## Coupling Through Emergent Conservation Laws (Part 7)

#### John Baez, Jonathan Lorand, Blake Pollard, and Maru Sarazola

Last time we examined ATP hydrolysis as a simple example of coupling through emergent conservation laws, but the phenomenon is more general. A slightly more complicated example is the urea cycle. The first metabolic cycle to be discovered, it is used by land-dwelling vertebrates to convert ammonia, which is highly toxic, to urea for excretion. Now we'll find 11 conserved quantities in the urea cycle, including 7 emergent ones.

(Yes, this post is about mathematics of piss!)

The urea cycle consists of five reactions:

$$\begin{array}{rcl} \mathrm{NH}_3 + \mathrm{HCO}_3^- + 2 \mathrm{ATP} &\longleftrightarrow& \mathrm{carbamoyl \; phosphate} + 2 \mathrm{ADP} + \mathrm{P}_{\mathrm{i}} \\ \\ \mathrm{A}_1 + \mathrm{carbamoyl \; phosphate} &\longleftrightarrow& \mathrm{A}_2 + \mathrm{P}_{\mathrm{i}} \\ \\ \mathrm{A}_2 + \mathrm{aspartate}+ \mathrm{ATP} &\longleftrightarrow& \mathrm{A}_3 + \mathrm{AMP} + \mathrm{PP}_{\mathrm{i}} \\ \\ \mathrm{A}_3 &\longleftrightarrow& \mathrm{A}_4 + \mathrm{fumarate} \\ \\ \mathrm{A}_4 + \mathrm{H}_2\mathrm{O} &\longleftrightarrow & \mathrm{A}_1 + \mathrm{urea} \end{array}$$

Ammonia ($$\mathrm{NH}_3$$) and carbonate ($$\mathrm{HCO}_3^-$$) enter in the first reaction, along with ATP. The four remaining reactions form a cycle in which four similar species A1, A2, A3, A4 cycle around, each transformed into the next. In case you're curious, these species are:

One atom of nitrogen from carbamoyl phosphate and one from aspartate enter this cycle, and they are incorporated in urea, which then leaves the cycle.

As you can see above, argininosuccinate is the largest of the four molecules that cycle around. It's formed when citrulline combines with aspartate, which looks like this:

Argininosuccinate then breaks down to form arginine and fumarate:

All this is powered by two exergonic reactions: the hydrolysis of ATP to ADP and phosphate (Pi) and the hydrolysis of ATP to adenosine monophosphate (AMP) and a compound with two phosphorus atoms, pyrophosphate (PPi). Thus, we are seeing a more elaborate example of an endergonic process coupled to ATP hydrolysis. The most interesting new feature is the use of a cycle.

Since inflows and outflows are crucial to the purpose of the urea cycle, a full analysis requires treating this cycle as an open chemical reaction network. However, we can gain some insight into coupling just by studying the emergent conservation laws present in this network, ignoring inflows and outflows.

There are a total of 16 species in the urea cycle. There are 5 forward reactions, which are easily seen to have linearly independent reaction vectors. Thus, the stoichiometric subspace has dimension 5. There must therefore be 11 linearly independent conserved quantities.

Some of these conserved quantities can be explained by fundamental laws of chemistry. All the species involved are made of five different atoms: carbon, hydrogen, oxygen, nitrogen and phosphorus. The conserved quantity

$$3[\mathrm{ATP}] + 2[\mathrm{ADP}] + [\mathrm{AMP}] + 2 [\mathrm{PP}_{\mathrm{i}}] + [\mathrm{P}_{\mathrm{i}}] + [\mathrm{carbamoyl \; phosphate}]$$

expresses conservation of phosphorus. The conserved quantity

$$[\mathrm{NH}_3] + [\mathrm{carbamoyl \; phosphate}] + [\mathrm{aspartate}] + 2[\mathrm{urea}] + 2[\mathrm{A}_1] + 3[\mathrm{A}_2] + 4[\mathrm{A}_3] + 4[\mathrm{A}_4]$$

expresses conservation of nitrogen. Conservation of oxygen and carbon give still more complicated conserved quantities. Conservation of hydrogen and conservation of charge are not really valid laws in this context, because all the reactions are occurring in water, where it is easy for protons (H+) and electrons to come and go. So, four linearly independent 'fundamental' conserved quantities are relevant to the urea cycle.

There must therefore be seven other linearly independent conserved quantities that are emergent: that is, not conserved in every possible reaction, but conserved by those in the urea cycle. A computer calculation shows that we can use these:

A)  $$[\mathrm{ATP}] + [\mathrm{ADP}] + [\mathrm{AMP}]$$, due to conservation of adenosine by all reactions in the urea cycle.

B) $$[\mathrm{H}_2\mathrm{O}] + [\mathrm{urea}]$$, since the only reaction in the urea cycle involving either $$\mathrm{H}_2\mathrm{O}$$ or $$\mathrm{urea}$$ has $$\mathrm{H}_2\mathrm{O}$$ as a reactant and $$\mathrm{urea}$$ as a product.

C) $$[\mathrm{aspartate}] + [\mathrm{PP}_{\mathrm{i}}]$$, since the only reaction involving either $$\mathrm{aspartate}$$ or $$\mathrm{PP}_{\mathrm{i}}$$ has $$\mathrm{aspartate}$$ as a reactant and $$\mathrm{PP}_{\mathrm{i}}$$ as a product.

D) $$2[\mathrm{NH}_3] + [\mathrm{ADP}]$$, since the only reaction involving either $$\mathrm{NH}_3$$ or $$\mathrm{ADP}$$ has $$\mathrm{NH}_3$$ as a reactant and $$2\mathrm{ADP}$$ as a product.

E)  $$2[\mathrm{HCO}_3^-] + [\mathrm{ADP}]$$, since the only reaction involving either $$\mathrm{HCO}_3^-$$ or $$\mathrm{ADP}$$ has $$\mathrm{HCO}_3^-$$ as a reactant and $$2\mathrm{ADP}$$ as a product.

F)  $$[\mathrm{A}_3] + [\mathrm{fumarate}] - [\mathrm{PP}_{\mathrm{i}}]$$, since these species are involved only in the third and fourth reactions of the urea cycle, and this quantity is conserved in both those reactions.

G)  $$[\mathrm{A}_1] + [\mathrm{A}_2] + [\mathrm{A}_3] + [\mathrm{A}_4]$$, since these species cycle around the last four reactions, and they are not involved in the first.

These emergent conservation laws prevent either form of ATP hydrolysis from occurring on its own: the reaction

$$\mathrm{ATP} + \mathrm{H}_2\mathrm{O} \longrightarrow \mathrm{ADP} + \mathrm{P}_{\mathrm{i}}$$

violates conservation of quantities B), D) and E), while

$$\mathrm{ATP} + \mathrm{H}_2\mathrm{O} \longrightarrow\mathrm{AMP} + \mathrm{PP}_{\mathrm{i}}$$

violates conservation of quantities B), C) and F). (In these reactions we are neglecting $$\mathrm{H}^+$$ ions, since as mentioned these are freely available in water.)

Indeed, any linear combination of these two forms of ATP hydrolysis is prohibited. But since this requires only two emergent conservation laws, the presence of seven is a bit of a puzzle. Conserved quantity C) prevents the destruction of aspartate without the production of an equal amount of $$\mathrm{PP}_{\mathrm{i}}$$, conserved quantity D) prevents the destruction of $$\mathrm{NH}_3$$ without the production of an equal amount of $$\mathrm{ADP}$$, and so on. But there seems to be more coupling than is strictly 'required'. Of course, many factors besides coupling are involved in an evolutionarily advantageous reaction network.

### Further directions

Our paper, similar to these blog articles but with some more equations and fewer pictures, is here:

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

As a slight hint at further directions to explore, here's an interesting quote:

"It is generally believed that enzyme-free prebiotic reactions typically go wild and produce many side products," says Pasquale Stano, an organic chemist at the University of Salento, Italy.

Emergent conservation laws limit the number of side products! For more, see:

• Melissae Fellet, Enzyme-free reaction cycles hint at primitive precursor to metabolism, Chemistry World, 10 January 2018.

This is about an artificially created cycle similar to the citric acid cycle, which air-breathing organisms use to 'burn' foods and create ATP.

In our final post, we'll take a look at the citric acid cycle and its emergent conservation laws. This material is more rough than the rest, and it didn't find its way into our paper on the arXiv, but we put a fair amount of work into it — so, we'll blog about it!