## Coupling Through Emergent Conservation Laws (Part 2)

#### John Baez, 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 $$\mathrm{X}$$, $$\mathrm{Y}$$ and $$\mathrm{XY}$$ 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 $$\mathrm{X}$$, 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.