## Abstract

Glycolysis, an ancient energy-processing pathway, can operate either under an efficient but slow regime or, alternatively, under a dissipative but fast-working regime. Trading an increase in efficiency for a decrease in rate represents a cooperative behavior, while a dissipative metabolism can be regarded as a cheating strategy. Herein, using irreversible thermodynamic principles and methods derived from game theory, we investigate whether, and under what conditions, the interplay between these two metabolic strategies may have promoted the clustering of undifferentiated cells. In the current model, multicellularity implies the loss of motility, which represents a hindrance rather than a improvement when competing with mobile single-celled organisms. Despite that, when considering glycolysis as the only energy-processing pathway, we conclude that cells endowed with a low basal anabolic metabolism may have benefited from clustering when faced to compete with cells exhibiting a high anabolic activity. The current results suggest that the transition to multicellularity may have taken place much earlier than hitherto thought, providing support for an extended period of Precambrian metazoan diversification.

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## Acknowledgments

The author thanks Alicia Esteban del Valle, Juan Antonio Pérez-Claros, and Miguel Angel Medina for thoughtful discussions. The author is also grateful to three anonymous referees who have helped to improve the original manuscript. This work was supported by the Universidad de Málaga and Grant CGL2007-65010 from the Ministerio de Educación y Ciencia de España.

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## Appendix: Insight into the Thermodynamic Working Regimes

### Appendix: Insight into the Thermodynamic Working Regimes

In the framework of nonequilibrium thermodynamics, the glycolytic fluxes of ATP formation (*J*
_{1}) and glucose consumption (*J*
_{2}) are proportional to the chemical affinities for ATP synthesis (*X*
_{1}) and glucose breakdown (*X*
_{2}):

where the proportionality coefficients (L_{ij} ≥ 0) incorporate kinetic (enzymatic) attributes. The rate at which energy is extracted from glucose is referred to as input power (*W*
_{
i
}), which is computed by multiplying the energy released per mole of glucose brokendown (*X*
_{
2
}) by the flux of glucose consumption (*J*
_{
2
}): *W*
_{
i
} = *J*
_{
2
}
*X*
_{
2
}. Under physiological conditions, the chemical affinity for ATP synthesis takes negative values, slowing down glucose consumption. Therefore, the maximum attainable input power is L_{22}X_{2}
^{2}, achieved under the hypothetical condition that ATP synthesis and glucose breakdown are uncoupled, L_{12} = 0 (Aledo 2007). On the other hand, the rate at which energy is conserved in the form of ATP is referred to as output power, *W*
_{
o
}, which can be calculated as the rate of ATP formation times the energy needed to produce 1 mol of ATP: *W*
_{
o
}
* = *−*J*
_{
1
}
*X*
_{
1
}
*.* Since X_{1} is negative (energy that has to be provided), the negative sign in the former equation has been arbitrarily introduced to give a positive output power.

Although far from equilibrium we lose the mathematical guarantee of linearity, this does not mean that linear flow-force relations cannot be established. In fact, empirical and theoretical analyses suggest that linear relations between fluxes and forces are much more common than perhaps expected (Westerhoff and van Dam 1987).

For our purposes, it will be convenient to express the output power as a function of the efficiency. To this end, and considering that the efficiency is defined as minus the ratio between input and output power, we can start by writing *W*
_{
o
}
* = ηJ*
_{
2
}
*X*
_{
2
}
*.* Now, substituting J_{2} by (A2), we obtain

This equation can be simplified taking into account some constraints that enforce certain relationships between their variables. In this respect, two interesting concepts that assist in the study of linear energy converters are the degree of coupling, q = L_{12}/(L_{11}L_{22})^{1/2}, and the phenomenological stoichiometry, Z = (L_{11}/L_{22})^{1/2}. For glycolysis, where ATP formation and glucose breakdown are tightly coupled processes (q = 1), with a fixed stoichiometry of two molecules of ATP formed per molecule of glucose split (Z = *J*
_{
1
}
*/J*
_{
2
} = 2), the following equalities hold: −*η*/2 = *X*
_{
1
}
*/X*
_{
2
} and L_{12} = (L_{11}L_{22})^{1/2}. When translating these substitutions into Eq. (A3), we obtain *W*
_{
o
} = *η X*
_{
2
}
^{2} ((L_{11}L_{22})^{1/2} (−*η*/2) + L_{22}), an expression that can be reorganized to yield *W*
_{
o
}
* = η*
*X*
_{
2
}
^{2}L_{22} ((L_{11}/L_{22})^{1/2} (−*η*/2) + 1), which can be further simplified when considering the constraint Z = (L_{11}/L_{22})^{1/2} = 2:

Thus, the output power is a negative quadratic function of the efficiency. The value of *η* for which a maximun output power is achieved can now be calculated:

In the current paper, cells performing under conditions optimizing the output power are said to exhibit a DF phenotype. Although the ability to maximize the output power is an important trait, for biological systems a high degree of fitness may imply not only high output powers but also high efficiencies (Aledo et al. 2007). Consequently, we also considered a function representing a compromise between high output power and high efficiency. Such a function can be obtained, as proposed by Stucki (1980), multiplying the efficiency by the output power. The resulting new function is referred to as efficient output power, EW_{o} (Stucki 1980; Aledo et al. 2007):

It is easy to prove that *η* = 2/3 represents an optimum for this finction. Organisms that operate under conditions optimizing the efficient output power are said to exhibit an ES regime.

In order to calculate the output power obtained by DF and EF organisms, we only have to evaluate the function (A4) at the points *η* = 1/2 and *η* = 2/3. In this way, it can be concluded that the output power of DF and EF cells are 1/4 and 2/9 of the maximum attainable input power, respectively.

In addition to the output power and efficiency, another relevant factor is the dissipation function, Φ. This thermodynamic function can be formulated as the sum of the products of all the fluxes and their corresponding driving forces. In our model:

where J_{3} is the flux of all the ATP-consuming processes lumped together. Since these processes are driven by the hydrolysis of ATP, X_{3} = −X_{1}. Thus, J_{3} = −L_{33}X_{1} and (A7) can be expressed as

According to the definitions given previously, *J*
_{
1
}
*X*
_{
1
} = −*W*
_{
o
}, *J*
_{
2
}
*X*
_{
2
}
* =* *W*
_{
o
}/*η*, and *X*
_{
1
} = −*ηX*
_{
2
}/2. Thus, (A8) can be rewritten Φ* = *−*W*
_{
o
}
* + W*
_{
o
}/*η + *L_{33}
*η*
^{2}
*X*
_{
2
}
^{2}/4. Finally, when we substitute the output power by its corresponding function of the efficiency, the result is

In order to evaluate Φ in the cases of DF cells (*η* = 1/2) and ES cells (*η* = 2/3), we need to know the value of L_{33}/L_{22}. To this aim we will take advantage of the minimum entropy production theorem, which claims that in the steady state the entropy production must be minimal. That is, *d*Φ*/dη* = 0 = 2(1 + L_{33}/4L_{22}) *η* − 2, from which we obtain the following constraint: *η *= 4 L_{22} /(4 L_{22} + L_{33}). Herein, it may be convenient to remember that DF and ES organisms operate at efficiencies of 1/2 and 2/3, respectively. This means that L_{33} = 4 L_{22} for dissipative cells, whereas in the case of efficient organisms the relation is L_{33} = 2 L_{22} (Aledo and Esteban del Valle 2005). Bearing these considerations in mind, Eq. A9 leads to a straightforward conclusion: The dissipation functions of DF and EF cells are one-half and one-third of the maximum attainable input power, respectively.

An intuitive (graphical) insight into the thermodynamics underlying these working regimes (DF vs. ES) can be provided by plotting the output power and dissipative functions against the efficiency [*W*
_{
o
} = L_{22}X_{2}
^{2} (−*η*
^{2} + *η*), solid line; Φ_{
DF
}
* *= L_{22}X_{2}
^{2} (2*η*
^{2} − 2*η* + 1), dashed line; Φ_{
ES
} = L_{22}X_{2}
^{2} (1.5*η*
^{2} − 2*η* + 1), dotted-dashed line; the *y* axis is in units of L_{22}X_{2}
^{2}]; it can be graphically deduced that an ES regime represents an excellent compromise between high output power and low entropy production. In other words, an ES strategy trades a low decrease in output power for a high decrease in entropy production.

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Aledo, J.C. An Early and Anaerobic Scenario for the Transition to Undifferentiated Multicellularity.
*J Mol Evol* **67**, 145–153 (2008). https://doi.org/10.1007/s00239-008-9128-y

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DOI: https://doi.org/10.1007/s00239-008-9128-y