Switching Between Cooperation and Competition in the Use of Extracellular Glucose

Abstract

This paper addresses some questions related to the evolution of cooperative behaviors, in the context of energetic metabolism. Glycolysis can perform either under a dissipative working regime suitable for rapid proliferation or under an efficient regime that entails a good modus operandi under conditions of glucose shortage. A cellular mechanism allowing switching between these two regimes may represent an evolutionary achievement. Thus, we have explored the conditions that might have favored the emergence of such an accommodative mechanism. Because of an inevitable conflict for resources between individual interests and the common good, rapid and inefficient use of glucose is always favored by natural selection in spatially homogeneous environment, regardless of the external conditions. In contrast, when the space is structured, the behavior of the system is determined by its free energy content. If the fuel is abundant, the dissipative strategy dominates the space. However, under famine conditions the efficient regime represents an evolutionary stable strategy in a Harmony game. Between these two extreme situations, both metabolic regimes are engaged in a Prisoner’s Dilemma game, where the output depends on the extracellular free energy. The energy transition values that lead from one domain to another have been calculated. We conclude that an accommodative mechanism permitting alternation between dissipative and efficient regimes might have evolved in heterogeneous and highly fluctuating environments. Overall, the current work shows how evolutionary optimization and game-theoretical approaches can be complementary in providing useful insights into biochemical systems.

Appendices

Appendix A: Insight into the Thermodynamic Trade-off Between Output Power and Efficiency

In the framework of nonequilibrium thermodynamics, glycolytic fluxes can be described by the following equations (Esteban del Valle and Aledo 2006):

$$ J_1 \, = \,L_{11} \,X_1 \, + \,L_{12} \,X_2 $$

(A1)

$$ J_2 \, = \,L_{12} \,X_1 \, + \,L_{22} \,X_2 $$

(A2)

where L _ij ≥ 0 are phenomenological coefficients that incorporate kinetic (enzymatic) attributes. Equation (A2) can be interpreted as follows: glucose consumption is favored by its driving force (L₂₂ X ₂ > 0). In contrast, a high ATP concentration will lead to a very negative value of X ₁, slowing down glucose consumption by a factor of L₁₂ X ₁. Therefore, when either X ₁ = 0 or ATP synthesis and glucose breakdown are uncoupled (L₁₂ = 0), the consumption of glucose becomes maximal and is given by J ₂ = L₂₂ X ₂. Under these hypothetical conditions, the maximum attainable input power is L₂₂ X ²₂ . It should be noted that the efficiency corresponding to this hypothetical situation is zero.

In order to stress the trade-off existing in glycolysis between the output power and the efficiency, the remainder of this section is devoted to express the output power as a function of the efficiency.

Bearing in mind the definition of output power and Eqs. (1), (A1), and (A2), we are entitled to write

$$ W_o \, = \,\eta J_2 X_2 \, = \,\eta X_2^2 \,\left( {L_{12} \,\left( {{{X_1 } \mathord{\left/ {\vphantom {{X_1 } {X_2 }}} \right. \kern-\nulldelimiterspace} {X_2 }}} \right) + L_{22} } \right)$$

(A3)

This equation can be simplified taking into account some constraints that enforce certain relationships between their variables.

In this respect, an interesting concept introduced by Kedem and Caplan (1965) to assist in the study of linear energy converters is the degree of coupling, q = L₁₂/(L₁₁L₂₂)^1/2, which represents a dimensionless measure of how tightly the driven process is coupled to the driver process. Another useful concept is the so-called phenomenological stoichiometry, defined as Z = (L₁₁/L₂₂)^1/2. Although Z must not be confused with the mechanistic stoichiometry, for values of q close to 1 the phenomenological and the mechanistic stoichiometries coincide (Stucki 1980). In the particular case of glycolysis, where ATP formation and glucose breakdown are tightly coupled processes, with a fixed stoichiometry of two molecules of ATP formed per molecule of glucose split, there are three constraints that need to be satisfied: (i) q = 1, (ii) J ₁ /J ₂ = 2, and (iii) Z = (L₁₁/L₂₂)^1/2 = 2.

Bearing the first constraint in mind, L₁₂ can be substituted for (L₁₁L₂₂)^1/2. On the other hand, constraint (ii) allows us to rewrite Eq. (1) as –η/2 = X ₁ /X _2.

When translating these substitutions into Eq. (A3) we obtain W _o = ηX ²₂ ((L₁₁L₂₂)^1/2 (–η/2) + L₂₂), an expression that can be reorganized to yield W _o = ηX ²₂ L₂₂( (L₁₁/L₂₂)^1/2 (–η/2) + 1), which can be further simplified when considering constraint (iii):

$$ W_o \, = \,L_{22} X_2^2 \left( { - \eta ^2 \, + \,\eta } \right)$$

(A4)

If the output power is normalized dividing by the maximum attainable input power, we obtain Eq. (2) from the main text.

Appendix B: Role of Entropy Production

In addition to W _o and η, another relevant factor is the dissipation function or rate of entropy production, Φ, which can be interpreted as the rate at which the energy available for biological processes is reduced. This thermodynamic function can be formulated as the sum of the products of all the fluxes and their corresponding driving forces (Kondepudi and Prigogine 1999):

$$ \Phi \, = \,J_1 X_1 \, + \,J_2 X_2 \, + \,J_3 X_3 $$

(B1)

Herein, J ₃ is the flux of all the ATP-consuming processes lumped together (Fig. 1A). These processes are driven by the hydrolysis of ATP, that is, X ₃ = –X ₁. Thus, J ₃ = –L₃₃ X ₁ (Stucki 1980). Therefore, Eq. (B1) can be expressed as

$$ \Phi \, = \,J_1 X_1 \, + \,J_2 X_2 \, + \,L_{33} X_1^2 $$

(B2)

According to the definitions given above, J ₁ X ₁ = –W _o , J ₂ X ₂ = W _o /η and X ₁ = –ηX ₂/2. Now, substituting into Eq. (B2), we obtain Ф = –W _o + W _o /η + L₃₃ X ²₂ η²/4. When Eq. (A4) is taken in consideration, the dissipation function can be rewritten as

$$ \Phi \, = \,L_{22} X_2^2 \left[ {\left( {1\, + \,L_{33} /4L_{22} } \right)\eta ^2 \, - \,2\eta \, + \,1} \right] $$

(B3)

In order to assess the contribution of dissipative and efficient cells to the local entropy production, we need to know, in each case, the ratio L₃₃/L₄₄. To this purpose, Prigogine’s theorem of the minimum entropy production can be helpful. According to this theorem, in the steady state the entropy production must be minimal. In formal terms: dΦ/dη = 0 = 2(1 + L₃₃/4L₂₂) η – 2, from which we obtain the following constraint:

$$ \eta \, = \,4\,L_{22} /\left( {4\,L_{22} \, + \,L_{33} } \right)$$

(B4)

Herein, it may be convenient to remember that DMOP and EMOP organisms operate at efficiencies of 1/2 and 2/3, respectively. This means that L₃₃ = 4L₂₂ for dissipative cells, whereas in the case of efficient organisms the relation is L₃₃ = 2L₂₂. Bearing these considerations in mind, Eq. (B3) leads to a straightforward conclusion: the contributions of DMOP and EMOP cells to the local entropy production are different (Φ_DMOP = 1/2 > Φ_EMOP = 1/3).