## Coupling Through Emergent Conservation Laws (Part 1)

#### John Baez, 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:

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:

$$\begin{array}{cccc} \mathrm{X} + \mathrm{ATP} & \leftrightarrow & \mathrm{ADP} + \mathrm{XP}_{\mathrm{i}} & \quad \quad (1) \\ \mathrm{XP}_{\mathrm{i}} + \mathrm{Y} & \leftrightarrow & \mathrm{XY} + \mathrm{P}_{\mathrm{i}} & \quad \quad (2) \end{array}$$

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:

$$\begin{array}{cccc} \mathrm{X} + \mathrm{Y} &\leftrightarrow & \mathrm{XY} & \quad \quad \quad (3) \\ \mathrm{ATP} &\leftrightarrow & \mathrm{ADP} + \mathrm{P}_{\mathrm{i}} & \quad \quad \quad (4) \end{array}$$

The first reaction here is just the synthesis of $$\mathrm{XY}$$ from $$\mathrm{X}$$ and $$\mathrm{Y}$$. 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!