reaction rate functions to each reaction (Fig. 3 B and C). During the simulation, we increase the Cig2 synthesis rate ($k_{19}$) stepwise at a specific point. This perturbation causes CDK-Cig2 concentration to increase, and it converged over time to a new steady-state value higher than the original (Fig. 3B). In contrast, this perturbation causes a transient change in CDK-Cdc13 concentration, but it quickly returns to the original value, and the steady-state value does not change from before the perturbation (Fig. 3B). Similarly, when the Cdc13 synthesis rate ($k_7$) is increased, CDK-Cdc13 concentration increases and converges to a new steady-state value higher than the original. (Fig. 3C). In contrast, CDK-Cig2 concentration temporarily increases, but quickly returns to its original value, and the steady-state value does not change from before the perturbation (Fig. 3C).

We also demonstrate how the concentrations of CDK–cyclin complexes change as the cell cycle progresses due to changes in reaction parameters (Fig. 3D). As the cell cycle progresses, the reaction parameter for Cig2 synthesis ($k_{19}$) is thought to first increase and then decrease (23, 28, 29); thus, we change $k_{19}$ accordingly. CDK-Cig2 concentration increases and decreases in response to the changes in the Cig2 synthesis rate, but CDK-Cdc13 concentration shows only a slight change. Similarly, the reaction parameter for Cdc13 synthesis ($k_7$) is thought to change in the next stage of the cell cycle (23, 26, 28); thus, we change $k_7$ in the later stage. During this phase, CDK-Cdc13 concentration increases and decreases in accordance with the changes in the Cdc13 synthesis rate, but CDK-Cig2 concentration only shows a slight change. In summary, if the known network information is correct, the concentrations of CDK-Cig2 and CDK-Cdc13 are controlled independently of each other (Fig. 3F).

Independent Control Is Achieved by Distinctive Buffering Structures.

We next aim to understand the mechanisms underlying independent control of CDK-Cig2 and CDK-Cdc13, specifically from the viewpoints of buffering structures (Fig. 3A). We identified two crucial buffering structures that play a pivotal role in the control of CDK–cyclin complexes. One is the buffering structure containing CDK-Cig2 but not CDK-Cdc13 (Fig. 3A and SI Appendix, Fig. S2A, green region, BS1), and the other is the buffering structure containing CDK-Cdc13 but not CDK-Cig2 (Fig. 3A and SI Appendix, Fig. S2A, blue region, BS2). BS1 indicates that any change in reaction parameters within BS1 does not affect the steady-state concentration of CDK-Cdc13, while influencing that of CDK-Cig2. For example, since the reaction parameter for Cig2 synthesis ($k_{19}$) is contained within BS1, activating transcription and translation of Cig2 cannot affect the steady-state concentration of CDK-Cdc13, consistent with the SSA results. Similarly, BS2 indicates that any parameter change in BS2 does not affect the steady-state concentration of CDK-Cig2, while influencing CDK-Cdc13. For example, activating the Cdc13 synthesis rate ($k_7$) cannot affect the steady-state concentration of CDK-Cig2.

All reactions in the system are contained within either of these two buffering structures. A mathematical proof guarantees that the intersection of these two buffering structures is also a buffering structure (35). Since CDK-Cig2 and CDK-Cdc13 exist outside this intersection, the reactions within the intersection do not affect either of these two complexes. Within BS1 and BS2, we found other buffering structures that do not contain either CDK-Cig2 or CDK-Cdc13 (Fig. 3A, gray and purple regions), further narrowing down the set of reactions that influence either CDK-Cig2 or CDK-Cdc13. These results indicate that independent control of the two complexes is achieved through two distinct buffering structures, one containing CDK-Cig2 and the other containing CDK-Cdc13.

The Effect of a Change in a Conserved Quantity.

In addition to changes in reaction rate parameters, SSA can also be used to analyze the response of a system caused by changes in conserved quantities within the system (9). In this cell cycle network, the total concentration of CDK, i.e., the sum of free CDK and its binding complexes, is set to be constant over time (SI Appendix, Fig. S3A), consistent with a previous experimental report (21). In reality, the concentration of this molecule may not be perfectly constant. To make sure that this uncertainty does not affect our conclusions, we performed the following analysis. Using SSA, we determined the qualitative changes in the concentration of each chemical when the total CDK amount changes. Among the six chemicals containing CDK, SSA predicted that only free CDK is affected by the change in the total CDK amount, while the other five, including CDK-Cig2 and CDK-Cdc13, remain unaffected. The results of the numerical simulation also confirmed the SSA prediction (SI Appendix, Fig. S3 B and C): The increase or decrease in the total amount of CDK does not affect the concentrations of either of these complexes, and only the concentration of free CDK changes.

These results are attributable to the existence of a buffering structure that includes CDK but not the other five CDK-containing complexes (SI Appendix, Fig. S3D). This buffering structure is a part of the intersection of BS1 and BS2. This result can be rephrased as follows: Although we may intuitively expect that an increase in the total CDK amount would upregulate both CDK-Cig2 and CDK-Cdc13, this is not the case at all; rather, neither CDK-Cig2 nor CDK-Cdc13 concentrations change. In other words, the amount of CDK expression cannot be a driving factor for either the ${\mathrm{G}}_1$-S or ${\mathrm{G}}_2$-M transitions.

Cells require a minimum total CDK amount to maintain a steady state, and the threshold can be calculated theoretically. When the total CDK amount is reduced below this thres hold, the steady state disappears (SI Appendix, Fig. S3C and section S4).

The Simplified Network Shows the Same Behavior as the Original Network.

We considered that the essence of the yeast cell cycle system lies in the following properties. A reaction system comprising three chemicals (A, B, C) satisfies the following conditions: i) the total amount of chemical A is conserved; ii) chemical B is synthesized, binds to A, subsequently dissociates from A, and is then degraded; iii) chemical C is similarly synthesized, binds to A, subsequently dissociates from A, and is then degraded. Then, we constructed the simplest cell cycle system consisting only of cyclin synthesis, CDK–cyclin complex formation, and dissociation, and performed SSA (Fig. 3E). When we performed SSA on this system, it exhibited exactly the same behavior as the original system. Similar to the original cell cycle system, we found that the parameters affecting the concentration of CDK-Cig2 and those affecting the concentration of CDK-Cdc13 did not overlap, indicating that these complexes are controlled independently of each other. Moreover, the total amount of CDK does not affect the concentration of either of these two complexes, and cells require a minimum total CDK amount to sustain a steady state (See SI Appendix, section 1.4 for details). These results are attributable to the fact that the simplified system retains the same BS1 (containing CDK-Cig2 but not CDK-Cdc13) and BS2 (containing CDK-Cdc13 but not CDK-Cig2) as the original system (Fig. 3E).

Experimental Perturbations Confirm That CDK-Cdc13 Is Controlled Independently from CDK-Cig2.

Do the behaviors predicted theoretically from the network structure actually occur in the fission yeast cell cycle system? We examined the effects of various perturbations on the concentrations of CDK–cyclin complexes using fluorescence cross-correlation spectroscopy (FCCS) (24, 36, 37) (See SI Appendix, section 2.3 for details). To eliminate the effects of cell cycle–dependent changes in CDK–cyclin complex concentrations, we used cell length as an indicator of cell cycle stage (21, 38) and selected subpopulations corresponding to key phases: >10.5 μm (G2-M phase) for CDK-Cdc13 measurement and <10.5 μm (${\mathrm{G}}_1$-S phase) for CDK-Cig2 measurement.

We first tested the existence of BS2, which includes CDK-Cdc13 but not CDK-Cig2 (Fig. 3A, blue). If this buffering structure exists, the perturbation to $k_7$ (the Cdc13 synthesis rate) should affect the concentration of CDK-Cdc13 but not CDK-Cig2 (Figs. 3 A and B and 4A). Additionally, since this buffering structure contains the conserved quantity CDK, perturbing the total CDK amount should not affect CDK-Cig2 (SI Appendix, Fig. S3 B and C and Fig. 4D). When the synthesis rate of Cdc13 was increased experimentally, FCCS analysis showed an increase in CDK-Cdc13 concentration, but the concentration of CDK-Cig2 did not change. (Fig. 4B and C). FCCS analysis also showed that the total CDK amount did not change the concentration of CDK-Cig2 (Fig. 4E). These results indicates that BS2 exists and that CDK-Cdc13 can be controlled independently from CDK-Cig2 in real fission yeast cells.

Fig. 4.

Experimental validation of theoretical predictions. (A) Scheme of the theoretical predictions regarding the effects of increasing $k_7$ (the synthesis rate of Cdc13) on the concentrations of CDK-Cig2 and CDK-Cdc13. (B and C) The concentrations of CDK-Cig2 (B) and CDK-Cdc13 (C) for control cells (Ctrl) and cells in which $k_7$ was experimentally increased. (D) Scheme of a theoretical prediction regarding the effects of increasing $d_{\mathit{cdk}}$ (the total amount of CDK, see SI Appendix, Fig. S3) on the concentration of CDK-Cig2. (E) The concentration of CDK-Cig2 for control cells (Ctrl) and cells in which $d_{\mathit{cdk}}$ was experimentally increased. (F) Scheme of the theoretical predictions regarding the effects of increasing $k_{19}$ (the transcription rate of Cig2) on the concentration of CDK-Cig2 and CDK-Cdc13. (G and H) The concentrations of CDK-Cig2 (G) and CDK-Cdc13 (H) for control cells (Ctrl) and cells in which $k_{19}$ was experimentally increased. (I) Scheme of a theoretical prediction regarding the effects of increasing $d_{\mathit{cdk}}$ on the concentration of CDK-Cdc13. (J) The concentration of CDK-Cdc13 for control cells (Ctrl) and cells in which $d_{\mathit{cdk}}$ was experimentally increased. (K) Summary of experimental results.

CDK-Cig2 Is Not Independently Controlled from CDK-Cdc13.

We next tested the existence of BS1, which includes CDK-Cig2 but not CDK-Cdc13 (Fig. 3A, green). If this buffering structure exists, the perturbation to $k_{19}$ (the Cig2 synthesis rate) should affect the concentration of CDK-Cig2 but not CDK-Cdc13 (Figs. 3 B and F and 4F). Additionally, since this buffering structure contains the conserved quantity CDK, the perturbation to the total CDK amount should not affect CDK-Cdc13 (SI Appendix, Fig. S3 B and C and Fig. 4I). When the synthesis rate of Cig2 was increased experimentally, FCCS analysis showed an increase in CDK-Cig2 concentration. (Fig. 4G). However, it also led to a decrease in the concentration of CDK-Cdc13 (Fig. 4H). FCCS analysis also showed that increasing the CDK amount increased the concentration of CDK-Cdc13 (Fig. 4J). These findings show that BS1 does not exist in the fission yeast cell cycle system (Fig. 4K).

Finding a Pathway by Comparing Predictions of the Topology-Based Theory with Experiments.

Our theoretical predictions were derived from the network structure alone, without depending on any choices about reaction rate functions or parameters. If the network information is correct, the predicted behavior should always be observed. Therefore, the fact that some of the theoretical predictions are inconsistent with the experimental results means that the network information we used differs from the actual network. All the reactions included in the network are supported by sufficient experimental evidence (23, 39–41), so we are certain that there exist unidentified reactions that are not included in the current network information.

To identify the missing reactions, we conducted a theoretical search for the missing network information. The unidentified reaction must satisfy the following condition: By adding the reaction, BS1 is destroyed while BS2 is maintained. In general, for a buffering structure to be destroyed, a reaction (or regulatory) arrow must be added from the chemical inside the buffering structure to the outside (42). In contrast, adding an arrow that goes into a chemical within a buffering structure never breaks the buffering structure. When adding a reaction arrow connecting two chemicals within a buffer structure, the number of reactions increases, and (in most cases) the number of cycles increases as well; therefore, the structure remains a buffer structure. In the case of the cell cycle system, there are cases where the number of cycles does not increase with the addition of arrows, but we did not consider these because all of them are biologically unrealistic. Therefore, here we only need to examine outgoing arrows emanating from the chemicals within BS1 to the outside.

We introduced outgoing reactions for each chemical in the network and conducted SSA. Among the 13 possible network modifications, only one modification breaks BS1 while maintaining BS2. This modification is the addition of the degradation pathway for the Cdc13 monomer (Fig. 5A and SI Appendix, Fig. S6 and section S3). Logically, an outgoing arrow may be realized by a regulatory arrow instead of a reaction arrow. We also examined all possible regulatory arrows, and listed those that satisfies the condition (SI Appendix, Fig. S7). However, since these regulatory arrows are biologically unrealistic, we will not consider them further in this paper. In the following, we focused on the possibility of the existence of the degradation pathway for the Cdc13 monomer. We refer to the network including this modification as the “modified network” (Fig. 5B). In the original network, Cdc13 is assumed to be ubiquitinated only when it binds to CDK in an anaphase-promoting complex (APC/C)-dependent manner. In the modified network, Cdc13 that does not form a complex with CDK should be degraded. This breaks BS1 but not BS2, fully explaining the observed behavior (SI Appendix, Fig. S4). The same conclusion applies to the simplified cell cycle network: Only the addition of a degradation reaction for the Cdc13 monomer breaks BS1 while maintaining BS2 (Fig. 5C and SI Appendix, Fig. S5).

Fig. 5.

Theoretically identifying missing reactions in the cell cycle network. (A) We considered network modifications where a degradation reaction is added to each of the chemicals that originally do not have a degradation reaction. Each modification was then evaluated for whether it satisfied the condition that BS1 is destroyed and BS2 is maintained. We find that only the addition of degradation of the free Cdc13, which is not bound to CDK, (Cdc13→0) satisfies the condition, thereby explaining the experimental results. (B and C) Graphical representations of the modified cell cycle network (B), and modified simplified network (C). A red reaction, representing the degradation of Cdc13 that does not bind to CDK, is added. This modification causes BS1 to disappear, while BS2 is maintained.

Experimental Validation of the Predicted Pathway.

We finally examined whether this predicted pathway (degradation of the Cdc13 monomer) exists in actual fission yeast cells. Given that APC/C primarily degrades Cdc13 bound to CDK, as shown in Fig. 3A, the degradation of the Cdc13 monomer may not depend on APC/C. We thus hypothesized that an APC/C-independent Cdc13 degradation pathway exists. To test this hypothesis, we conducted a cycloheximide (CHX) chase analysis (26, 43) for cdc2 (cdk)-miRFP; cdc13-mNeonGreen (mNG) cells, and cells expressing mNG not fused to Cdc13 (negative control). CHX is a translation inhibitor that blocks new protein synthesis, allowing us to monitor the degradation of proteins over time. After adding CHX, we measured the fluorescence intensity of mNG at various time points for each cell. Given that APC/C becomes active just before anaphase in M phase (41), we used image analysis to distinguish cells before nuclear division (${\mathrm{G}}_2$ phase) from those after nuclear division (M phase). (Fig. 6A). The degradation rate of Cdc13 in ${\mathrm{G}}_2$ phase was lower than that in M phase. However, the degradation rate of Cdc13 in ${\mathrm{G}}_2$ phase was still higher than that of mNG or CDK. (Fig. 6 B, D, and E), suggesting that Cdc13 is actively degraded in ${\mathrm{G}}_2$ phase, where APC/C is inactive. No significant increase in cell size was observed after CHX addition, indicating that the decline in the Cdc13 concentration reflects an actual degradation process rather than dilution due to cell growth (SI Appendix, Fig. S8). Consistent with this observation, pharmacological inhibition of APC/C did not decrease the degradation rate of Cdc13 in ${\mathrm{G}}_2$ phase (Fig. 6 C–E). These results confirm the existence of an APC/C-independent Cdc13 degradation pathway, which was predicted by comparing the SSA expectations with experimental results.

Fig. 6.

Cdc13 is degraded in non-M phase in an APC-independent manner. (A) Schematic representation of the experimental strategy to measure the degradation rate of Cdc13. Cdc13-mNeonGreen (mNG); cdk-miRFP cells were used, and mNG not fused to Cdc13 was used as a control. Using image analysis, we distinguished cells that did not go through nuclear division (${\mathrm{G}}_2$ cells) and cells that went through nuclear division (M cells). (B) Time course for the relative fluorescence levels of control (gray) and Cdc13-mNG (blue) for ${\mathrm{G}}_2$ cells under the treatment of CHX. Cdc13-mNG for M cells (light blue) is also shown. CHX is added at time 0. Time-lapse images were acquired using confocal microscopy. $N=35$ cells for $G_2$ cells, $N=17$ for M cells, and $N=13$ for a control. (C) Time course for the relative fluorescence levels of control (gray) and Cdc13-mNeonGreen (blue) under the treatment of CHX with the APC inhibitor Apcin. CHX and Apcin are added at time 0. $N=42$ cells. (D) Average time course for the mNG, CDK (${\mathrm{G}}_2$ phase, Apcin -), and Cdc13 levels in (B and C) are overlaid. (E) The degradation rates of mNG, CDK, and Cdc13 of each cell estimated by fitting the data in (B and C) to an exponential decay model. Horizontal bars are means. ***$P<0.0001$ by Student’s t test adjusted by the Bonferroni method for multiple comparisons.

Discussion

We clearly show that the mechanism of independent control arises solely from the structural properties of the reaction network. Many theoretical approaches have attempted to understand complex biological systems in terms of network structures (17, 44), but many of them require assumptions about specific functional forms, and cannot be said to be purely topology-based. We have developed several purely topology-based theories (7–10, 45–51), and SSA is one of them. The concept of a buffering structure mathematically predicts that network topology creates independent control of multiple functions emerging from a single connected network. This study clearly demonstrates the physiological link between the mathematical concept of a buffering structure and molecular basis that enables independent control of interconnected systems in living cells.

Here, we state the general conditions for “independent control by buffering structures” more precisely. i) Suppose there are two buffering structures (BS1 and BS2) that are not in an inclusion relationship, and that BS1 and BS2 contain chemical $X_1$ and $X_2$, respectively. The two BSs may intersect or may contain smaller buffering structures. However, ii) assume that $X_1$ is contained within BS1 but not within a smaller buffering structure in BS1 (hereafter referred to as “$X_1$ is directly contained within BS1”), and similarly that $X_2$ is also directly contained within BS2. When these two conditions are satisfied, parameters directly contained within BS1 affect the steady state of $X_1$ but do not affect that of $X_2$, and parameters directly contained within BS2 affect the steady state of $X_2$ but do not affect that of $X_1$. Mathematically, changes in parameters directly contained within BS1 generically induce nonzero changes in $X_1$, except for special parameter choices (i.e., changes in the steady state $X_1$ are not constitutively zero). However, for biologically observable chemical changes, we may need to assume that these parameters have “sufficiently large effects” on the target chemicals.

We also emphasize that the above-mentioned condition is not only a sufficient condition but also a necessary condition for the independent control between two chemicals. We have previously proven the inverse theorem of the law of localization (10). This theorem proves that if there exists a chemical in a reaction system whose steady state does not respond at all to some parameter change, then a corresponding BS must exist. Therefore, when the steady states of $X_1$ and $X_2$ are independently controllable, there must exist a BS1 that directly includes $X_1$ but not $X_2$, and a BS2 that directly includes $X_2$ but not $X_1$.

We clarified that the transition through the ${\mathrm{G}}_2$-M checkpoint is controllable independently from the ${\mathrm{G}}_1$-S checkpoint. This is an achievement of successful collaboration between theory and experiment: predicted by theoretical analysis of network structure and verified by precise quantitative measurements. As the two CDK–cyclin complexes governing the two checkpoints share the same CDK, the competition hypothesis has been proposed, which states that the two cyclins may always compete with each other for CDK binding. Our results refute this hypothesis, strongly suggesting a different scheme from both theoretical and experimental viewpoints.

The fact that the ${\mathrm{G}}_2$-M checkpoint is controllable independently from the ${\mathrm{G}}_1$-S checkpoint might be important for cell function. The ${\mathrm{G}}_1$-S checkpoint and the ${\mathrm{G}}_2$-M checkpoint have different roles, determining the timing of the initiation of genome replication and that of mitosis, respectively. The fact that the control of the initiation of mitosis is independent from that of the initiation of replication is likely to be adaptive for proper cell function. In particular, even if the amount of Cdc13 inappropriately increases due to some expression fluctuation during the ${\mathrm{G}}_1$-S transition, it would never influence CDK-Cig2 and therefore normal ${\mathrm{G}}_1$-S transition would be realized. On the other hand, an irregular increase in Cig2 concentration not only affects CDK-Cig2 but also CDK-Cdc13. How should we understand this? In fission yeast, while the expression level of Cdc13 rapidly increases at a specific stage, Cig2 expression does not exhibit remarkable stage-specific changes (21). Therefore, even if an irregular fluctuation occurs in Cig2 expression, the magnitude of the expression change would be small and its effect would be negligible. For these reasons, the selective pressure to ensure that Cig2 expression changes do not affect CDK-Cdc13 might not have been very strong.

Combining theoretical predictions with experimental results, we also identified a reaction pathway that does not exist in the network data: the degradation of the Cdc13 monomer during G₂ phase, when APC/C is inactive. It was previously shown that Cdc13 transcription is active, but Cdc13 protein abundance is low in ${\mathrm{G}}_2$ phase (26). However, the mechanism maintaining low Cdc13 protein in this phase has been unclear. Our findings suggest that this pathway functions to maintain low levels of Cdc13 prior to the ${\mathrm{G}}_2$-M transition, thereby to prevent premature mitosis. The significantly higher degradation rate of Cdc13 compared to CDK suggests that this degradation is a targeted process, rather than nonspecific protein turnover caused by damage or oxidation. We hypothesize that Cdc13 contains a degron motif. This can be tested in future work by generating deletion mutants that lack specific regions of Cdc13 and assessing their stability.

Analysis of the simplified cell cycle network indicates that independent regulation is not specific to the network we used, and have the potential to universally appear in all possible cell cycle systems (Fig. 3E). In many species, the cell cycle is controlled by the dynamics of CDK–cyclin complexes, where the levels of CDKs remain relatively constant, while the levels of cyclins change over time (19). In such systems, it is highly likely that multiple, nonoverlapping buffering structures will appear. We expect that independent control of the cell cycle through buffering structures is universal across species.

Applying SSA to the cell cycle systems of other species, including humans, is a potential direction for future research. Unlike fission yeast, mammalian cells have multiple CDKs, including CDK1, CDK2, CDK4, and CDK6. Among these, only CDK1—the counterpart to fission yeast’s Cdc2—is essential, since the cell cycle progresses almost normally in most cell types even in the absence of other CDKs (52–54). CDK1 binds to all cyclins when other CDKs are absent (54). This suggests that fundamental regulatory mechanisms governing the cell cycle are conserved between fission yeast and mammals. This raises the possibility that buffering structures also play a crucial role in the mammalian cell cycle, just as they do in fission yeast. Notably, since SSA is applicable to any network, it will be a powerful tool to investigate this hypothesis.

By iterating between the theoretical predictions and experimental validation, we were even able to update the cell cycle network information that we initially relied on. This was possible because SSA is a topology-based theory that derives predictions from network information alone. If we had used mathematical modeling approaches assuming reaction rate functions and parameter values, a mismatch between theoretical predictions and experimental validation would have required us to consider the possibility that the model assumptions were incorrect. By using SSA, we are able to conclude that discrepancies in the known network information must exist whenever the predicted behavior is not observed experimentally. Furthermore, since SSA can immediately determine the response to any perturbations based on network structure, we are able to comprehensively explore possible network modifications and determine the modifications required to achieve the observed behavior. We believe that we have proposed a systems biology framework that advances the elucidation of systems through theoretical predictions and experimental validation.

Materials and Methods

Theoretical Methods.

Structural sensitivity analysis (SSA) and a buffering structure.

The dynamics of a chemical reaction system.

We consider a chemical reaction system consisting of M chemicals $(X_1,\dots ,X_M)$ and N reactions $(R_1,\dots ,R_N)$, where reaction $R_n$ is $\sum _{m=1}^M{y}_m^n{X}_m ⟶ \sum _{m=1}^M{\overline{y}}_m^n{X}_m$. The coefficients $y_m^n$ and ${\overline{y}}_m^n$ are the numbers of chemical $X_m$ participating in the nth reaction $R_n$ at the reactant and product stages, respectively. Dynamical changes in the concentrations of chemicals are described by

$$\begin{array}{c}\hfill \frac{d{x}_m(t)}{\mathit{dt}}={\displaystyle \sum _{n=1}^N}({\overline{y}}_m^n-y_m^n)r_n={\displaystyle \sum _{n=1}^N}{\nu }_{\mathit{mn}}{r}_n.\end{array}$$ [1]

Here $x_m$ ($m=1,\dots ,M$) denotes the concentration of the chemical species $X_m$. The reaction rate (flux) of reaction $R_n$ is $r_n(k_n; \mathit{x})$, which depends on the reaction parameter $k_n$ and the concentrations $\mathit{x}={(x_1,\dots ,x_M)}^{\mathrm{T}}$. The stoichiometric coefficient ${\nu }_{\mathit{mn}}={\overline{y}}_m^n-y_m^n$ represents the change in the amount of chemical m after one reaction $R_n$ occurs. Eq. 1 can be written in vector form as $\frac{d\mathit{x}}{\mathit{dt}}=\nu \mathit{r}$, where $\mathit{r}={(r_1,\dots ,r_N)}^{\mathrm{T}}$, and ν is the stoichiometric matrix whose $(m,n)$-th entry is ${\nu }_{\mathit{mn}}$.

The specific form of the rate function $r_n$ is usually unknown, but it is natural that $r_n$ increases with the concentration of its substrates. Thus, we assume that:

$$\begin{array}{c}\hfill \frac{\partial r_n}{\partial x_m}=\{\begin{array}{cc}>0,\hfill & \text{if} y_m^n>0 \text{or} m\in M_n^{+},\hfill \\ <0,\hfill & \text{if} m\in M_n^{-},\hfill \\ =0,\hfill & \text{otherwise},\hfill \end{array}\end{array}$$ [2]

where $M_n^{+}$ and $M_n^{-}$ are the sets of chemicals that are not substrates of reaction $R_n$ but regulate it positively or negatively, respectively. Common kinetics such as mass-action or Michaelis–Menten satisfy this condition.

Furthermore, each $r_n$ depends on its own reaction parameter $k_n$, which can be the activity or amount of the enzyme catalyzing reaction $R_n$, i.e., $\frac{\partial r_n}{\partial k_n}\ne 0$.

Structural sensitivity analysis (SSA) (7).

Here, for simplicity, we review SSA for systems without conserved quantities, i.e., $dim[ker({\nu }^{\mathrm{T}})]=0$ (see SI Appendix, section S1 for the general cases). We now consider a steady state of Eq. 1. At steady state, $\nu \mathit{r}=0$. Hence, $\mathit{r}=\sum _{p=1}^K{\mu }^p{\mathit{c}}^p,$ where ${\mathit{c}}^p$ ($p=1,\dots ,K$) are basis vectors of $ker(\nu )$ (cycles).

We assume that the system reaches a new steady state after a small change in enzyme activity $k_{n^{\ast }}$ ($k_{n^{\ast }} \to k_{n^{\ast }}+\delta k_{n^{\ast }}$). The change in reaction rate $r_n$ due to this perturbation is expressed as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\delta }_{n^{\ast }}{r}_n=[\frac{\partial r_n}{\partial k_{n^{\ast }}}{\delta }_{n,n^{\ast }}+{\displaystyle \sum _{m=1}^M}\frac{\partial r_n}{\partial x_m}\frac{\partial x_m}{\partial k_{n^{\ast }}}]\delta k_{n^{\ast }},\\ \hfill (n,n^{\ast }=1,\dots ,N),\end{array}\end{array}$$ [3]

where ${\delta }_{n,n^{\ast }}$ is the Kronecker delta. All derivatives are evaluated at the steady state. At the same time,

$$\begin{array}{c}\hfill {\delta }_{n^{\ast }}{r}_n={\displaystyle \sum _{p=1}^K}{\delta }_{n^{\ast }}{\mu }^p c_n^p,\end{array}$$ [4]

where ${\delta }_{n^{\ast }}{\mu }^p$ is the change in ${\mu }^p$ caused by the perturbation of $k_{n^{\ast }}$. Combining Eqs. 3 and 4, we obtain:

Here, $r_{n,m}:=\frac{\partial r_n}{\partial x_m}$ and ${\delta }_{n^{\ast }}{x}_m:=\frac{\partial x_m}{\partial k_{n^{\ast }}}\delta k_{n^{\ast }}$. All matrices in Eq. 5 are square. Let $A:=(r_{n,m} | -c^p)$ denote the first matrix on the left-hand side. The right-hand side is a diagonal matrix with nonzero diagonal elements. Assuming A is invertible, we obtain:

$$\begin{array}{c}\hfill S:=\left(\begin{array}{ccc}{\delta }_1{x}_1& \cdots & {\delta }_N{x}_1\\ ⋮& \ddots & ⋮\\ {\delta }_1{x}_M& \cdots & {\delta }_N{x}_M\\ {\delta }_1{\mu }^1& \cdots & {\delta }_N{\mu }^1\\ ⋮& \ddots & ⋮\\ {\delta }_1{\mu }^K& \cdots & {\delta }_N{\mu }^K\end{array}\right)\propto -A^{-1},\end{array}$$ [6]

where $S\propto -A^{-1}$ means $S_{\mathit{ij}}\ne 0\iff {(-A^{-1})}_{\mathit{ij}}\ne 0$. Each element of S describes the steady-state response of $x_m$ (or ${\mu }^p$) to a perturbation to the reaction parameter $k_{n^{\ast }}$.

The responses of a system to parameter perturbations are thus determined from the network structure via Eq. 6, according to one of two criteria depending on the case.

1. The presence or absence of a response is solely determined by the network structure. The value of $r_{n,m}$ is nonzero when chemical m is a substrate of reaction $R_n$ (or when m regulates the rate of $R_n$). Otherwise, $r_{n,m}$ is identically zero. In other words, the pattern of zero and nonzero components in the left part of the matrix A is determined solely by the structure of the reaction network, and is independent of the choices of the parameters or reaction rate functions. The right part of A (the basis vectors of the stoichiometric matrix) is also determined only by the network structure. Hence, the zero/nonzero pattern of each element in $S\propto -A^{-1}$ is likewise determined solely from the network topology. Since we are interested only in qualitative responses, we hereafter let $S:=-A^{-1}$.

2. Determining the direction (sign) of a response. In some cases, we can not only determine whether a response exists, but also infer the direction (sign) of the response. For example, if $k_n$ represents the activity of the enzyme catalyzing reaction $R_n$, it is natural to assume $\frac{\partial r_n}{\partial k_n}>0$. In this case, from Eq. 5, the sign of each entry of S directly corresponds to the sign of the system’s response.

Example: The chemical reaction system in Fig. 1C and SI Appendix, Fig. S1A consists of three reactions: $R_1: 0\stackrel{k_1}{\to }P$, $R_2: P\stackrel{k_2}{\to }Q$, and $R_3: Q\stackrel{k_3}{\to }0$. Since $ker(\nu )$ is spanned by ${(1,1,1)}^{\mathrm{T}}$, the matrix A is computed as

$$A=\left(\begin{array}{ccc}0& 0& -1\\ r_{2,P}& 0& -1\\ 0& r_{3,Q}& -1\end{array}\right).$$

By inverting A, we obtain

Since $r_{2,P}$ and $r_{3,Q}$ are positive, the signs of the entries of S are

The second column of S indicates that an increase in $k_2$ causes a decrease in the steady-state concentration of chemical P, whereas the steady-state concentration of Q remains unaffected (Fig. 1 C and D and SI Appendix, Fig. S1A).We applied SSA to the cell cycle network following the same procedure (SI Appendix, section S1).

Law of localization and buffering structure.

We briefly review the law of localization and buffering structure (8, 9) for the systems without conserved quantities (see SI Appendix for the general cases).

Definition of a buffering structure.

For a given chemical reaction network, we consider a subset of the network $\mathrm{\Gamma }=({\mathfrak{m}}_{\mathrm{\Gamma }},{\mathfrak{r}}_{\mathrm{\Gamma }})$, where ${\mathfrak{m}}_{\mathrm{\Gamma }}$ is a subset of chemical species and ${\mathfrak{r}}_{\mathrm{\Gamma }}$ is a subset of reactions. We call Γ a buffering structure if Γ satisfies the following two conditions:

1. ${\mathfrak{r}}_{\mathrm{\Gamma }}$ encompasses all reactions whose reaction rates depend on the concentration of the chemicals within ${\mathfrak{m}}_{\mathrm{\Gamma }}$, i.e., all reaction arrows that leave from ${\mathfrak{m}}_{\mathrm{\Gamma }}$ are included in ${\mathfrak{r}}_{\mathrm{\Gamma }}$ (“output-completeness”).

2. $\lambda (\mathrm{\Gamma })=0$.

Here, the index $\lambda (\mathrm{\Gamma })$ is defined as

$$\begin{array}{c}\hfill \lambda (\mathrm{\Gamma }):=-|{\mathfrak{m}}_{\mathrm{\Gamma }}|+|{\mathfrak{r}}_{\mathrm{\Gamma }}|-N({\mathfrak{r}}_{\mathrm{\Gamma }}).\end{array}$$ [7]

$N({\mathfrak{r}}_{\mathrm{\Gamma }})$ is the number of cycles consisting of reactions in ${\mathfrak{r}}_{\mathrm{\Gamma }}$ (see SI Appendix for the rigorous definition).

Law of localization (8, 9).

The steady-state chemical concentrations and reaction fluxes outside of a buffering structure $\mathrm{\Gamma }=({\mathfrak{m}}_{\mathrm{\Gamma }},{\mathfrak{r}}_{\mathrm{\Gamma }})$ do not change under any perturbation of the reaction rate parameters in ${\mathfrak{r}}_{\mathrm{\Gamma }}$.

This theorem follows directly from SSA (SI Appendix).

An example of a buffering structure.

In the chemical reaction system in Fig. 1C and SI Appendix, Fig. S1A, $\mathrm{\Gamma }=\{P,R_2\}$ is a buffering structure, since

1. Γ is output-complete (the reaction arrow that leaves from P is included in Γ),

2. $\lambda (\mathrm{\Gamma })=0 (|{\mathfrak{m}}_{\mathrm{\Gamma }}|=1, |{\mathfrak{r}}_{\mathrm{\Gamma }}|=1, N({\mathfrak{r}}_{\mathrm{\Gamma }})=0)$.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Code used for the theoretical analyses is available on GitHub at https://github.com/YuheiYamauchi/cell-cycle-BS (55). Experimental data are available in the SSBD:repository at https://doi.org/10.24631/ssbd.repos.2026.08.540 (56).