21 February, 2021


Talc is one of the softest minerals—its hardness defines a ‘1’ on Moh’s scale of hardness. I just learned its structure at the molecular level, and I can’t resist showing it to you.

Talc is layers of octahedra sandwiched in tetrahedra!

Since these sandwiches form separate parallel sheets, I bet they can easily slide past each other. That must be why talc is so soft.

The octahedra are magnesium oxide and the tetrahedra are silicon oxide… with some hydroxyl groups attached to liven things up. The overall formula is Mg3Si4O10(OH)2. It’s called ‘hydrated magnesium silicate’.

The image here was created by Materialscientist and placed on Wikicommons under a Creative Commons Attribution-Share Alike 3.0 Unported license.

Stretched Water

3 October, 2020

The physics of water is endlessly fascinating. The phase diagram of water at positive temperature and pressure is already remarkably complex, as shown in this diagram by Martin Chaplin:

Click for a larger version. And read this post of mine for more:


But it turns out there’s more: water is also interesting at negative pressure.

I don’t know why I never wondered about this! But people study stretched water, essentially putting a piston of water under tension and measuring its properties.

You probably know one weird thing about water: ice floats. Unlike most liquids, water at standard pressure reaches its maximum density above the freezing point, at about 4 °C. And for any fixed pressure, you can try to find the temperature at which water reaches its maximum density. You get a curve of density maxima in the pressure-temperature plane. And with stretched water experiments, you can even study this curve for negative pressures:

This graph is from here:

• Gaël Pallares, Miguel A. Gonzalez, Jose Luis F. Abascal, Chantal Valeriani, and Frédéric Caupin, Equation of state for water and its line of density maxima down to -120 MPa, Physical Chemistry Chemical Physics 18 (2016), 5896–5900.

-120 MPa is about -1200 times atmospheric pressure.

This is just the tip of the iceberg. I’m reading some papers and discovering lots of amazing things that I barely understand:

• Stacey L. Meadley and C. Austen Angell, Water and its relatives: the stable, supercooled and particularly the stretched, regimes.

• Jeremy C. Palmer, Peter H. Poole, Francesco Sciortino and Pablo G. Debenedetti, Advances in computational studies of the liquid–liquid transition in water and water-like models, Chemical Reviews 118 (2018), 9129–9151.

I would like to learn about some of these things and explain them. But for now, let me just quote the second paper to illustrate how strange water actually is:

Water is ubiquitous and yet also unusual. It is central to life, climate, agriculture, and industry, and an understanding of its properties is key in essentially all of the disciplines of the natural sciences and engineering. At the same time, and despite its apparent molecular simplicity, water is a highly unusual substance, possessing bulk properties that differ greatly, and often qualitatively, from those of other compounds. As a consequence, water has long been the subject of intense scientific scrutiny.

In this review, we describe the development and current status of the proposal that a liquid−liquid transition (LLT) occurs in deeply supercooled water. The focus of this review is on computational work, but we also summarize the relevant experimental and theoretical background. Since first proposed in 1992, this hypothesis has generated considerable interest and debate. In particular, in the past few years several works have challenged the evidence obtained from computer simulations of the ST2 model of water that support in principle the existence of an LLT, proposing instead that what was previously interpreted as an LLT is in fact ice crystallization. This challenge to the LLT hypothesis has stimulated a significant amount of new work aimed at resolving the controversy and to better understand the nature of an LLT in water-like computer models.

Unambiguously resolving this debate, it has been shown recently that the code used in the studies that most sharply challenge the LLT hypothesis contains a serious conceptual error that prevented the authors from properly characterizing the phase behavior of the ST2 water model. Nonetheless, the burst of renewed activity focusing on simulations of an LLT in water has yielded considerable new insights. Here, we review this recent work, which clearly demonstrates that an LLT is a well-defined and readily observed phenomenon in computer simulations of water-like models and is unambiguously distinguished from the crystal−liquid phase transition.

Yes, you heard that right: a phase transition between two phases of liquid water below the freezing point!

Both these phases are metastable: pretty quickly the water will freeze. But apparently it still makes some sense to talk about phases, and a phase transition between them!

What does this have to do with stretched water? I’m not sure, but apparently understanding this stuff is connected to understanding water at negative pressures. It also involves the ‘liquid-vapor spinodal’.


The liquid-vapor spinodal is another curve in the pressure-temperature plane. As far as I can tell, it works like this: when the pressure drops below this curve, the liquid—which is already unstable: it would evaporate given time—suddenly forms bubbles of vapor.

At negative pressures the liquid-vapor spinodal has a pretty intuitive meaning: if you stretch water too much, it breaks!

There’s a conjecture due to a guy named Robin J. Speedy, which implies the liquid-vapor spinodal intersects the curve of density maxima! And it does so at negative pressures. I don’t really understand the significance of this, but it sounds cool. Super-cool.

Here’s what Palmer, Poole, Sciortino and Debenedetti have to say about this:

The development of a thermodynamically self-consistent picture of the behavior of the deeply supercooled liquid that correctly predicts these experimental observations remains at the center of research on water. While a number of competing scenarios have been advanced over the years, the fact that consensus continues to be elusive demonstrates the complexity of the theoretical problem and the difficulty of the experiments required to distinguish between scenarios.

One of the first of these scenarios, Speedy’s “stability limit conjecture” (SLC), exemplifies the challenge. As formulated by Speedy, and comprehensively analyzed by Debenedetti and D’Antonio, the SLC proposes that water’s line of density maxima in the P−T plane intersects the liquid−vapor spinodal at negative pressure. At such an intersection, thermodynamics requires that the spinodal pass through a minimum and reappear in the positive pressure region under deeply supercooled conditions. Interestingly, this scenario has recently been observed in a numerical study of model colloidal particles. The apparent power law behavior of water’s response functions is predicted by the SLC in terms of the approach to the line of thermodynamic singularities found at the spinodal.

Although the SLC has recently been shown to be thermodynamically incompatible with other features of the supercooled water phase diagram, it played a key role in the development of new scenarios. The SLC also pointed out the importance of considering the behavior of “stretched” water at negative pressure, a regime in which the liquid is metastable with respect to the nucleation of bubbles of the vapor phase. The properties of stretched water have been probed directly in several innovative experiments which continue to generate results that may help discriminate among the competing scenarios that have been formulated to explain the thermodynamic behavior of supercooled water.

Banning Lead in Wetlands

27 September, 2020

An European Union commission has voted to ban the use of lead ammunition near wetlands and waterways! The proposal now needs to be approved by the European Parliament and Council. They are expected to approve the ban. If so, it will go into effect in 2022. The same commission, called REACH, may debate a complete ban on lead ammunition and fishing weights later this year.

Why does this matter? The European Chemicals Agency has estimated that as many as 1.5 million aquatic birds die annually from lead poisoning because they swallow some of the 5000 tonnes of lead shot that land in European wetlands each year. Water birds are more likely to be poisoned by lead because they mistake small lead shot pellets for stones they deliberately ingest to help grind their food.

In fact, about 20,000 tonnes of lead shot is fired each year in the EU, and 60,000 in the US. Eating game shot with lead is not good for you—but also, even low levels of lead in the environment can cause health damage and negative changes in behavior.

How much lead is too much? This is a tricky question, so I’ll just give some data. In the U.S., the geometric mean of the blood lead level among adults was 1.2 micrograms per deciliter (μg/dL) in 2009–2010. Blood lead concentrations in poisoning victims ranges from 30-80 µg/dL in children exposed to lead paint in older houses, 80–100 µg/dL in people working with pottery glazes, 90–140 µg/dL in individuals consuming contaminated herbal medicines, 110–140 µg/dL in indoor shooting range instructors and as high as 330 µg/dL in those drinking fruit juices from glazed earthenware containers!

The amount of lead that US children are exposed to has been dropping, thanks to improved regulations:

However, what seem like low levels now may be high in the grand scheme of things. The amount of lead has increased by a factor of about 300 in the Greenland ice sheet during the past 3000 years. Most of this is due to industrial emissions:

• Amy Ng and Clair Patterson, Natural concentrations of lead in ancient Arctic and Antarctic ice, Geochimica et Cosmochimica Acta 45 (1981), 2109–2121.

Postdocs in Categories and Chemistry

24 February, 2020

The University of Southern Denmark wants to hire several postdocs who will use category theory to design enzymes. This sounds like a wonderful job for people who like programming, chemistry and categories—and especially double pushout rewriting. The application deadline is 20 March 2020. The project is described here and the official job announcement is here.

I’ve seen Christoph Flamm, Daniel Merkle, Peter Sadler give talks on this project in Luxembourg, and it’s really fascinating. They’re using double pushout rewriting (as shown in the picture above) and other categorical techniques to design sequences of chemical reactions that accomplish desired tasks.

Let me quote a bit from the job announcement:

Several two-year postdoc positions starting 1 July 2020 are available at the University of Southern Denmark (SDU) for research on an exciting project in algorithmic cheminformatics supported by the Novo Nordisk Foundation: “From Category Theory to Enzyme Design – Unleashing the Potential of Computational Systems Chemistry”. We are seeking highly motivated individuals with a PhD in computer science, computational chemistry/biochemistry, or related areas. The ideal candidate has familiarity with several of the following areas: concurrency theory, graph transformation, algorithmics, algorithm engineering, systems chemistry, systems biology, metabolic engineering, or enzyme design. Solid competences in programming and ease with formal thinking are prerequisites.

The project is based on the novel application of formalisms, algorithms, and computational methods from computer science to questions in systems biology and systems chemistry. We aim to expand these methods and their formal foundations, create efficient algorithms and implementations of them, and use these implementations for research in understanding the catalytic chemistry of enzymes.

The Algorithmic Cheminformatics group at the Department of Mathematics and Computer Science at SDU offers a dynamic research environment, comprising two full professors, an assistant professor, and several students. The project includes partnerships with Harvard Medical School (Department of Systems Biology) and the University of Vienna (Institute for Theoretical Chemistry), who will host the postdoctoral researcher for extended visits.

The research group is located at the main campus of SDU, which is located in Odense, the third largest town in Denmark, 1:20 hours by train west of Copenhagen. Denmark is known for its high standards of living, free health care, and competitive salaries. Applicants from outside Denmark are eligible for substantial tax reductions.

For further information contact Professor Daniel Merkle (e-mail: daniel@imada.sdu.dk, phone: +45 6550 2322).

Here’s a bit from the project description. (There’s much more here.)

The proposed project builds on a new and powerful methodology that strikes a balance between chemical detail and computational efficiency. The approach lies at the intersection of classical chemistry, present-day systems chemistry and biology, computer science, and category theory. It adapts techniques from the analysis of actual (mechanistic) causality in concurrency theory to the chemical and biological setting. Because of this blend of intellectual and technical influences, we name the approach computational systems chemistry (CSC). The term “computational” emphasizes both the deployment of computational tools in the service of practical applications and of theoretical concepts at the foundation of computation in support of reasoning and understanding. The goal of this exploratory project is to provide a proof-of-concept toward the long-term goal of tackling many significant questions in large and combinatorially complex CRNs that could not be addressed by other means. In particular, CSC shows promise for generating new technological ideas through theoretical rigor. This exploratory project is to be considered as initial steps towards establishing this highly promising area through the following specific objectives:

• Integrate and unify algorithmic ideas and best practices from two existing platforms. One platform was conceived, designed, and implemented for organic chemistry by the lead PI and his group in Denmark as well as the chemistry partner from University of Vienna. The other platform draws on the theory of concurrency and was designed and implemented for protein- protein interaction networks supporting cellular signaling and decision-making processes by the partner from Harvard Medical School and his collaborators. The combination is ripe with potential synergies as both platforms are formally rooted in category theory.

• Demonstrate a proof-of-concept (PoC) using a biochemical driving project. The goal of this exploratory project is the analysis and design of enzymes whose catalytic site is viewed as a small (catalytic) reaction network in its own right. Such enzymes can then be used in the design of reaction networks.

• Train the next generation of scientists for CSC: This will enable the transition towards a large-scale implementation of our approaches to tackle key societal challenges, such as the development of personalized medicine, the monitoring of pollution, and the achievement of a more environmentally friendly and sustainable network of industrial synthesis.

We argue that CSC is in a position today similar to where bioinformatics and computational biology were a few decades ago and that it has similarly huge potential. The long-term vision is to unleash that potential.

Metal-Organic Frameworks

11 March, 2019

I’ve been talking about new technologies for fighting climate change, with an emphasis on negative carbon emissions. Now let’s begin looking at one technology in more detail. This will take a few articles. I want to start with the basics.

A metal-organic framework or MOF is a molecular structure built from metal atoms and organic compounds. There are many kinds. They can be 3-dimensional, like this one made by scientists at CSIRO in Australia:

And they can be full of microscopic holes, giving them an enormous surface area! For example, here’s a diagram of a MOF with yellow and orange balls showing the holes:

In fact, one gram of the stuff can have a surface area of more than 12,000 square meters!

Gas molecules like to sit inside these holes. So, perhaps surprisingly at first, you can pack a lot more gas in a cylinder containing a MOF than you can in an empty cylinder at the same pressure!

This lets us store gases using MOFs—like carbon dioxide, but also hydrogen, methane and others. And importantly, you can also get the gas molecules out of the MOF without enormous amounts of energy. Also, you can craft MOFs with different hole sizes and different chemical properties, so they attract some gases much more than others.

So, we can imagine various applications suited to fighting climate change! One is carbon capture and storage, where you want a substance that eagerly latches onto CO2 molecules, but can also easily be persuaded to let them go. But another is hydrogen or methane storage for the purpose of fuel. Methane releases less CO2 than gasoline does when it burns, per unit amount of energy—and hydrogen releases none at all. That’s why some advocate a hydrogen economy.

Could hydrogen-powered cars be better than battery-powered cars, someday? I don’t know. But never mind—such issues, though important, aren’t what I want to talk about now. I just want to quote something about methane storage in MOFs, to give you a sense of the state of the art.

• Mark Peplow, Metal-organic framework compound sets methane storage record, C&EN, 11 December 2017.

Cars powered by methane emit less CO2 than gasoline guzzlers, but they need expensive tanks and compressors to carry the gas at about 250 atm. Certain metal-organic framework (MOF) compounds—made from a lattice of metal-based nodes linked by organic struts—can store methane at lower pressures because the gas molecules pack tightly inside their pores.

So MOFs, in principle, could enable methane-powered cars to use cheaper, lighter, and safer tanks. But in practical tests, no material has met a U.S. Department of Energy (DOE) gas storage target of 263 cm3 of methane per cm3 of adsorbent at room temperature and 64 atm, enough to match the capacity of high-pressure tanks.

A team led by David Fairen-Jimenez at the University of Cambridge has now developed a synthesis method that endows a well-known MOF with a capacity of 259 cm3 of methane per cm3 under those conditions, at least 50% higher than its nearest rival. “It’s definitely a significant result,” says Jarad A. Mason at Harvard University, who works with MOFs and other materials for energy applications and was not involved in the research. “Capacity has been one of the biggest stumbling blocks.”

Only about two-thirds of the MOF’s methane was released when the pressure dropped to 6 atm, a minimum pressure needed to sustain a decent flow of gas from a tank. But this still provides the highest methane delivery capacity of any bulk adsorbent.

A couple things are worth noting here. First, the process of a molecule sticking to a surface is called adsorption, not to be confused with absorption. Second, notice that using MOFs they managed to compress methane by a factor of 259 at a pressure of just 64 atmospheres. If we tried the same trick without MOFs we would need a pressure of 259 atmospheres!

But MOFs are not only good at holding gases, they’re good at sucking them up, which is really the flip side of the same coin: gas molecules avidly seek to sit inside the little holes of your MOF. So people are also using MOFs to build highly sensitive detectors for specific kinds of gases:

Tunable porous MOF materials interface with electrodes to sound the alarm at the first sniff of hydrogen sulfide, Phys.Org, 7 March 2017.

And some MOFs work in water, too—so people are trying to use them as water filters, sort of a high-tech version of zeolites, the minerals that inspired people to invent MOFs in the first place. Zeolites have an impressive variety of crystal structures:

and so on… but MOFs seem to be more adjustable in their structure and chemical properties.

Looking more broadly at future applications, we can imagine MOFs will be important in a host of technologies where we want a substance with lots of microscopic holes that are eager to hold specific molecules. I have a feeling that the most powerful applications of MOFs will come when other technologies mature. For example: projecting forward to a time when we get really good nanotechnology, we can imagine MOFs as useful “storage lockers” for molecular robots.

But next time I’ll talk about what we can do now, or soon, to capture carbon dioxide with MOFs.

In the meantime: can you imagine some cool things we could do with MOFs? This may feed your imagination:

• Wikipedia, Metal-organic frameworks.

Toric Geometry in Reaction Networks

3 July, 2018

I want to figure out how to use toric geometry in chemistry. This is a good intro to toric geometry:

• William Fulton, Introduction to Toric Varieties, Princeton U. Press, 1993.

and this is a great explanation of how it shows up in chemistry:

• Mercedes Perez Millan, Alicia Dickenstein, Anne Shiu and Carsten Conradi, Chemical reaction systems with toric steady states, Bulletin of Mathematical Biology 74 (2012), 1027–1065.

You don’t need to read Fulton’s book to understand this paper! But you don’t need to read either to understand what I’m about to say. It’s very simple.

Suppose we have a bunch of chemical reactions. For example, just one:

\mathrm{A} \mathrel{\substack{\alpha_{\rightarrow} \\\longleftrightarrow\\ \alpha_{\leftarrow}}} \mathrm{B} + \mathrm{C}

or more precisely two: the forward reaction

\mathrm{A} \to \mathrm{B} + \mathrm{C}

with its rate constant \alpha_\to, and the reverse reaction

\mathrm{B} + \mathrm{C} \to \mathrm{A}

with its rate rate constant \alpha_{\rightarrow}. Then as I recently explained, these reactions are in a detailed balanced equilibrium when

\alpha_{\to} [\mathrm{A}] = \alpha_{\rightarrow} [\mathrm{B}] [\mathrm{C}]

This says the forward reaction is happening at the same rate as the reverse reaction.

Note: we have three variables, the concentrations [\mathrm{A}], [\mathrm{B}] and [\mathrm{C}], and they obey a polynomial equation. But it’s a special kind of polynomial equation! It just says that one monomial—a product of variables, times a constant—equals another monomial. That’s the kind of equation that’s allowed in toric geometry.

Let’s look at another example:

\mathrm{B} + \mathrm{C} \mathrel{\substack{\beta_{\rightarrow} \\\longleftrightarrow\\ \beta_{\leftarrow}}} \mathrm{D} + \mathrm{D} + \mathrm{A}

Now we have a detailed balance equilibrium when

\beta_{\to} [\mathrm{B}] [\mathrm{C}] = \beta_{\leftarrow} [\mathrm{D}]^2 [\mathrm{A}]

Again, one monomial equals another monomial.

Now let’s look at a bigger reaction network, formed by combining the two so far:

\mathrm{A} \mathrel{\substack{\alpha_{\rightarrow} \\\longleftrightarrow\\ \alpha_{\leftarrow}}} \mathrm{B} + \mathrm{C}    \mathrel{\substack{\beta_{\rightarrow} \\\longleftrightarrow\\ \beta_{\leftarrow}}} \mathrm{D} + \mathrm{D} + \mathrm{A}

Detailed balance is a very strong condition: it says that each reaction is occurring at the same rate as its reverse. So, it happens when

\alpha_{\to} [\mathrm{A}] = \alpha_{\rightarrow} [\mathrm{B}] [\mathrm{C}]


\beta_{\to} [\mathrm{B}] [\mathrm{C}] = \beta_{\leftarrow} [\mathrm{D}]^2 [\mathrm{A}]

So, we can have more than one equation, but all of them simply equate two monomials. That’s how it always works in a detailed balanced equilibrium.

Definition. An affine toric variety is a subset of \mathbb{R}^n defined by a system of equations, each of which equates two monomials in the coordinates x_1, \dots, x_n.

So, if we ignore the restriction that our variables should be ≥ 0, the space of detailed balanced equilibria for a reaction network where every reaction is reversible is an affine toric variety. And the point is, there’s a lot one can say about such spaces!

A simple example of an affine toric variety is the twisted cubic, which is the subset

\{ (x,x^2,x^3) \} \subset \mathbb{R}^3

Here it is, as drawn by Claudio Rocchini:

I may say more about this, but today I just wanted to get the ball rolling.

Puzzle. What’s a reaction network whose detailed balanced equilibrium equations give the twisted cubic?

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 6)

1 July, 2018

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

Now let’s think about emergent conservation laws!

When a heavy rock connected to a lighter one by a pulley falls down and pulls up the lighter one, you’re seeing an emergent conservation law:

Here the height of the heavy rock plus the height of light one is a constant. That’s a conservation law! It forces some of the potential energy lost by one rock to be transferred to the other. But it’s not a fundamental conservation law, built into the fabric of physics. It’s an emergent law that holds only thanks to the clever design of the pulley. If the rope broke, this law would be broken too.

It’s not surprising that biology uses similar tricks. But let’s see exactly how it works. First let’s look at all four reactions we’ve been studying:

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

It’s easy to check that the rate equations for these reactions have the following conserved quantities, that is, quantities that are constant in time:

A) [\mathrm{X}] + [\mathrm{XP}_{\mathrm{i}} ] + [\mathrm{XY}], due to the conservation of X.

B) [\mathrm{Y}] + [\mathrm{XY}], due to the conservation of Y.

C) 3[\mathrm{ATP}] +[\mathrm{XP}_{\mathrm{i}} ] +[\mathrm{P}_{\mathrm{i}}] +2[\mathrm{ADP}], due to the conservation of phosphorus.

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

Moreover, these quantities, and their linear combinations, are the only conserved quantities for reactions (1)–(4).

To see this, we use some standard ideas from reaction network theory. Consider the 7-dimensional space with orthonormal basis given by the species in our reaction network:

\mathrm{ATP}, \mathrm{ADP}, \mathrm{P}_{\mathrm{i}}, \mathrm{XP}_{\mathrm{i}}, \mathrm{X}, \mathrm{Y}, \mathrm{XY}

We can think of complexes like \mathrm{ADP} + \mathrm{P}_{\mathrm{i}} as vectors in this space. An arbitrary choice of the concentrations of all species also defines a vector in this space. Furthermore, any reaction involving these species defines a vector in this space, namely the sum of the products minus the sum of the reactants. This is called the reaction vector of this reaction. Reactions (1)–(4) give these reaction vectors:

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

Any change in concentrations caused by these reactions must lie in the stoichiometric subspace: that is, the space spanned by the reaction vectors. Since these vectors obey one nontrivial relation:

v_\alpha + v_\beta = v_\gamma + v_\delta

the stochiometric subspace is 3-dimensional. Therefore, the space of conserved quantities must be 4-dimensional, since these specify the constraints on allowed changes in concentrations.

Now let’s compare the situation where ‘coupling’ occurs! For this we consider only reactions (3) and (4):

Now the stoichiometric subspace is 2-dimensional, since v_\gamma and v_\delta are linearly independent. Thus, the space of conserved quantities becomes 5-dimensional! Indeed, we can find an additional conserved quantity:

E) [\mathrm{Y} ] +[\mathrm{P}_{\mathrm{i}}]

that is linearly independent from the four conserved quantities we had before. It does not derive from the conservation of a particular molecular component. In other words, conservation of this quantity is not a fundamental law of chemistry. Instead, it is an emergent conservation law, which holds thanks to the workings of the cell! It holds in situations where the rate constants of reactions catalyzed by the cell’s enzymes are so much larger than those of other reactions that we can ignore those other reactions.

And remember from last time: these are precisely the situations where we have coupling.

Indeed, the emergent conserved quantity E) precisely captures the phenomenon of coupling! The only way for ATP to form ADP + Pi without changing this quantity is for Y to be consumed in the same amount as Pi is created… thus forming the desired product XY.

Next time we’ll look at a more complicated example from biology: the urea cycle.


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 }


\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.