Does maximal entropy production play a role in the evolution of biological complexity? A biological point of view

Abstract

A considerable literature has developed around the concept that the complexity of biological organisms, and the development of ever increasing complexity during biological evolution, is driven, in some way, by the maximization of entropy production (MEP). Most of these studies deal in very general terms with the living state and do not examine specific biological models. In the present study, we discuss some of the basic postulates of MEP as they are applied to living organisms. It is concluded that the MEP ideas seem to have little relevance either for the development of biological complexity or biological evolution.

Appendices

Appendix 1

1.1 Thermodynamic entropy

The thermodynamic, or phenomenological, definition of entropy, due to Clausius, S_C, refers to reversible heat transfer ($\delta Q_{\text{rev}}$) from the surroundings into a system at temperature T

$${\text{d}}S_{\text{C}} = \frac{{\delta Q_{\text{rev}} }}{T}\;{\text{and}}\;S_{\text{c}} = \mathop \int \limits_{A}^{B} \frac{{\delta Q_{\text{rev}} }}{T}.$$

(3)

S_C is the increase in entropy of the system, where the integral is taken between any two arbitrary states. It is universally known that it is the Clausius function (Eq. 3) which defines entropy and gave rise to the second law. For a reversible transformation between two states $\Delta S_{\text{rev}} = 0$, and for an irreversible transformation between the same two state as the reversible process $\Delta S_{\text{irrev}} \ge 0$. The second law states that for any transformation, $\Delta S_{\text{tot}} \ge 0$, where S_tot is the total entropy change of the system plus the environment (surroundings).

Of central importance is the so-called fundamental thermodynamic equation, due to Gibbs:

$${\text{d}}U = T{\text{d}}S - p{\text{d}}V + \mu {\text{d}}N,$$

(4)

where U is the internal energy, S is the entropy, T is the temperature, p is the pressure, V is the volume, µ is the chemical potential (the Gibbs free energy per mole at constant volume) and N is the number of particles.

It should be specified that the word “reversible” in physics and chemistry is often used differently. In physics, it means that a process may be carried out by infinitesimal changes (microscopically reversible), in which it is considered to not deviate significantly from equilibrium where the principle of detailed balance applies. An irreversible process does not have this constraint. In chemistry, a reversible process has no requirement for detailed balance and the reactants continually generate the products and the products continually generate the reactants, albeit at different rates when the system is out of equilibrium.

1.2 Statistical entropy

While the thermodynamic entropy relates to macroscopic systems, statistical entropy purports to provide a microscopic description in terms of the permutations that a system of many particles may adopt. It is defined in a completely different manner with respect to thermodynamic entropy. The basic definition (Eq. 5), due to Boltzmann (S_B), is given by the logarithm of the maximum number of accessible microstates (particle permutations in position and momentum; W), which are accessible in the system phase space volume of an isolated ensemble of non-interacting, classical particles in equilibrium and where each microstate has the same energy

$$S_{\text{B}} = k_{\text{B}} \ln W.$$

(5)

It is this equation, and not Eq. 3, which introduces the concept of order/disorder into entropy. At equilibrium, the system is in a state of maximum disorder (greatest number of microstates) and S_B attains its maximal value.

A more general definition of the statistical entropy is that provided by Gibbs (S_G), which is valid for the so-called canonical ensemble. Energy may be exchanged with an environmental bath (closed system). It recognizes that the microstates may differ in energy, as distinct from S_B,

$$S_{\text{G}} = - k_{\text{B}} \mathop \sum \limits_{i} p_{i} \ln p_{i} .$$

(6)

The microstate probability, p_i, ignoring energy degeneracy, is usually defined as $p_{i} = e^{{ - \beta E_{i} }} /\mathop \sum \nolimits_{i} e^{{ - \beta E_{i} }}$, where E_i is the internal energy of each i-th microstate. The denominator in this probability expression is the canonical partition function for an energetically discrete and equilibrated system in which the mean macrostate internal energy is $E = \mathop \sum \nolimits_{i} p_{i} E_{i}$.

It is usually assumed that S_C = S_B = S_G, i.e., the thermodynamic and statistical entropies are equivalent. This has, in fact, been demonstrated in the case of S_C = S_G for the special case in which the mass, or number of particles (N), of the system remains constant (Jaynes 1965). However, in the case in which entropy is expressed “extensively”, i.e., it scales with the extensive variables (E, V, N), this equivalence may not be the case (e.g., Versteegh and Dieks 2011).

Crooks entropy production fluctuation

This approach is based on probability theory for non-equilibrium, stochastic, microscopically reversible systems and, as can be seen from Eq. 7, it states that the logarithm of the probability of the forward, P_F, and reverse, P_R, transitions is given by the entropy production, ω,

$$\ln \frac{{P_{\text{F}} \left( \omega \right)}}{{P_{\text{R}} \left( { - \omega } \right)}} = \omega ,$$

(7)

and is not restricted only to systems which have undergone a perturbation to near equilibrium conditions. Thus, it is a non-equilibrium analysis associated with entropy production which is free of the LEA.

Based on the Crooks (1999) reasoning, England (2013) applied it to the problem of bacterial replication. The macrosystem  is defined as that in which the probability of one bacterial replicator (P_F) producing a second bacterium is considered in the forward (F) direction, and then considering the reverse process (P_R). In the words of England “The first of these probabilities P(I → II) gives the likelihood that a system prepared according to I is observed to satisfy II after time τ. The second probability P(II → I) gives the likelihood that after another interval of τ that the same system would be observed again to satisfy I. Putting these two quantities together and taking their ratio thus quantifies for us the irreversibility of spontaneously propagating from I to II”. This leads to the equation

$$\frac{1}{T}\Delta Q\left( {I \to II} \right) + \ln \frac{{P\left( {{\text{II}} \to {\text{I}}} \right)}}{{P\left( {{\text{I}} \to {\text{II}}} \right)}} + \Delta S_{\text{int}} \ge 0,$$

(8)

where $\Delta S_{\text{int}}$ is the internal entropy change for the forward reaction and the first term is the change in entropy of the bath. While we consider this reasoning sound for time reversible systems, as stressed by Crooks (1999) and also by England (2013), the bacterial reproduction case is not, and cannot ever be, time reversible.

1.3 Non-equilibrium thermodynamics and the MEP

Non-equilibrium thermodynamics is based on a number of propositions taken from classical equilibrium thermodynamics. For example, the fundamental Gibbs equation (Eq. 4) is assumed to apply. In particular, it is usually based on the so-called local equilibrium assumption (LEA). This limits its usefulness to non-equilibrium systems which are close to equilibrium, while many biological systems are far from equilibrium. In the LEA it is assumed that within the macroscopic non-equilibrium system, there will be small volume elements, dV, in a near equilibrium state and therefore with a definable entropy. The total entropy of the macroscopic system is obtained by integration over all small volume elements. Recently, Ben-Naim (2018) has cast doubt on this aspect. For chemical reactions, there will be interactions between the small volume elements (boundary effects), and integration to the macrosystem to obtain the non-equilibrium entropy is suggested to be incorrect. The point is made that as the volume elements increase in size, the boundary effects decrease and become increasingly irrelevant. Be that as it may, the LEA forms the basis of classical non-equilibrium thermodynamics and the MEP.

MEP utilizes the linear assumption between forces, X, and fluxes, J, (Eq. 9). In the LEA, due to the pioneering work of Prigogine (1945),

$$\frac{{{\text{d}}S_{i} }}{{{\text{d}}t}} = \mathop \sum \limits_{i} X_{i} J_{i} \;{\text{with}},\;J_{i} = \mathop \sum \limits_{k} L_{i,k} X_{k} ,$$

(9)

where X is the generalized and constant thermodynamic force, L are the rate parameters and J is then the constant fluxes and ${\text{d}}S_{i} /{\text{d}}t$, often written as σ, is the rate of entropy production by the system due to irreversible processes. Whereas in many physical systems X may be some physical gradient, for (bio)chemical reactions X is usually considered to be the free energy difference between products and substrates (ΔG) (e.g., Velasco et al. 2011; Kondepudi and Prigogine 1998).

It is also proposed that entropy may be represented as the time variable (t) of the equilibrium entropies (Eqs. 3–5). For example, and considering Eq. 5,

$$S\left( t \right) = - k_{\text{B}} \mathop \sum \limits_{i} p_{i} \left( t \right)\ln p_{i} \left( t \right),$$

(10)

where p_i is now considered to be a non-equilibrium probability (Tsallis 1988).

Non-equilibrium thermodynamics is concerned with fluxes of matter and energy and is usually based on equation

$$\frac{{{\text{d}}S}}{{{\text{d}}t}} = \frac{{{\text{d}}S_{\text{i}} }}{{{\text{d}}t}} + \frac{{{\text{d}}S_{\text{e}} }}{{{\text{d}}t}},$$

(11)

where dS/dt is the time derivative of the entropy of the macrosystem, dS_i/dt is the rate of entropy production by the system due to irreversible processes within the macrosystem and dS_e/dt is the rate of entropy transfer with the environment.

As pointed out by Velasco et al. (2011), the term “entropy production” is not always clear. For non-equilibrium systems in the local equilibrium assumption (LEA), these authors provide convincing reasoning that it is the “excess entropy production”, where “excess entropy production” means the heat dissipated by the system and which is unavailable to perform work. If the SS is an equilibrium state, then both dS_e/dt and dS_i/dt = 0. If the SS is a non-equilibrium state, then $- \frac{{{\text{d}}S_{\text{e}} }}{{{\text{d}}t}} = \frac{{{\text{d}}S_{\text{i}} }}{{{\text{d}}t}} > 0$.

MEP theory is based on the constancy of the thermodynamic force (X), and so for a (bio)chemical reaction ΔG is assumed to be constant. For a generalized (bio)chemical reaction (A ↔ B), and considering a closed, non-equilibrium system, the thermodynamic force (ΔG) is given by the well-known expression:

$$\Delta G = \Delta G^{0} + Rt\ln \frac{\left[ B \right]}{\left[ A \right]},$$

(12)

where ΔG is the free energy change for any concentrations of reactants and products; ΔG⁰ is the standard free energy change where all components are present at equal molarities; the terms in the square brackets are the concentrations of the reactants and products, which may assume any value. R is the gas constant; T is the temperature. This is a simple way of representing a non-equilibrium state in a chemical macrosystem.

Let us now consider this reaction as one step in an “open” sequence of reactions (see also Beard and Qian 2007). Thus, the generalized (bio)chemical reaction may now be written as → A ↔ B → . This non-equilibrium reaction will satisfy the condition of a constant thermodynamic force (ΔG) only when the [B]/[A] ratio is constant in time, i.e., at the SS. This shows, in a simple way, that the SS is essential for MEP.

In MEP theory, the thermodynamic force is assumed to be constant and the system attains a steady state (SS) due to Le Chậtelier’s principle. The SS may be thought of as being analogous to the equilibrium state in equilibrium thermodynamics, i.e., in the open, non-equilibrium system the spontaneous movement is toward an SS.

Appendix 2

Let us consider a closed system where a reaction transforming the component A into component B occurs:

$$\begin{aligned} A_{{\mathop \leftarrow \limits_{{k_{ - 1} }} }}^{{\mathop \to \limits^{{ k_{1 } }} }} B \hfill \\ \mathop \to \limits^{ J } \hfill \\ \end{aligned}$$

where k₁ and k₋₁ are the forward and backward rate constants. We put the A and B concentration $\left[ A \right] \equiv C_{\text{A}}$, $\left[ B \right] \equiv C_{\text{B}}$. Pressure, p, and temperature, T, are constant.

The kinetics, with the assumption of the mass action law, are represented by a system of two differential equations:

$$\begin{aligned} \frac{{{\text{d}}C_{\text{A}} }}{{{\text{d}}t}} = - k_{1} C_{\text{A}} + k_{ - 1} C_{\text{B}} , \hfill \\ \frac{{{\text{d}}C_{\text{B}} }}{{{\text{d}}t}} = k_{1} C_{\text{A}} - k_{ - 1} C_{\text{B}} , \hfill \\ \end{aligned}$$

(13)

and two reaction flows (reaction rates) with two kinetic constants are defined and one macroscopic flow, J, is present

$$J = - \frac{{{\text{d}}C_{\text{A}} }}{{{\text{d}}t}} = \frac{{{\text{d}}C_{\text{B}} }}{{{\text{d}}t}}.$$

At equilibrium, $J = 0$ and then $k_{1} C_{\text{A}}^{\text{eq}} = k_{ - 1} C_{\text{B}}^{\text{eq}}$ → $\frac{{C_{\text{B}}^{\text{eq}} }}{{C_{\text{A}}^{\text{eq}} }} = \frac{{k_{1} }}{{k_{ - 1} }} \equiv K$ (equilibrium constant).

The intention is to find an explicit form for the rate of entropy production $\sigma = JX$, where J is the reaction flux and X is the thermodynamic generalized force (e.g., Glansdorff and Prigogine 1971). We consider that all the conditions for the validity of this entropy production representation are satisfied.

In every non-equilibrium instant, we have

$$J = k_{1} C_{\text{A}} - k_{ - 1} C_{\text{B}} ,$$

(14)

and, as no exchange of matter with the surroundings is present (closed system),

$$C_{\text{A}} + C_{\text{B}} = C_{\text{A}}^{\text{eq}} + C_{\text{B}}^{\text{eq}} .$$

(15)

The concentration of A and B can be written as

$$\begin{aligned} C_{\text{A}} = C_{\text{A}}^{\text{eq}} + \Delta C_{\text{A}} \hfill \\ C_{\text{B}} = C_{\text{B}}^{\text{eq}} + \Delta C_{\text{B}} , \hfill \\ \end{aligned}$$

with $\Delta C_{\text{A}} = - \Delta C_{\text{B}}$ (see Eq. 15) and the flux J can then be explicitly written as

$$\begin{aligned} J & = k_{1} \left( {C_{\text{A}}^{\text{eq}} + \Delta C_{\text{A}} } \right) - k_{ - 1} \left( {C_{\text{B}}^{\text{eq}} + \Delta C_{\text{B}} } \right) \\ = \Delta C_{\text{A}} \left( {k_{1} + k_{ - 1} } \right) = k_{1} \Delta C_{\text{A}} \left( {1 + \frac{1}{K}} \right). \\ \end{aligned}$$

(16)

Now, we look for an explicit representation of the generalized force X that appears in the symbolic expression for the entropy production.

The “thermodynamic force” X can be written in terms of the chemical potentials μ as well as the Gibbs free energy $\Delta G$

$$X = \frac{{\mu_{\text{A}} - \mu_{\text{B}} }}{T} = \frac{\Delta G}{T} ,$$

where $\Delta G = - RT\ln \left( {K\frac{{C_{\text{A}} }}{{C_{\text{B}} }}} \right)$ and then

$$\begin{aligned} X & = - R\ln \left( {K\frac{{C_{\text{A}}^{\text{eq}} + \Delta C_{\text{A}} }}{{C_{\text{B}}^{\text{eq}} + \Delta C_{\text{B}} }}} \right) = - R\ln \left( {\frac{{1 + \frac{{\Delta C_{\text{A}} }}{{C_{\text{A}}^{\text{eq}} }}}}{{1 + \frac{{\Delta C_{\text{B}} }}{{C_{\text{B}}^{\text{eq}} }}}}} \right) \\ & = - R\ln \left( {1 + \frac{{\Delta C_{\text{A}} }}{{C_{\text{A}}^{\text{eq}} }}} \right) + R\ln \left( {1 + \frac{{\Delta C_{\text{B}} }}{{C_{\text{B}}^{\text{eq}} }}} \right). \\ \end{aligned}$$

(17)

If the system is near equilibrium,

$$\frac{{\Delta C_{\text{A}} }}{{C_{\text{A}}^{\text{eq}} }} \ll 1 ; \frac{{\Delta C_{\text{B}} }}{{C_{\text{B}}^{\text{eq}} }} \ll 1$$

and then

$$\ln \left( {1 + \frac{{\Delta C_{\text{A}} }}{{C_{\text{A}}^{\text{eq}} }}} \right) \cong \frac{{\Delta C_{\text{A}} }}{{C_{\text{A}}^{\text{eq}} }},$$

so that the thermodynamic force X is written as

$$\begin{aligned} X & = R\left( {\frac{{\Delta C_{\text{A}} }}{{C_{\text{A}}^{\text{eq}} }} - \frac{{\Delta C_{\text{B}} }}{{C_{\text{B}}^{\text{eq}} }}} \right) = R\left( {\frac{{\Delta C_{\text{A}} }}{{{\text{C}}_{\text{A}}^{\text{eq}} }} + \frac{{\Delta C_{\text{A}} }}{{C_{\text{B}}^{\text{eq}} }}} \right) = R\frac{{\Delta C_{\text{A}} }}{{C_{\text{A}}^{\text{eq}} }}\left( {1 + \frac{{C_{\text{A}}^{\text{eq}} }}{{C_{\text{B}}^{\text{eq}} }}} \right) \\ & = R\frac{{\Delta C_{\text{A}} }}{{C_{\text{A}}^{\text{eq}} }}\left( {1 + \frac{1}{K}} \right). \\ \end{aligned}$$

(18)

Finally, combining 16 and 18, the “entropy production rate” is

$$\sigma = JX = R\frac{{\Delta C_{\text{A}}^{2} }}{{C_{\text{A}}^{\text{eq}} }}k_{1} \left( {1 + \frac{1}{K}} \right)^{2} = R\frac{{\Delta C_{\text{A}}^{2} }}{{C_{\text{A}}^{\text{eq}} }}\frac{{\left( {k_{1} + k_{ - 1} } \right)^{2} }}{{k_{1} }},$$

(19)

where it can be seen that in the local equilibrium assumption (LEA) σ is a function of both the forward and reverse kinetic rate constants, and does not yield the single rate constants.