Applied Category Theory Course: Collaborative Design

13 July, 2018


In my online course we’re reading the fourth chapter of Fong and Spivak’s book Seven Sketches. Chapter 4 is about collaborative design: building big projects from smaller parts. This is based on work by Andrea Censi:

• Andrea Censi, A mathematical theory of co-design.

The main mathematical content of this chapter is the theory of enriched profunctors. We’ll mainly talk about enriched profunctors between categories enriched in monoidal preorders. The picture above shows what one of these looks like!

Here are my lectures so far:

Lecture 55 – Chapter 4: Enriched Profunctors and Collaborative Design
Lecture 56 – Chapter 4: Feasibility Relations
Lecture 57 – Chapter 4: Feasibility Relations
Lecture 58 – Chapter 4: Composing Feasibility Relations
Lecture 59 – Chapter 4: Cost-Enriched Profunctors
Lecture 60 – Chapter 4: Closed Monoidal Preorders
Lecture 61 – Chapter 4: Closed Monoidal Preorders
Lecture 62 – Chapter 4: Constructing Enriched Categories
Lecture 63 – Chapter 4: Composing Enriched Profunctors
Lecture 64 – Chapter 4: The Category of Enriched Profunctors
Lecture 65 – Chapter 4: Collaborative Design
Lecture 66 – Chapter 4: Collaborative Design
Lecture 67 – Chapter 4: Feedback in Collaborative Design
Lecture 68 – Chapter 4: Feedback in Collaborative Design
Lecture 69 – Chapter 4: Feedback in Collaborative Design
Lecture 70 – Chapter 4: Tensoring Enriched Profunctors


Beyond Classical Bayesian Networks

7 July, 2018

In the final installment of the Applied Category Theory seminar, two students explain a category-theoretic approach to Bayesian networks and their generalizations. Check it out:

• Pablo Andres-Martinez and Sophie Raynor, Beyond Classical Bayesian networks, The n-Category Café, 7 July 2018.

Pablo Andres-Martinez is a postdoc at the University of Edinburgh, working in the cool-sounding Centre for Doctoral Training in Pervasive Parallelism. Sophie Raynor works at Hoppinger. Their blog article discusses this paper:

• Joe Henson, Raymond Lal and Matthew F. Pusey, Theory-independent limits on correlations from generalized Bayesian networks, New Journal of Physics 16 (2014), 113043.


Coupling Through Emergent Conservation Laws (Part 8)

3 July, 2018

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

To wrap up this series, let’s look at an even more elaborate cycle of reactions featuring emergent conservation laws: the citric acid cycle. Here’s a picture of it from Stryer’s textbook Biochemistry:

I’ll warn you right now that we won’t draw any grand conclusions from this example: that’s why we left it out of our paper. Instead we’ll leave you with some questions we don’t know how to answer.

All known aerobic organisms use the citric cycle to convert energy derived from food into other useful forms. This cycle couples an exergonic reaction, the conversion of acetyl-CoA to CoA-SH, to endergonic reactions that produce ATP and a chemical called NADH.

The citric acid cycle can be described at various levels of detail, but at one level it consists of ten reactions:

\begin{array}{rcl}   \mathrm{A}_1 + \text{acetyl-CoA} + \mathrm{H}_2\mathrm{O} & \longleftrightarrow &  \mathrm{A}_2 + \text{CoA-SH}  \\  \\   \mathrm{A}_2 & \longleftrightarrow &  \mathrm{A}_3 + \mathrm{H}_2\mathrm{O} \\  \\  \mathrm{A}_3 + \mathrm{H}_2\mathrm{O} & \longleftrightarrow &   \mathrm{A}_4 \\  \\   \mathrm{A}_4 + \mathrm{NAD}^+  & \longleftrightarrow &  \mathrm{A}_5 + \mathrm{NADH} + \mathrm{H}^+  \\  \\   \mathrm{A}_5 + \mathrm{H}^+ & \longleftrightarrow &  \mathrm{A}_6 + \textrm{CO}_2 \\  \\  \mathrm{A}_6 + \mathrm{NAD}^+ + \text{CoA-SH} & \longleftrightarrow &  \mathrm{A}_7 + \mathrm{NADH} + \mathrm{H}^+ + \textrm{CO}_2 \\  \\   \mathrm{A}_7 + \mathrm{ADP} + \mathrm{P}_{\mathrm{i}}   & \longleftrightarrow &  \mathrm{A}_8 + \text{CoA-SH} + \mathrm{ATP} \\  \\   \mathrm{A}_8 + \mathrm{FAD} & \longleftrightarrow &  \mathrm{A}_9 + \mathrm{FADH}_2 \\  \\  \mathrm{A}_9 + \mathrm{H}_2\mathrm{O}  & \longleftrightarrow &  \mathrm{A}_{10} \\  \\  \mathrm{A}_{10} + \mathrm{NAD}^+  & \longleftrightarrow &  \mathrm{A}_1 + \mathrm{NADH} + \mathrm{H}^+  \end{array}

Here \mathrm{A}_1, \dots, \mathrm{A}_{10} are abbreviations for species that cycle around, each being transformed into the next. It doesn’t really matter for what we’ll be doing, but in case you’re curious:

\mathrm{A}_1= oxaloacetate,
\mathrm{A}_2= citrate,
\mathrm{A}_3= cis-aconitate,
\mathrm{A}_4= isocitrate,
\mathrm{A}_5= oxalosuccinate,
\mathrm{A}_6= α-ketoglutarate,
\mathrm{A}_7= succinyl-CoA,
\mathrm{A}_8= succinate,
\mathrm{A}_9= fumarate,
\mathrm{A}_{10}= L-malate.

In reality, the citric acid cycle also involves inflows of reactants such as acetyl-CoA, which is produced by metabolism, as well as outflows of both useful products such as ADP and NADH and waste products such as CO2. Thus, a full analysis requires treating this cycle as an open chemical reaction network, where species flow in and out. However, we can gain some insight just by studying the emergent conservation laws present in this network, ignoring inflows and outflows—so let’s do that!

There are a total of 22 species in the citric acid cycle. There are 10 forward reactions. We can see that their vectors are all linearly independent as follows. Since each reaction turns \mathrm{A}_i into \mathrm{A}_{i+1}, where we count modulo 10, it is easy to see that any nine of the reaction vectors are linearly independent. Whichever one we choose to ‘close the cycle’ could in theory be linearly dependent on the rest. However, it is easy to see that the vector for this reaction

\mathrm{A}_8 + \mathrm{FAD} \longleftrightarrow \mathrm{A}_9 + \mathrm{FADH}_2

is linearly independent from the rest, because only this one involves FAD. So, all 10 reaction vectors are linearly independent, and the stoichiometric subspace has dimension 10.

Since 22 – 10 = 12, there must be 12 linearly independent conserved quantities. Some of these conservation laws are ‘fundamental’, at least by the standards of chemistry. All the species involved are made of 6 different atoms (carbon, hydrogen, oxygen, nitrogen, phosphorus and sulfur), and conservation of charge provides another fundamental conserved quantity, for a total of 7.

(In our example from last time we didn’t keep track of conservation of hydrogen and charge, because both \mathrm{H}^+ and e^- ions are freely available in water… but we studied the citric acid cycle when we were younger, more energetic and less wise, so we kept careful track of hydrogen and charge, and made sure that all the reactions conserved these. So, we’ll have 7 fundamental conserved quantities.)

For example, the conserved quantity

[\text{acetyl-CoA}] + [\text{CoA-SH}] + [\mathrm{A}_7]

arises from the fact that \text{acetyl-CoA}, \text{CoA-SH} and \mathrm{A}_7 contain a single sulfur atom, while none of the other species involved contain sulfur.

Similarly, the conserved quantity

3[\mathrm{ATP}] + 2[\mathrm{ADP}] + [\mathrm{P}_{\mathrm{i}}] + 2[\mathrm{FAD}] +2[\mathrm{FADH}_2]

expresses conservation of phosphorus.

Besides the 7 fundamental conserved quantities, there must also be 5 linearly independent emergent conserved quantities: that is, quantities that are not conserved in every possible chemical reaction, but remain constant in every reaction in the citric acid cycle. We can use these 5 quantities:

[\mathrm{ATP}] + [\mathrm{ADP}], due to the conservation of adenosine.

[\mathrm{FAD}] + [\mathrm{FADH}_2], due to conservation of flavin adenine dinucleotide.

[\mathrm{NAD}^+] + [\mathrm{NADH}], due to conservation of nicotinamide adenine dinucleotide.

[\mathrm{A}_1] + \cdots + [\mathrm{A}_{10}]. This expresses the fact that in the citric acid cycle each species [\mathrm{A}_i] is transformed to the next, modulo 10.

[\text{acetyl-CoA}] + [\mathrm{A}_1] + \cdots + [\mathrm{A}_7] + [\text{CoA-SH}]. It can be checked by hand that each reaction in the citric acid cycle conserves this quantity. This expresses the fact that during the first 7 reactions of the citric acid cycle, one molecule of \text{acetyl-CoA} is destroyed and one molecule of \text{CoA-SH} is formed.

Of course, other conserved quantities can be formed as linear combinations of fundamental and emergent conserved quantities, often in nonobvious ways. An example is

3 [\text{acetyl-CoA}] + 3 [\mathrm{A}_2] + 3[\mathrm{A}_3] + 3[\mathrm{A}_4] + 2[\mathrm{A}_5] +
2[\mathrm{A}_6] + [\mathrm{A}_7] + [\mathrm{A}_8] + [\mathrm{A}_9] + [\mathrm{A}_{10}] + [\mathrm{NADH}]

which expresses the fact that in each turn of the citric acid cycle, one molecule of \text{acetyl-CoA} is destroyed and three of \mathrm{NADH} are formed. It is easier to check by hand that this quantity is conserved than to express it as an explicit linear combination of the 12 conserved quantities we have listed so far.

Finally, we bit you a fond farewell and leave you with this question: what exactly do the 7 emergent conservation laws do? In our previous two examples (ATP hydrolysis and the urea cycle) there were certain undesired reactions involving just the species we listed which were forbidden by the emergent conservation laws. In this case I don’t see any of those. But there are other important processes, involving additional species, that are forbidden. For example, if you let acetyl-CoA sit in water it will ‘hydrolyze’ as follows:

\text{acetyl-CoA} + \mathrm{H}_2\mathrm{O} \longleftrightarrow \text{CoA-SH} + \text{acetate} + \text{H}^+

So, it’s turning into CoA-SH and some other stuff, somewhat as does in the citric acid cycle, but in a way that doesn’t do anything ‘useful’: no ATP or NADH is created in this process. This is one of the things the citric acid cycle tries to prevent.

(Remember, a reaction being ‘forbidden by emergent conservation laws’ doesn’t mean it’s absolutely forbidden. It just means that it happens much more slowly than the catalyzed reactions we are listing in our reaction network.)

Unfortunately acetate and \text{H}^+ aren’t on the list of species we’re considering. We could add them. If we added them, and perhaps other species, could we get a setup where every emergent conservation law could be seen as preventing a specific unwanted reaction that’s chemically allowed?

Ideally the dimension of the space of emergent conservation laws would match the dimension of the space spanned by reaction vectors of unwanted reactions, so ‘everything would be accounted for’. But even in the simpler example of the urea cycle, we didn’t achieve this perfect match.

 


 
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.


Coupling Through Emergent Conservation Laws (Part 7)

2 July, 2018

joint post with 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 looks like this:

We’ll focus on this portion:

\mathrm{NH}_3 + \mathrm{HCO}_3^- + 2 \mathrm{ATP} \leftrightarrow \mathrm{carbamoyl \; phosphate} + 2 \mathrm{ADP} + \mathrm{P}_{\mathrm{i}}

\mathrm{A}_1 + \mathrm{carbamoyl \; phosphate} \leftrightarrow \mathrm{A}_2 + \mathrm{P}_{\mathrm{i}}

\mathrm{A}_2 + \mathrm{aspartate}+ \mathrm{ATP} \leftrightarrow \mathrm{A}_3 + \mathrm{AMP} + \mathrm{PP}_{\mathrm{i}}

\mathrm{A}_3  \leftrightarrow \mathrm{A}_4 + \mathrm{fumarate}

\mathrm{A}_4 + \mathrm{H}_2\mathrm{O} \leftrightarrow \mathrm{A}_1 + \mathrm{urea}

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:

• A1 = ornithine:

• A2 = citrulline:

• A3 = argininosuccinate:

• A3 = arginine:

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!

 


 
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.


Coupling Through Emergent Conservation Laws (Part 5)

30 June, 2018

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

Coupling is the way biology makes reactions that ‘want’ to happen push forward desirable reactions that don’t want to happen. Coupling is achieved through the action of enzymes—but in a subtle way. An enzyme can increase the rate constant of a reaction. However, it cannot change the ratio of forward to reverse rate constants, since that is fixed by the difference of free energies, as we saw in Part 2:

\displaystyle{ \frac{\alpha_\to}{\alpha_\leftarrow} = e^{-\Delta {G^\circ}/RT} }    \qquad

and the presence of an enzyme does not change this.

Indeed, if an enzyme could change this ratio, there would be no need for coupling! For example, increasing the ratio \alpha_\rightarrow/\alpha_\leftarrow in the reaction

\mathrm{X} + \mathrm{Y} \mathrel{\substack{\alpha_{\rightarrow} \\\longleftrightarrow\\ \alpha_{\leftarrow}}} \mathrm{XY}

would favor the formation of XY, as desired. But this option is not available.

Instead, to achieve coupling, the cell uses enyzmes to greatly increase both the forward and reverse rate constants for some reactions while leaving those for others unchanged!

Let’s see how this works. In our example, the cell is trying to couple ATP hydrolysis to the formation of the molecule XY from two smaller parts X and Y. These reactions don’t help do that:

\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}

but these do:

\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}

So, the cell uses enzymes to make the rate constants for reactions (3) and (4) much bigger than for (1) and (2). In this situation we can ignore reactions (1) and (2) and still have a good approximate description of the dynamics, at least for sufficiently short time scales.

Thus, we shall study quasiequilibria, namely steady states of the rate equation for reactions (3) and (4) but not (1) and (2). In this approximation, the relevant Petri net becomes this:

Now it is impossible for ATP to turn into ADP + Pi without X + Y also turning into XY. As we shall see, this is the key to coupling!

In quasiequilibrium states, we shall find a nontrivial relation between the ratios [\mathrm{XY}]/[\mathrm{X}][\mathrm{Y}] and [\mathrm{ATP}]/[\mathrm{ADP}][\mathrm{P}_{\mathrm{i}}]. This lets the cell increase the amount of XY that gets made by increasing the amount of ATP present.

Of course, this is just part of the full story. Over longer time scales, reactions (1) and (2) become important. They would drive the system toward a true equilibrium, and destroy coupling, if there were not an inflow of the reactants ATP, X and Y and an outflow of the products Pi and XY. To take these inflows and outflows into account, we need the theory of ‘open’ reaction networks… which is something I’m very interested in!

But this is beyond our scope here. We only consider reactions (3) and (4), which give the following rate equation:

\begin{array}{ccl}   \dot{[\mathrm{X}]} & = & -\gamma_\to [\mathrm{X}][\mathrm{ATP}] + \gamma_\leftarrow [\mathrm{ADP}][\mathrm{XP}_{\mathrm{i}} ]  \\  \\  \dot{[\mathrm{Y}]} & = & -\delta_\to [\mathrm{XP}_{\mathrm{i}} ][\mathrm{Y}] +\delta_\leftarrow [\mathrm{XY}][\mathrm{P}_{\mathrm{i}}]  \\ \\  \dot{[\mathrm{XY}]} & = &\delta_\to [\mathrm{XP}_{\mathrm{i}} ][\mathrm{Y}] -\delta_\leftarrow [\mathrm{XY}][\mathrm{P}_{\mathrm{i}}]  \\ \\  \dot{[\mathrm{ATP}]} & = & -\gamma_\to [\mathrm{X}][\mathrm{ATP}] + \gamma_\leftarrow [\mathrm{ADP}][\mathrm{XP}_{\mathrm{i}} ]  \\ \\  \dot{[\mathrm{ADP}]} & =& \gamma_\to [\mathrm{X}][\mathrm{ATP}] - \gamma_\leftarrow [\mathrm{ADP}][\mathrm{XP}_{\mathrm{i}} ]  \\  \\  \dot{[\mathrm{P}_{\mathrm{i}}]} & = & \delta_\to [\mathrm{XP}_{\mathrm{i}} ][\mathrm{Y}] -\delta_\leftarrow [\mathrm{XY}][\mathrm{P}_{\mathrm{i}}]  \\ \\  \dot{[\mathrm{XP}_{\mathrm{i}} ]} & = & \gamma_\to [\mathrm{X}][\mathrm{ATP}] - \gamma_\leftarrow [\mathrm{ADP}][\mathrm{XP}_{\mathrm{i}} ] \\ \\ && -\delta_\to [\mathrm{XP}_{\mathrm{i}}][\mathrm{Y}] +\delta_\leftarrow [\mathrm{XY}][\mathrm{P}_{\mathrm{i}}]  \end{array}

Quasiequilibria occur when all these time derivatives vanish. This happens when

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

This pair of equations is equivalent to

\displaystyle{ \frac{\gamma_\to}{\gamma_\leftarrow}\frac{[\mathrm{X}][\mathrm{ATP}]}{[\mathrm{ADP}]}=[\mathrm{XP}_{\mathrm{i}} ]  =\frac{\delta_\leftarrow}{\delta_\to}\frac{[\mathrm{XY}][\mathrm{P}_{\mathrm{i}}]}{[\mathrm{Y}]} }

and it implies

\displaystyle{ \frac{[\mathrm{XY}]}{[\mathrm{X}][\mathrm{Y}]}  = \frac{\gamma_\to}{\gamma_\leftarrow}\frac{\delta_\to}{\delta_\leftarrow} \frac{[\mathrm{ATP}]}{[\mathrm{ADP}][\mathrm{P}_{\mathrm{i}}]} }

If we forget about the species XPi (whose presence is crucial for the coupling to happen, but whose concentration we do not care about), the quasiequilibrium does not impose any conditions other than the above relation. We conclude that, under these circumstances and assuming we can increase the ratio

\displaystyle{ \frac{[\mathrm{ATP}]}{[\mathrm{ADP}][\mathrm{P}_{\mathrm{i}}]} }

it is possible to increase the ratio

\displaystyle{\frac{[\mathrm{XY}]}{[\mathrm{X}][\mathrm{Y}]} }

This constitutes ‘coupling’.

We can say a bit more, since we can express the ratios of forward and reverse rate constants in terms of exponentials of free energy differences using the laws of thermodynamics, as explained in Part 2. Reactions (1) and (2), taken together, convert X + Y + ATP to XY + ADP + Pi. So do reactions (3) and (4) taken together. Thus, these two pairs of reactions involve the same total change in free energy, so

\displaystyle{          \frac{\alpha_\to}{\alpha_\leftarrow}\frac{\beta_\to}{\beta_\leftarrow} =   \frac{\gamma_\to}{\gamma_\leftarrow}\frac{\delta_\to}{\delta_\leftarrow} }

But we’re also assuming ATP hydrolysis is so strongly exergonic that

\displaystyle{ \frac{\beta_\to}{\beta_\leftarrow} \gg \frac{\alpha_\leftarrow}{\alpha_\to}  }

This implies that

\displaystyle{    \frac{\gamma_\to}{\gamma_\leftarrow}\frac{\delta_\to}{\delta_\leftarrow} \gg 1 }

Thus,

\displaystyle{ \frac{[\mathrm{XY}]}{[\mathrm{X}][\mathrm{Y}]}  \gg \frac{[\mathrm{ATP}]}{[\mathrm{ADP}][\mathrm{P}_{\mathrm{i}}]} }

This is why coupling to ATP hydrolysis is so good at driving the synthesis of XY from X and Y! Ultimately, this inequality arises from the fact that the decrease in free energy for the reaction

\mathrm{ATP} \to \mathrm{ADP} + \mathrm{P}_{\mathrm{i}}

greatly exceeds the increase in free energy for the reaction

\mathrm{X} + \mathrm{Y} \to \mathrm{XY}

But this fact can only be put to use in the right conditions. We need to be in a ‘quasiequilibrium’ state, where fast reactions have reached a steady state but not slow ones. And we need fast reactions to have this property: they can only turn ATP into ADP + Pi if they also turn X + Y into XY. Under these conditions, we have ‘coupling’.

Next time we’ll see how coupling relies on an ’emergent conservation law’.

 


 
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.


Coupling Through Emergent Conservation Laws (Part 4)

29 June, 2018

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.


Coupling Through Emergent Conservation Laws (Part 3)

28 June, 2018

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

Last time we gave a quick intro to the chemistry and thermodynamics we’ll use to understand ‘coupling’. Now let’s really get started!

Suppose that we are in a setting in which some reaction

\mathrm{X} + \mathrm{Y} \mathrel{\substack{\alpha_{\rightarrow} \\\longleftrightarrow\\ \alpha_{\leftarrow}}} \mathrm{XY}

takes place. Let’s also assume we are interested in the production of \mathrm{XY} from \mathrm{X} and \mathrm{Y}, but that in our system, the reverse reaction is favored to happen. This means that that reverse rate constant exceeds the forward one, let’s say by a lot:

\alpha_\leftarrow \gg \alpha_\to

so that in equilibrium, the concentrations of the species will satisfy

\displaystyle{ \frac{[\mathrm{XY}]}{[\mathrm{X}][\mathrm{Y}]}\ll 1 }

which we assume undesirable. How can we influence this ratio to get a more desired outcome?

This is where coupling comes into play. Informally, we think of the coupling of two reactions as a process in which an endergonic reaction—one which does not ‘want’ to happen—is combined with an exergonic reaction—one that does ‘want’ to happen—in a way that improves the products-to-reactants concentrations ratio of the first reaction.

An important example of coupling, and one we will focus on, involves ATP hydrolysis:

\mathrm{ATP} + \mathrm{H}_2\mathrm{O} \mathrel{\substack{\beta_{\rightarrow} \\\longleftrightarrow\\ \beta_{\leftarrow}}} \mathrm{ADP} + \mathrm{P}_{\mathrm{i}} + \mathrm{H}^+

where ATP (adenosine triphosphate) reacts with a water molecule. Typically, this reaction results in ADP (adenosine diphosphate), a phosphate ion \mathrm{P}_{\mathrm{i}} and a hydrogen ion \mathrm{H}^+. To simplify calculations, we will replace the above equation with

\mathrm{ATP}  \mathrel{\substack{\beta_{\rightarrow} \\\longleftrightarrow\\ \beta_{\leftarrow}}} \mathrm{ADP} + \mathrm{P}_{\mathrm{i}}

since suppressing the bookkeeping of hydrogen and oxygen atoms in this manner will not affect our main points.

One reason ATP hydrolysis is good for coupling is that this reaction is strongly exergonic:

\beta_\to \gg \beta_\leftarrow

and in fact so much that

\displaystyle{ \frac{\beta_\to}{\beta_\leftarrow} \gg \frac{\alpha_\leftarrow}{\alpha_\to}  }

Yet this fact alone is insufficient to explain coupling!

To see why, suppose our system consists merely of the two reactions

\begin{array}{ccc}  \mathrm{X} + \mathrm{Y}   & \mathrel{\substack{\alpha_{\rightarrow} \\\longleftrightarrow\\ \alpha_{\leftarrow}}} & \mathrm{XY} \\ \\  \mathrm{ATP} & \mathrel{\substack{\beta_{\rightarrow} \\\longleftrightarrow\\ \beta_{\leftarrow}}} &  \mathrm{ADP} + \mathrm{P}_{\mathrm{i}} \label{beta}  \end{array}

happening in parallel. We can study the concentrations in equilibrium to see that one reaction has no influence on the other. Indeed, the rate equation for this reaction network is

\begin{array}{ccl}  \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}]\\ \\  \dot{[\mathrm{ATP}]} & =& -\beta_\to [\mathrm{ATP}]+\beta_\leftarrow [\mathrm{ADP}][\mathrm{P}_{\mathrm{i}}]\\ \\  \dot{[\mathrm{ADP}]} & = &\beta_\to [\mathrm{ATP}]-\beta_\leftarrow [\mathrm{ADP}][\mathrm{P}_{\mathrm{i}}]\\ \\  \dot{[\mathrm{P}_{\mathrm{i}}]} & = &\beta_\to [\mathrm{ATP}]-\beta_\leftarrow [\mathrm{ADP}][\mathrm{P}_{\mathrm{i}}]  \end{array}

When concentrations are constant, these are equivalent to the relations

\displaystyle{  \frac{[\mathrm{XY}]}{[\mathrm{X}][\mathrm{Y}]} = \frac{\alpha_\to}{\alpha_\leftarrow} \ \ \text{ and } \ \ \frac{[\mathrm{ADP}][\mathrm{P}_{\mathrm{i}}]}{[\mathrm{ATP}]} = \frac{\beta_\to}{\beta_\leftarrow} }

We thus see that ATP hydrolysis is in no way affecting the ratio of [\mathrm{XY}] to [\mathrm{X}][\mathrm{Y}]. Intuitively, there is no coupling because the two reactions proceed independently. This ‘independence’ is clearly visible if we draw the reaction network as a so-called Petri net:

So what really happens when we are in the presence of coupling? Stay tuned for the next episode!

By the way, here’s what ATP hydrolysis looks like in a bit more detail, from a website at Loreto College:


 


 
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.