## Coupling Through Emergent Conservation Laws (Part 4)

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

We’ve been trying to understand coupling: how a chemical reaction that ‘wants to happen’ because it decreases the amount of free energy can drive forward a chemical reaction that increases free energy.

For coupling to occur, the reactant species in both reactions must interact in some way. Indeed, in real-world examples where ATP hydrolysis is coupled to the formation of larger molecule $\mathrm{XY}$ from parts $\mathrm{X}$ and $\mathrm{Y},$ it is observed that, aside from the reactions we discussed last time:

$\begin{array}{cclc} \mathrm{X} + \mathrm{Y} & \mathrel{\substack{\alpha_{\rightarrow} \\\longleftrightarrow\\ \alpha_{\leftarrow}}} & \mathrm{XY} & \qquad (1) \\ \\ \mathrm{ATP} & \mathrel{\substack{\beta_{\rightarrow} \\\longleftrightarrow\\ \beta_{\leftarrow}}} & \mathrm{ADP} + \mathrm{P}_{\mathrm{i}} & \qquad (2) \end{array}$

two other reactions (and their reverses) take place:

$\begin{array}{cclc} \mathrm{X} + \mathrm{ATP} & \mathrel{\substack{\gamma_{\rightarrow} \\\longleftrightarrow\\ \gamma_{\leftarrow}}} & \mathrm{ADP} + \mathrm{XP}_{\mathrm{i}} & (3) \\ \\ \mathrm{XP}_{\mathrm{i}} +\mathrm{Y} & \mathrel{\substack{\delta_{\rightarrow} \\\longleftrightarrow\\ \delta_{\leftarrow}}} & \mathrm{XY} + \mathrm{P}_{\mathrm{i}} & (4) \end{array}$

We can picture all four reactions (1-4) in a single Petri net as follows:

Taking into account this more complicated set of reactions, which are interacting with each other, is still not enough to explain the phenomenon of coupling. To see this, let’s consider the rate equation for the system comprised of all four reactions. To write it down neatly, let’s introduce reaction velocities that say the rate at which each forward reaction is taking place, minus the rate of the reverse reaction:

$\begin{array}{ccl} J_\alpha &=& \alpha_\to [\mathrm{X}][\mathrm{Y}] - \alpha_\leftarrow [\mathrm{XY}] \\ \\ J_\beta &=& \beta_\to [\mathrm{ATP}] - \beta_\leftarrow [\mathrm{ADP}] [\mathrm{P}_{\mathrm{i}}] \\ \\ J_\gamma &=& \gamma_\to [\mathrm{ATP}] [\mathrm{X}] - \gamma_\leftarrow [\mathrm{ADP}] [\mathrm{XP}_{\mathrm{i}} ] \\ \\ J_\delta &=& \delta_\to [\mathrm{XP}_{\mathrm{i}} ] [\mathrm{Y}] - \delta_\leftarrow [\mathrm{XY}] [\mathrm{P}_{\mathrm{i}}] \end{array}$

All these follow from the law of mass action, which we explained in Part 2. Remember, this says that any reaction occurs at a rate equal to its rate constant times the product of the concentrations of the species involved. So, for example, this reaction

$\mathrm{XP}_{\mathrm{i}} +\mathrm{Y} \mathrel{\substack{\delta_{\rightarrow} \\\longleftrightarrow\\ \delta_{\leftarrow}}} \mathrm{XY} + \mathrm{P}_{\mathrm{i}}$

goes forward at a rate equal to $\delta_\rightarrow [\mathrm{XP}_{\mathrm{i}}][\mathrm{Y}]$, while the reverse reaction occurs at a rate equal to $\delta_\leftarrow [\mathrm{ADP}] [\mathrm{P}_{\mathrm{i}}].$ So, its reaction velocity is

$J_\delta = \delta_\to [\mathrm{XP}_{\mathrm{i}} ] [\mathrm{Y}] - \delta_\leftarrow [\mathrm{XY}] [\mathrm{P}_{\mathrm{i}}]$

In terms of these reaction velocities, we can write the rate equation as follows:

$\begin{array}{ccl} \dot{[\mathrm{X}]} & = & -J_\alpha - J_\gamma \\ \\ \dot{[\mathrm{Y}]} & = & -J_\alpha - J_\delta \\ \\ \dot{[\mathrm{XY}]} & = & J_\alpha + J_\delta \\ \\ \dot{[\mathrm{ATP}]} & = & -J_\beta - J_\gamma \\ \\ \dot{[\mathrm{ADP}]} & = & J_\beta + J_\gamma \\ \\ \dot{[\mathrm{P}_{\mathrm{i}}]} & = & J_\beta + J_\delta \\ \\ \dot{[\mathrm{XP}_{\mathrm{i}} ]} & = & J_\gamma -J_\delta \end{array}$

This makes sense if you think a bit: it says how each reaction contributes to the formation or destruction of each species.

In a steady state, all these time derivatives are zero, so we must have

$J_\alpha = J_\beta = -J_\gamma = - J_\delta$

Furthermore, in a detailed balanced equilibrium, every reaction occurs at the same rate as its reverse reaction, so all four reaction velocities vanish! In thermodynamics, a system that’s truly in equilibrium obeys this sort of detailed balance condition.

When all the reaction velocities vanish, we have:

$\begin{array}{ccl} \displaystyle{ \frac{[\mathrm{XY}]}{[\mathrm{X}][\mathrm{Y}]} } &=& \displaystyle{ \frac{\alpha_\to}{\alpha_\leftarrow} } \\ \\ \displaystyle{ \frac{[\mathrm{ADP}][\mathrm{P}_{\mathrm{i}}]}{[\mathrm{ATP}]} } &=& \displaystyle{ \frac{\beta_\to}{\beta_\leftarrow} } \\ \\ \displaystyle{ \frac{[\mathrm{ADP}] [\mathrm{XP}_{\mathrm{i}} ]}{[\mathrm{ATP}][\mathrm{X}]} } &=& \displaystyle{ \frac{\gamma_\to}{\gamma_\leftarrow} } \\ \\ \displaystyle{ \frac{[\mathrm{XY}][\mathrm{P}_{\mathrm{i}}]}{[\mathrm{XP}_{\mathrm{i}} ][\mathrm{Y}]} } &=& \displaystyle{ \frac{\delta_\to}{\delta_\leftarrow} } \end{array}$

Thus, even when the reactants interact, there can be no coupling if the whole system is in equilibrium, since then the ratio $[\mathrm{XY}]/[\mathrm{X}][\mathrm{Y}]$ is still forced to be $\alpha_\to/\alpha_\leftarrow$. This is obvious to anyone who truly understands what Boltzmann and Gibbs did. But here we saw it in detail.

The moral is that coupling cannot occur in equilibrium. But how, precisely, does coupling occur? Stay tuned!

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.

### 13 Responses to Coupling Through Emergent Conservation Laws (Part 4)

1. This sounds like the biochemical equivalent of a gravitational slingshot.

• John Baez says:

A lot like that! I’ve been thinking about pulley where a large falling mass pulls up a smaller one. There’s an ’emergent conservation law’ in that system, too!

2. Tim D says:

Shouldn’t reaction (4), and a few paragraphs later where this reaction is repeated as an example, be

X P_i + Y $\leftrightarrow$ P_i + XY

X P_i + Y $\leftrightarrow$ ADP + P_i

?

• John Baez says:

Yes, sorry!

Thanks for catching that: for some reason typesetting these reactions was difficult, and I did a bit too much cut-and-pasting.

• Toby Bartels says:

There's still a typo in the last paragraph: ‘can out arise’ should be ‘cannot arise’.

• John Baez says:

Thanks! Most of these corrections will help the final paper, not just the blog articles.

3. Peter Lichtner says:

It seems that the four reaction are linearly dependent: namely R2-R3+R1=R4, implying that there are only 3 independent mass action relations: K1 K2/K3 = K4, where K_i = equilibrium constant of ith reaction.

• John Baez says:

Yes, they are linearly dependent, and this is important!

4. domenico says:

I think that it is possibile to write the Hamiltonian of the system, for example the X dynamic is:
$\dot X=\alpha_{\rightarrow} X \cdot Y-\alpha_{\leftarrow} XY$
the Hamiltonian of this system is:
$H= p_{X} [\alpha_{\rightarrow} X \cdot Y-\alpha_{\leftarrow} XY ]$
So that there is a dynamic equation for the momentum:
$\dot p_{X}=- \alpha_{\rightarrow} Y p_{X}$
if there are invariances of the Hamiltonian, then there are conservation laws.

I am writing on my tablet, so that I cannot be accurate.

• domenico says:

I think that it is possibile to write a quantum Hamiltonian from the classical Hamiltonian and to obtain an amplitude dynamic for the concentrations

• John Baez says:

Is your idea that the canonically conjugate variables are $X, Y, XY$ and some momenta $p_X, p_Y, p_{XY}$ ? Without specifying the canonically conjugate, I can’t work out Hamilton’s equations from your proposed Hamiltonian.

Also: why does your Hamiltonian involve variables $X, Y, XY$ and $p_X$ but not $p_Y$ or $p_{XY}$? There should be complete symmetry between $X$ and $Y$ here.

• domenico says:

I write only a termine of the Hamiltonian, H(X), because of this term give me the reaction velocity equation, and this dynamics happen for each arbitrary momentum dynamics (the dynamics of $p_X$ can be arbitrary, and one see only the X dynamics).
I must add the term for $\cdot Y$, $\cdot XY$ and the
others reaction velocity, so that H=H(X)+H(Y)+\cdots, but I wrote a single term like an example.
If I add $p_X p_X$ to the Hamiltonian, this term don’t change the concentration dynamics, like each arbitrary function $f(p_X,p_Y,p_{XY},\cdots)$, but it couild be interesting for the quantization; it is like the Einstein coefficient for the absorption and emissioni of chemical compounds, bit this is only a coarse idea.

• domenico says:

I make a great, great error, this is an Hamiltonian without kinetic energy, there are only terms in the form

$H = \sum_i F_i(X,Y,XY,\cdots) p_i + G(X,Y,XY,\cdots)$

where $F_i$ are reaction velocity equations, $G$ is an arbitrary function, and $F_X$ is for example

$\alpha_{\rightarrow} X \cdot Y-\alpha_{\leftarrow} XY$

and $p_X$ the corresponding momentum. The Hamiltonian don’t give more information on the reaction velocities system, it seems only a more compact mode to write the system.

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