## Coupling Through Emergent Conservation Laws (Part 1)

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

In the cell, chemical reactions are often ‘coupled’ so that reactions that release energy drive reactions that are biologically useful but involve an increase in energy. But how, exactly, does coupling work?

Much is known about this question, but the literature is also full of vague explanations and oversimplifications. Coupling cannot occur in equilibrium; it arises in open systems, where the concentrations of certain chemicals are held out of equilibrium due to flows in and out. One might thus suspect that the simplest mathematical treatment of this phenomenon would involve non-equilibrium steady states of open systems. However, Bazhin has shown that some crucial aspects of coupling arise in an even simpler framework:

• Nicolai Bazhin, The essence of ATP coupling, ISRN Biochemistry 2012 (2012), article 827604.

He considers ‘quasi-equilibrium’ states, where fast reactions have come into equilibrium and slow ones are neglected. He shows that coupling occurs already in this simple approximation.

In this series of blog articles we’ll do two things. First, we’ll review Bazhin’s work in a way that readers with no training in biology or chemistry should be able to follow. (But if you get stuck, ask questions!) Second, we’ll explain a fact that seems to have received insufficient attention: in many cases, coupling relies on emergent conservation laws.

Conservation laws are important throughout science. Besides those that are built into the fabric of physics, such as conservation of energy and momentum, there are also many ’emergent’ conservation laws that hold approximately in certain circumstances. Often these arise when processes that change a given quantity happen very slowly. For example, the most common isotope of uranium decays into lead with a half-life of about 4 billion years—but for the purposes of chemical experiments in the laboratory, it is useful to treat the amount of uranium as a conserved quantity.

The emergent conservation laws involved in biochemical coupling are of a different nature. Instead of making the processes that violate these laws happen more slowly, the cell uses enzymes to make other processes happen more quickly. At the time scales relevant to cellular metabolism, the fast processes dominate, while slowly changing quantities are effectively conserved. By a suitable choice of these emergent conserved quantities, the cell ensures that certain reactions that release energy can only occur when other ‘desired’ reactions occur. To be sure, this is only approximately true, on sufficiently short time scales. But this approximation is enlightening!

Following Bazhin, our main example involves ATP hydrolysis. We consider this following schema for a whole family of reactions:

$\begin{array}{ccc} \mathrm{X} + \mathrm{ATP} & \longleftrightarrow & \mathrm{ADP} + \mathrm{XP}_{\mathrm{i}} \qquad (1) \\ \mathrm{XP}_{\mathrm{i}} + \mathrm{Y} & \longleftrightarrow & \mathrm{XY} + \mathrm{P}_{\mathrm{i}} \,\;\;\;\;\qquad (2) \end{array}$

Some concrete examples of this schema include:

• The synthesis of glutamine (XY) from glutamate (X) and ammonium (Y). This is part of the important glutamate-glutamine cycle in the central nervous system.

• The synthesis of sucrose (XY) from glucose (X) and fructose (Y). This is one of many processes whereby plants synthesize more complex sugars and starches from simpler building-blocks.

In these and other examples, the two reactions, taken together, have the effect of synthesizing a larger molecule XY out of two parts X and Y while ATP is broken down to ADP and the phosphate ion Pi Thus, they have the same net effect as this other pair of reactions:

$\begin{array}{ccc} \mathrm{X} + \mathrm{Y} &\longleftrightarrow & \mathrm{XY} \;\;\;\quad \quad \qquad (3) \\ \mathrm{ATP} &\longleftrightarrow & \mathrm{ADP} + \mathrm{P}_{\mathrm{i}} \qquad (4) \end{array}$

The first reaction here is just the synthesis of XY from X and Y. The second is a deliberately simplified version of ATP hydrolysis. The first involves an increase of energy, while the second releases energy. But in the schema used in biology, these processes are ‘coupled’ so that ATP can only break down to ADP + Pi if X and Y combine to form XY.

As we shall see, this coupling crucially relies on a conserved quantity: the total number of Y molecules plus the total number of Pi ions is left unchanged by reactions (1) and (2). This fact is not a fundamental law of physics, nor even a general law of chemistry (such as conservation of phosphorus atoms). It is an emergent conservation law that holds approximately in special situations. Its approximate validity relies on the fact that the cell has enzymes that make reactions (1) and (2) occur more rapidly than reactions that violate this law, such as (3) and (4).

In the series to come, we’ll start by providing the tiny amount of chemistry and thermodynamics needed to understand what’s going on. Then we’ll raise the question “what is coupling?” Then we’ll study the reactions required for coupling ATP hydrolysis to the synthesis of XY from components X and Y, and explain why these reactions are not yet enough for coupling. Then we’ll show that coupling occurs in a ‘quasiequilibrium’ state where reactions (1) and (2), assumed much faster than the rest, have reached equilibrium, while the rest are neglected. And then we’ll explain the role of emergent conservation laws!

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.

### 7 Responses to Coupling Through Emergent Conservation Laws (Part 1)

1. Toby Bartels says:

I can’t see the reactions listed after the paragraph ending with ‘such as these:’. I presume that they’re just the second pair of reactions again. Speaking of which, every time in the aforementioned paragraph when you refer to reactions ‘above’, it at first sounds like you’re referring to the second pair, but you must really be referring to the first pair.

• John Baez says:

Hi there, long time no see! I hope you’re doing okay.

I accidentally left out the equations after “such as these.”

All your guesses about what I meant are right. In the PDF version of this paper I use equation numbers to refer to previous equations, and some of these problems go away. I think equation numbers work badly on this WordPress version of LaTeX, and they seem a bit stuffy for a blog article, but I’ll try ’em.

• Toby Bartels says:

It seems to work. There's one place where you still have ‘the two reactions in the above schema’ ambiguously.

I’m doing fine. I read all of the top-level posts here, even though I don’t comment much.

2. I don’t understand the utility of using, “emergent conservation laws,” at least, in the case of the illustrative reaction.

To me the most informative and valuable information would be about how the enzyme and the reactants interact, and about how/if the enzyme concentration changes.

• John Baez says:

The question we’re trying to address is: what’s coupling? Enzymes are important for speeding up certain reactions, and that’s crucial for coupling. But you could know everything about what enzymes you’ve got and how they interact with the reactants, and still not have a precise understanding of when coupling happens, when it does not happen, and more fundamentally what coupling is. Conversely, you could know nothing about the enzymes—just the rate constants for the reactions they catalyze—and still be able to figure out what coupling is and when it happens.

Don’t forget: we’re mathematicians, not chemists. There’s not a chance in hell that we’re going to discover anything new about specific enzymes and how they catalyze reactions. But we do have a chance of thinking about the rate equations of reaction networks and coming up with insights about what ‘coupling’ really is.

• penny b says:

As a nonexpert, the dumb question is why we cannot conceive of coupling as resolution of kinetic energy into potential to achieve closest, cheapest, nearest stability. Coupling is nearest resolution of displacement from rest (to potential).

3. • Jonathan Lorand, Classifying linear canonical relations.

• Jonathan Lorand and Alan Weinstein, Decomposition of (co)isotropic relations.

Lorand is working on a paper with Weinstein and Christian Herrman that delves deeper into these topics. I first met him at the ACT2018 school in Leiden, where we worked along with Blake Pollard, Fabrizio Genovese (shown below) and Maru Sarazola on biochemical coupling through emergent conservation laws. Right now he’s visiting UCR and working with me to dig deeper into these questions using symplectic geometry! This is a very tantalizing project that keeps on not quite working…

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