Coupling Through Emergent Conservation Laws (Part 2)

joint post with Jonathan Lorand, Blake Pollard, and Maru Sarazola

Here’s a little introduction to the chemistry and thermodynamics prerequisites for our work on ‘coupling’. Luckily, it’s fun stuff that everyone should know: a lot of the world runs on these principles!

We will be working with reaction networks. A reaction network consists of a set of reactions, for example

\mathrm{X}+\mathrm{Y}\longrightarrow \mathrm{XY}

Here X, Y and XY are the species involved, and we interpret this reaction as species X and Y combining to form species XY. We call X and Y the reactants and XY the product. Additive combinations of species, such as X + Y, are called complexes.

The law of mass action states that the rate at which a reaction occurs is proportional to the product of the concentrations of the reactants. The proportionality constant is called the rate constant; it is a positive real number associated to a reaction that depends on chemical properties of the reaction along with the temperature, the pH of the solution, the nature of any catalysts that may be present, and so on. Every reaction has a reverse reaction; that is, if X and Y combine to form XY, then XY can also split into X and Y. The reverse reaction has its own rate constant.

We can summarize this information by writing

\mathrm{X} + \mathrm{Y} \mathrel{\substack{\alpha_{\rightarrow} \\\longleftrightarrow\\ \alpha_{\leftarrow}}}  \mathrm{XY}

where \alpha_{\to} is the rate constant for X and Y to combine and form XY, while \alpha_\leftarrow is the rate constant for the reverse reaction.

As time passes and reactions occur, the concentration of each species will likely change. We can record this information in a collection of functions

[\mathrm{X}] \colon \mathbb{R} \to [0,\infty),

one for each species X, where \mathrm{X}(t) gives the concentration of the species \mathrm{X} at time t. This naturally leads one to consider the rate equation of a given reaction, which specifies the time evolution of these concentrations. The rate equation can be read off from the reaction network, and in the above example it is:

\begin{array}{ccc}  \dot{[\mathrm{X}]} & = & -\alpha_\to [\mathrm{X}][\mathrm{Y}]+\alpha_\leftarrow [\mathrm{XY}]\\  \dot{[\mathrm{Y}]} & = & -\alpha_\to [\mathrm{X}][\mathrm{Y}]+\alpha_\leftarrow [\mathrm{XY}]\\  \dot{[\mathrm{XY}]} & = & \alpha_\to [\mathrm{X}][\mathrm{Y}]-\alpha_\leftarrow [\mathrm{XY}]  \end{array}

Here \alpha_\to [\mathrm{X}] [\mathrm{Y}] is the rate at which the forward reaction is occurring; thanks to the law of mass action, this is the rate constant \alpha_\to times the product of the concentrations of X and Y. Similarly, \alpha_\leftarrow [\mathrm{XY}] is the rate at which the reverse reaction is occurring.

We say that a system is in detailed balanced equilibrium, or simply equilibrium, when every reaction occurs at the same rate as its reverse reaction. This implies that the concentration of each species is constant in time. In our example, the condition for equilibrium is

\displaystyle{ \frac{\alpha_\to}{\alpha_\leftarrow}=\frac{[\mathrm{XY}]}{[\mathrm{X}][\mathrm{Y}]} }

and the rate equation then implies that

\dot{[\mathrm{X}]} =  \dot{[\mathrm{Y}]} =\dot{[\mathrm{XY}]} = 0

The laws of thermodynamics determine the ratio of the forward and reverse rate constants. For any reaction at all, this ratio is

\displaystyle{ \frac{\alpha_\to}{\alpha_\leftarrow} = e^{-\Delta {G^\circ}/RT} }  \qquad \qquad \qquad (1)

where T is the temperature, R is the ideal gas constant, and \Delta {G^\circ} is the free energy change under standard conditions.

Note that if \Delta {G^\circ} < 0, then the rate constant of the forward reaction is larger than the rate constant of the reverse reaction:

\alpha_\to > \alpha_\leftarrow

In this case one may loosely say that the forward reaction ‘wants’ to happen ‘spontaneously’. Such a reaction is called exergonic. If on the other hand \Delta {G^\circ} > 0, then the forward reaction is ‘non-spontaneous’ and it is called endergonic.

The most important thing for us is that \Delta {G^\circ} takes a very simple form. Each species has a free energy. The free energy of a complex

\mathrm{A}_1 + \cdots + \mathrm{A}_m

is the sum of the free energies of the species \mathrm{A}_i. Given a reaction

\mathrm{A}_1 + \cdots + \mathrm{A}_m \longrightarrow \mathrm{B}_1 + \cdots + \mathrm{B}_n

the free energy change \Delta {G^\circ} for this reaction is the free energy of

\mathrm{B}_1 + \cdots + \mathrm{B}_n

minus the free energy of

\mathrm{A}_1 + \cdots + \mathrm{A}_m.

As a consequence, \Delta{G^\circ} is additive with respect to combining multiple reactions in either series or parallel. In particular, then, the law (1) imposes relations between ratios of rate constants: for example, if we have the following more complicated set of reactions

\mathrm{A} \mathrel{\substack{\alpha_{\rightarrow} \\\longleftrightarrow\\ \alpha_{\leftarrow}}} \mathrm{B}

\mathrm{B} \mathrel{\substack{\beta_{\rightarrow} \\\longleftrightarrow\\ \beta_{\leftarrow}}} \mathrm{C}

\mathrm{A} \mathrel{\substack{\gamma_{\rightarrow} \\\longleftrightarrow\\ \gamma_{\leftarrow}}} \mathrm{C}

then we must have

\displaystyle{    \frac{\gamma_\to}{\gamma_\leftarrow} = \frac{\alpha_\to}{\alpha_\leftarrow} \frac{\beta_\to}{\beta_\leftarrow} .  }

So, not only are the rate constant ratios of reactions determined by differences in free energy, but also nontrivial relations between these ratios can arise, depending on the structure of the system of reactions in question!

Okay—this is all the basic stuff we’ll need to know. Please ask questions! Next time we’ll go ahead and use this stuff to start thinking about how biology manages to make reactions that ‘want’ to happen push forward reactions that are useful but wouldn’t happen spontaneously on their own.


The paper:

• John Baez, Jonathan Lorand, Blake S. Pollard and Maru Sarazola,
Biochemical coupling through emergent conservation laws.

The blog series:

Part 1 – Introduction.

Part 2 – Review of reaction networks and equilibrium thermodynamics.

Part 3 – What is coupling?

Part 4 – Interactions.

Part 5 – Coupling in quasiequilibrium states.

Part 6 – Emergent conservation laws.

Part 7 – The urea cycle.

Part 8 – The citric acid cycle.

9 Responses to Coupling Through Emergent Conservation Laws (Part 2)

  1. […] Last time we talked about chemistry and thermodynamics. Now let’s use this to start thinking about coupling! […]

  2. Toby Bartels says:

    How do we handle units/dimensions in these equations? Everything's fine if everything is dimensionless, but I measure dimensionful concentrations with units such as moles per litre. Everything involving free energy is fine, and the rate equations are also fine if the rate constants are appropriately dimensionful, but then the two constants have different dimensions, so all of this talk of ratios of rate constants and whether one is larger than another doesn't make sense.

    • John Baez says:

      Hmm, good question! Here’s my initial thought. Since the fundamental law

      \displaystyle{ \frac{\alpha_\to}{\alpha_\leftarrow} = e^{-\Delta {G^\circ}/RT} }

      only makes sense if the forward and reverse reaction rate constants have the same dimensions, they’d better have the same dimensions. So, people had better measure concentrations in a way that makes this happen! I’ll look and see what they do. Perhaps “moles per mole of water” would work well for substances in aqueous solution, as we have in biology.

    • John Baez says:

      Okay, there’s a discussion of this issue starting on page 12 of these lecture slides:

      • David P. Goldenberg, Thermodynamics: Gibbs free energy, equilibrium constants and the entropy change for a bimolecular reaction.

      I’ll think about it.

      • Toby Bartels says:

        On page 13, this effectively says that in practice, we use the mole per litre (or molar, M) as the unit, pretends that this solves the problem, and then admits that it doesn't.

        Looking through Wikipedia eventually brought me to which says that the concentration [X] should really be replaced by the activity {X}, which is the concentration multiplied by an activity coefficient γ(X). Using the ratio of concentrations as the equilibrium constant is acceptable when you're primarily interested in calculating concentrations, as long as the activity coefficient is constant, which (they say) it often is. But for relating the equilibrium constant to free energy, you need to use the ratio of the activities directly.

        Over on they say that the activity coefficient is basically just a fudge factor to account for deviations from ideal conditions; ideally, the activity coefficient is always 1. This doesn't fit with the previous article, which implied that the activity coefficient is dimensionful. But the current article gives examples, which say that the activity is the activity coefficient times the molar fraction, and the molar fraction is basically your ‘moles per mole of water’ suggestion. (Actually, your suggestion is moles of X per moles of solvent, while the actual definition is moles of X per moles of solution, but these are usually approximately equal in practice. The latter is more reasonable in principle, since it doesn't require any assumptions about which of the other substances around are solutes and which are solvents. That said, it's not clear to me how much that matters if you're going to throw in a correction for deviations from ideal conditions anyway.)

        So in summary, while there are reasons to use [X] in practice, you need to use {X} to calculate free energy change, and {X} is roughly the molar fraction, which is roughly your suggestion. However, I only reached this conclusion by combining two Wikipedia articles that don't quite agree with each other and whose references I haven't checked, so take it with a grain of salt.

        • John Baez says:

          Thanks! It sounds like there are some foundational issues I should look into! Since these questions are old, and immensely practical, some smart people must have thought about them coherently. So, I should find some chemistry textbooks that are good on conceptual issues, and see what they have to say!

          Anyone have suggestions?

  3. […] Coupling is the way biology makes reactions that ‘want’ to happen push forward desirable reactions that don’t want to happen. Coupling is achieved through the action of enzymes—but in a subtle way. An enzyme can increase the rate constant of a reaction. However, it cannot change the ratio of forward to reverse rate constants, since that is fixed by the difference of free energies, as we saw in Part 2: […]

  4. I want to figure out how to use toric geometry in chemistry. This is a good intro to toric geometry:

    • William Fulton, Introduction to Toric Varieties, Princeton U. Press, 1993.

    and this is a great explanation of how it shows up in chemistry:

    • Mercedes Perez Millan, Alicia Dickenstein, Anne Shiu and Carsten Conradi, Chemical reaction systems with toric steady states, Bulletin of Mathematical Biology 74 (2012), 1027–1065.

  5. Alexandre Vaudrey says:

    The very definition of Gibbs energy involving entropy, which depends for each reactant, on its concentration (or on its partial pressure for a gas), is it really the standard Gibbs energy of reaction \Delta G^{\circ} that must be used in equation (1) for the calculation of rate constants? Such a standard quantity, if I’m not wrong, is calculated considering all the reactants at a standard pressure, but it’s not this way within the initial blend of reactants, isn’t it? If so, it seems to me that the system of differential equations is not linear anymore.

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