Synchronization in biological systems is widely observed in the natural world. This phenomenon is one of the collective behaviors thought to have a crucial role in maintaining the individual life or in giving benefits for communities. One of the well-known examples is synchronized flashing of male fireflies [1] [2] .

Glycolysis is a biological mechanism to decompose glucose and to store energy in the form of ATP. In this chemical process, synchronized variations can be seen for the concentrations of various metabolites [3] - [8] . For example, yeast cells exhibit glycolytic synchronized oscillations under certain conditions. A suspension of yeast cells with high population densities shows synchronization, whereas that with low densities does not. The emer- gence of the collective behavior such as synchronization may occur above a critical cell density.

The theoretical studies of glycolytic oscillations in yeast cells using differential equations started substantially when Sel’kov presented a simple kinetic model of an enzyme reaction with substrate inhibition and product ac- tivation [9] . His ordinary differential equation system with two variables showed the occurrence of periodic self- oscillations in glycolysis. Another starting point is the allosteric enzyme model by Goldbeter and Lefever and Goldbeter, which also consists of two variables that referred to ATP and ADP, respectively [10] [11] . They adopted a partial differential equation system, which afterward facilitated a large extent of investigations for pattern formation and spatiotemporal structures. As these examples, we exemplify the studies of target pattern formation and spatiotemporal chaos by Zhang et al. [12] and of inward rotating spiral waves by Straube et al. [13] . Meanwhile, the Sel’kov model is succeeded by Lavlova et al. in the study of inward and outward wave propagation [14] . Besides these two streams, Bier et al. proposed a simple conceptual model consisting of glu- cose and ATP for explanation of glycolytic synchronized oscillations [15] .

The epoch-making study was performed by Wolf and Heinrich in order to elucidate the mechanisms of syn- chronous behaviors in glycolytic oscillations [16] . Based on the detailed description of glycolytic reaction processes, they constructed a seven-variable model, by which the effect of intercellular coupling on oscillatory dynamics was theoretically analyzed. Further, Henson et al. connected this model with the cell-ensemble model- ling technique and showed that a large ensemble of about 1000 population cells were required to adequately capture complex dynamic behaviors in glycolytic oscillations [17] .

However, it seems that no model has succeeded in demonstrating perfect synchronization of a large number of yeast cells that oscillated with different phases and periods. We guess that one of the reasons is that these mod- els assumed acetaldehyde or pyruvate to be an intercellular communication substance for glycolytic synchroni- zation.

It is known that acetaldehyde mediates the synchronization of glycolytic oscillations [6] . Various experiments demonstrated that acetaldehyde had a very strong synchronization effect on a suspension of yeast cells [8] . However, these studies do not exclude the possibility that extracellular nucleotides such as ATP and ADP can play an important role in synchronization of glycolytic oscillations in yeast cells. These are signaling molecules contained in all tissues [18] . Other studies also showed that Saccharomyces cerevisiae released ATP to extra- cellular solutions, suggesting that extracellular ATP, ADP, AMP or adenosine played a role in yeast physiology as intercellular communicators [19] [20] .

Then, we attempted to exchange acetaldehyde or pyruvate to ATP or other metabolites in the original model by Wolf and Heinrich [16] . Our present study using these modified seven-variable models shows that glycolytic synchronization in yeast cells may occur via intercellular mediators such as ATP and other metabolites, and that ATP is the most effective synchronizer even under the condition where individual cells have quite different properties.

Our mathematical models are derived from the glycolytic oscillation model originally presented by Wolf and Heinrich [16] . Major modifications are the following three:

1) Extension to two-dimensional partial differential equation systems with the diffusion term,

2) Exchange of the intercellular coupling substance and,

3) Non-dimensionalization for variables and parameters.

The first modification enables us to observe directly various kinds of oscillations including the synchronous one, and the possibilities of synchronization significantly increase by the second modification. Moreover, the third modification, which leads to the reduction of parameter numbers, eases mathematical analyses of simulation models.

The schematic diagram of the model is sketched in Figure 1, where the simulation area consists of N × N par- titioned square compartments divided at regular intervals. One cell is embedded within each compartment [16] [21] , whose volumetric ratio φ to the compartment is one of the key control parameters of our models. In this ar- ticle, we adopt N = 11 in all simulations, resulting in the total cell number is N × N (=121). However, the num- ber of N can be decided arbitrarily.

Six major substances are contained within a cell, which are denoted as S₁, S₂, S₃, S₄, N₂ and A₃. These are regarded as independent variables, the meanings of which are explained in Table 1. In addition to these six, two more substances, N₁ and A₂, are also contained, which are connected by the relations, N₁ + N₂ = N and A₂ + A₃ = A, respectively. As N and A are constant, two variables, N₁ and A₂, are not independent, whose numbers are auto- matically identified. In the original model, it is assumed that only S₄ can permeate through the membrane, then, the substance in the external solution is labeled as [16] . The model of this type, via the S₄ intercellular me- diator, is also the starting point of our studies. The transmembrane coupling metabolite is later altered to the other one such as A₃, S₃, S₂ and N₂; thereafter, the variations of system behaviors are investigated.

The meanings of parameters used in this article are also explained in Table 1, most of which are recycles of those in the original model [16] . The dimensionless parameter values are listed in Table 2, which are obtained by non-dimensionalization processes such as

Most of variables and parameters were defined by Wolf and Heinrich [16] , which are reused in this article.

Numerical analyses are performed using these parameter values. A parameter k₇ is newly introduced for stabiliza- tion of the system. The diffusion coefficient d_F and the parameter f₀ that defines the amplitude of randomization are also employed. When f₀ = 0.1, for example, randomized items such as initial values and parameter values are scattered within the range from 90% to 110% around the above values. As for two control parameters, φ and d_F, two different values are provided in Model II, III and IV, while only d_F is varied in Model I. Reference values are converted by non-dimensionalization procedures from those in the original model by Wolf and Heinrich [16] .

Two parameters, k₁ and K_I, are eliminated by these processes [21] . Besides, the original parameter k is renamed r, and a new parameter k₇ is employed. The system is stabilized by the addition of k₇, especially when the trans- membrane substance is changed to A₃, S₃, S₂ or N₂. As a result, our seven-variable, two-dimensional reac- tion-diffusion model with S₄ being the intercellular mediator reads as follows.

Here, the diffusion term is incorporated in the last equation, which describes the intercellular coupling via S₄. This basic model described by Equations (1) is referred to as Model I in this article.

Next, we exchange the coupling substance in the external medium in order to examine other candidates that can produce synchronous oscillations. It should be noted that a new parameter k₇ takes an important role as a stabilizer in these models. For example, when the substance that functions as the communicator is A₃, the ma- thematical model is described such as

To be accurate, one more variable A₂ or should be added besides, however, it is confirmed that the addition of either variable hardly influences the simulation result. The system described by Equations (2) is re- ferred to as Model II in this article.

Similarly, we can construct the system with the intercellular communicator S₃ (Equations (3)) and the system with the intercellular communicator S₂ (Equations (4)). These are referred to as Model III and Model IV, respec- tively.

In above formulations of Equations (3) and Equations (4), several equations are omitted, which are the same as those in Equations (1). Moreover, we can construct Model V, where the intercellular mediator is N₂, in the same manner, formulations of which are not shown explicitly.

Although two parameters are eliminated by non-dimensionalization procedures, there are still twelve parame- ters in our models except for two control parameters d_F and φ. In principle, we try to use parameter values of the reference state converted from the original model [16] , which are listed in Table 2. If it is impossible to make stable limit cycle oscillations using these values, we adjust the values of such parameters as J₀, r and κ, which are thought to be comparatively easy to manipulate. The exception is Model V, where the value of k₅ is varied. Besides the reference [16] , the parameter values in the reference [17] are also referred to.

It is also important to choose properly the initial value of each variable. We determine these values in refer- ence to fixed points, which are calculated in advance. Table 3 is a list of initial values used to create synchron- ous oscillations. In the case of Model II, for example, the initial value of S₁ is fluctuated within the range from 4.68 (= 5.2 × 0.9) to (5.72 = 5.2 × 1.1), S₂ = 0.43 (= 4.3 × 0.1), and so on.

“FP” denotes the fixed point. For detailed calculations, see Appendix.

1) Glycolytic oscillations with different phases can be synchronized by means of intercellular coupling via such substances as S₄, A₃, S₃ and S₂, as demonstrated in Models I, II, III and IV. Meanwhile, synchronization of oscillations with different periods can be mediated by intercellular coupling substances such as A₃, S₃ and S₂, as demonstrated in Models II, III and IV. The latter synchronization is characterized by speedy convergence to the synchronous oscillatory state.

2) Among three candidates that can induce synchronization for different periods, A₃ could be the most re- sponsible for the phenomenon, because the normalized standard deviations of N₂ concentrations in Model II are the smallest compared with those in Model III or IV.

3) The transition from the asynchronous to the synchronous oscillation is observed with the increase in the diffusion coefficient d_F, which could be referred to the Kuramoto transition mechanism. On the other hand, the direct transition from the convergence to the fixed point to the synchronous oscillation is observed with the in- crease in the density parameter φ, which could be referred to the dynamical quorum sensing.

We thank Drs. K. Shibata, P. G. Søensen, S. C. Müller, M. J. B. Hauser, T. Yamaguchi, T. Yamamoto and S. Sasaki for helpful discussion and M. Denda for giving us information on ATP as a signaling molecule.

It is useful to know in advance the coordinates of fixed points for estimations of the initial values in numerical simulations. Fixed points are the solutions of the simultaneous equations in which all time derivatives equal zero. As for the fixed point of Model I, N₂ is calculated by solving the quadratic equation for the first time, then, those of S₂, S₄, , A₃, S₃, S₁ are computed in this order, such as

Here, the diffusion term is neglected.

For another example, the fixed point of Model II is computed, as follows.

Similarly, we can identify the coordinates of fixed points for Models III, IV and V as well. We would like to stress that coordinates of fixed points for each parameter value are calculated or identified precisely in all models presented in this article.