Supplementary Material for Transition to diversification by competition for multiple resources in a catalytic reaction network

X + Z + SX → 2X + Z, Z + X + SZ → 2Z + X. We denote the intrinsic catalytic activities of X, Y , and Z, respectively, by cX , cY and cZ . Each reaction to synthesize X, Y , and Z utilizes the resource SX , SY , and SZ , respectively, which are shared by cells. The intrinsic reaction rates for replicating the molecule species X are given by F X = VAf A XsX , F B X = VBf B XsX , respectively, for cell type A and B. Here, VA, and VB are the volumes of cell types A and B, respectively, and f X = cY x A Xx A Y , f B X = cZx B Xx B Z , where x I i = N I i /VI(i = X, Y, Z; I = A,B) and N I i is the number of molecule species i. The rates of replicating Y and Z are, respectively, given by F Y = VAf A Y sY , F B Z = VBf B Z sZ Here, f Y = cXx A Xx A Y , f B Z = cXx B Xx B Z . si = Si/S 0 i (i = X, Y, Z) is the normalized concentration of the resource where Si is the concentration and S i is introduced to normalize Si to one when Si = S 0 i . The rate equations of cell-types A and B for molecule species X are written as, dN X dt = F X = VAf A XsX , dN X dt = F X = VBf B XsX , (1)

respectively, for cell type A and B. Here, V A , and V B are the volumes of cell types A and B, respectively, and , B) and N I i is the number of molecule species i. The rates of replicating Y and Z are, respectively, given by is the normalized concentration of the resource where S i is the concentration and S 0 i is introduced to normalize S i to one when S i = S 0 i . The rate equations of cell-types A and B for molecule species X are written as, and, for Y and Z, The dynamics of resources S X , S Y , and S Z are respectively written as In the steady state, the value of s X ,s X , is written as, In Fig. 1, we shows X as a function of D with a set of parameters. For large D,s X → 1. As D is decreased,s X starts to decrease and deviates from one. For smaller D,s X decreases linearly as αD.
Similarly, for resources S Y and S Z the steady state values are written as The dynamics of each molecule species changes with the diffusion constant D. When D is sufficiently large,s X is approximately one. As D decreases and the resource is limited,s X deviates from one, and the maximum inflow rate of the resource is given by J X = DS 0 X (1 −s X ). In each range of D, we refer to (X − i) :S X is sufficiently supplied, and (X − ii) : S X is limited.
For each range, the right-hand-sides of the rate equations (1) can be written as where r A and r B = 1 − r A are the ratios of the resource being distributed to cell types A and B.
For the resources S Y and S Z , we assume the range changes simultaneously for both of Y and Z because, otherwise, either type with a limited resource will vanish; we refer to (YZ − i): S Y and S Z are sufficiently supplied, and (YZ − ii): S Y and S Z are limited. For each range, the right-hand-sides of the rate equations (2) can be written as As will be shown in subsequent subsections, we investigate four cases for the conditions (X, Y Z) = (i, i), (ii, i), (i, ii) and (ii, ii).
The volume V A , V B of each cell increases in proportional to the numbers of molecules as where R is a constant. Then the increase in volumes is given by Let us suppose that a cell divides when its volume exceeds a certain critical value and that the total volume of all cells is restricted by some constant ∑ V k = V T = const. Then the volume fraction of each type A and B follows where B. The case in which all the resources are sufficiently available: condition (X, YZ) = (i, i) For the condition (X, Y Z) = (i, i), when all resources are sufficiently available, the increases in molecule species are written as The solutions to these equations are (v A , v B ) = (1, 0) or (0, 1).
Hence, the fittest cell type, i.e., that with larger f i (i=A or B) dominates the population.

C. The case SX is limited: the condition (X, YZ) = (ii, i)
For the condition (X, Y Z) = (ii, i), when S X is limited, where J X is the maximum inflow of S X and r A and r B = (1 − r A ) are the ratios of S X distributed into A and B. Then, where j X = J X /V T . The stationary condition is written as The stationary condition has solutions (v A , v B ) = (1, 0), (0, 1). The solution (v A , v B ) = (1, 0) satisfies the condition with r A = 1. Linearizing the rate equation around the fixed point (1, 0) by writing v A = 1 − δ, we get satisfies the equation when r A = 0. By linearizing the rate equation around the fixed point (0, 1) and writing v A = δ, we obtain thus, both solutions are stable.

D. The case SY and SZ are limited: condition
For the condition (X, Y Z) = (i, ii) when S Y and S Z are limited, increases of molecule species are written as Then, The stationary state gives By linearizing the dynamics around the fixed point as and the solution is stable.
As will be shown below, 0 < v 0 A < 1, so that the two types coexist: Then the difference in the numerator and denominator in the expression of v 0 A is given by Here, so that the denominator is greater than the numerator. Thus, v 0 A < 1. Therefore, 0 < v 0 A < 1. By writing v A = v 0 A + δ, and linearizing the dynamics around the fixed point, we obtain for v B . The fixed-point solution is obtained in the same way as in the case f A X > f B X , by replacing A with B, and Y with Z. For the condition (X, Y Z) = (ii, ii), when all resources are limited, Then,

Again by writing v
follows, and the coexisting state is stable.

F. Numerical simulations
In this subsection, we present results of numerical simulations of the rate equations for the continuous concentration variables to show whether competition of limited resources results in dominance or coexistence of cells of type A and B. The results for stochastic simulations are shown in the main text.
We consider M tot cells of type A or B. The dynamics of the number N I i of molecule species i(i = X, Y, Z) in a cell-type I(I = A, B) is written as eqs. (1) and (2). The dynamics of each resource S i (i = X, Y, Z) is given by eqs. (3)-(5).
As an initial distribution of cell types, types A and B are each represented at 50%. In each type, the numbers N X , N Y in type A, and N X , N Z in type B are randomly assigned. A cell divides when V K = ∑ i N K i exceeds a threshold V max . At the division, the number of each molecule species N K i is divided between the two daughter cells with a normal distribution with average N K i /2 and variances N K i /4. The normal distribution approximates the random partition of molecules into two daughter cells.
The average time required for the initial condition to reach a state in which either of the types is extinct is shown in Fig. 2 which is consistent with the point where the resource S X starts to be limited in Fig. 1. Correspondingly, the average numbers of N X , N Y and N Z at division events start to deviate below that point (Fig. 3), and available resources show deviations between S X and S Y or S Z below that point (Fig.4).
At the point D = D * , consumption and inflow of resources are balanced and S X starts to compete. The condition for balance in S X is where V denotes the average volume of cells. By substituting V = 750, S X = 850, S 0 X = 1000, M tot = 100, D * is estimated as ∼ 100.
For S Y and S Z , competition starts roughly when the inflow is half that of S X because approximately half of M tot cells consume the respective resource. This suggests that below D = 0.5D * , all the resources are limited; thus, two types of cells coexist. This is consistent with our numerical observations that below D = 40, the types coexist.

II. WHEN DIFFERENT MOLECULE SPECIES CONSUME COMMON RESOURCES : THE CASE WITH KR < KM
In this section, we investigate the case with K R < K M where plural resource species are commonly used for the replication of different molecules X i .

A. Model
In the reaction the correspondence between molecule X j and resource Sĵ is randomly assigned and fixed throughout simulations. Hence, each resource species is used commonly for K M /K R reactions on the average.

B. Results
We show the number of molecule species in each cells (compositional diversity) and that in 10 or more cells (phenotypic diversity), respectively, in Fig. 5(A) for K R = 1, i.e., a single resource is consumed to replicate all the molecule species X i (i = 1, ..., K M ). In this case, neither compositional nor phenotypic diversity increases as the diffusion constant D decreases.
When two resource species are available(K R = 2; Fig. 5(B)), the phenotypic diversity increases as D decreases, but a clear increase is not discernible in the compositional diversity. In the case of small K R , the randomly determined reservoir concentrations, S i 0 ∈ [0, M tot ](i = 1, 2), are also relevant parameters to determine the point at which the resources become limited, in addition to the diffusion constant. Here, we also show the results for fixed S 0 i = M tot (i = 1, 2). In contrast to the K M = K R case, there is an increase in phenotypic diversity for D < 0.1, while even below the point, the number of remaining chemical species is approximately constant and does not increase as D is decreased. This result indicates that the diversity is bounded by the number of resource species: coexistence of at most two cell types is possible.
For larger K R (K R = 10 in Fig. 5(C), and K R = 100 in Fig. 5(D)), both compositional and phenotypic diversity increase as D is decreased because K R is sufficiently large so that it does not effectively restrict the number of cell types. The phenotypic diversity, i.e., the number of coexisting cell types increases as ∼ K R , but the increase is saturated for larger K R (see Fig. 5), which is also bounded by M tot . Indeed, as M tot is increased, the number increases (see below).

III. DEPENDENCE ON Mtot
We investigated the dependence of the diversity on the number of interacting cells, M tot , in the case K M = K R . The number of molecule species in each cells (compositional diversity) and that in 10 or more cells (phenotypic diversity) for M tot = 100, 200, and 300 is shown in Fig. 6.
While both measures of diversity increase for each M tot , as D is decreased, the increment depends on M tot . The increment in compositional diversity decreases as M tot is increased. On the other hand, the increment in phenotypic diversity with the decrease in D increases as M tot is increased. As shown in Fig. 3(C) of the main text, the two types As discussed in the previous section, multiple cell types (up to K R ) can coexist; thus, more types of cells can be present and are not eliminated from the system as M tot is increased, which results in the increase in phenotypic diversity. On the other hand, as the number of cell types increases, a greater number of resource species are competitive because the cell population can consume more resource species. This leads each individual cell to be more specialized with fewer components, which results in suppression of the increase in compositional diversity.

IV. SUPPLEMENTARY FIGURES
Figs 7 and 8 show similarities H ij among a period of division events, where the cell indices given by the x− and y− axes are rearranged so that the same (similar) types are clustered. The data are identical to the similarities represented in Fig. 2(II)(iii) and (III)(iii) of the main text, but the plots are given after rearrangement of cell indices.
Types II-A and II-B are clustered as shown at the top of Fig. 7. The similarities between cells of the same type have values close to 1, while types A and B have similarities around 0.6.
The types from III-A to III-F are clustered at the top of Fig. 8. The similarities between cells of the same type have values close to 1, and those between types A and B, and between D and E have positive values. For other pairs, the values are almost orthogonal.