The Ideal Monatomic Gas

15 July, 2021

Today at the Topos Institute, Sophie Libkind, Owen Lynch and I spent some time talking about thermodynamics, Carnot engines and the like. As a result, I want to work out for myself some basic facts about the ideal gas. This stuff is all well-known, but I’m having trouble finding exactly what I want—and no more, thank you—collected in one place.

Just for background, the Carnot cycle looks roughly like this:


This is actually a very inaccurate picture, but it gets the point across. We have a container of gas, and we make it execute a cyclic motion, so its pressure P and volume V trace out a loop in the plane. As you can see, this loop consists of four curves:

• In the first, from a to b, we put a container of gas in contact with a hot medium. Then we make it undergo isothermal expansion: that is, expansion at a constant temperature.

• In the second, from b to c, we insulate the container and let the gas undergo adiabatic reversible expansion: that is, expansion while no heat enters or leaves. The temperature drops, but merely because the container expands, not because heat leaves. It reaches a lower temperature. Then we remove the insulation.

• In the third, from c to d, we put the container in contact with a cold medium that matches its temperature. Then we make it undergo isothermal contraction: that is, contraction at a constant temperature.

• In the fourth, from d to a, we insulate the container and let the gas undergo adiabatic reversible contraction: that is, contraction while no heat enters or leaves. The temperature increases until it matches that of the hot medium. Then we remove the insulation.

The Carnot cycle is historically important because it’s an example of a heat engine that’s as efficient as possible: it give you the most work possible for the given amount of heat transferred from the hot medium to the cold medium. But I don’t want to get into that. I just want to figure out formulas for everything that’s going on here—including formulas for the four curves in this picture!

To get specific formulas, I’ll consider an ideal monatomic gas, meaning a gas made of individual atoms, like helium. Some features of an ideal gas, like the formula for energy as a function of temperature, depend on whether it’s monatomic.

As a quirky added bonus, I’d like to highlight how certain properties of the ideal monatomic gas depend on the dimension of space. There’s a certain chunk of the theory that doesn’t depend on the dimension of space, as long as you interpret ‘volume’ to mean the n-dimensional analogue of volume. But the number 3 shows up in the formula for the energy of the ideal monatomic gas. And this is because space is 3-dimensional! So just for fun, I’ll do the whole analysis in n dimensions.

There are four basic formulas we need to know.

First, we have the ideal gas law:

PV = NkT

where

P is the pressure.
V is the n-dimensional volume.
N is the number of molecules in a container of gas.
k is a constant called Boltzmann’s constant.
T is the temperature.

Second, we have a formula for the energy, or more precisely the internal energy, of a monatomic ideal gas:

U = \frac{n}{2} NkT

where

U is the internal energy.
n is the dimension of space.

The factor of n/2 shows up thanks to the equipartition theorem: classically, a harmonic oscillator at temperature T has expected energy equal to kT times its number of degrees of freedom. Very roughly, the point is that in n dimensions there are n different directions in which an atom can move around.

Third, we have a relation between internal energy, work and heat:

dU = \delta W + \delta Q

Here

dU is the differential of internal energy.
\delta W is the infinitesimal work done to the gas.
\delta Q is the infinitesimal heat transferred to the gas.

The intuition is simple: to increase the energy of some gas you can do work to it or transfer heat to it. But the math may seem a bit murky, so let me explain.

I emphasize ‘to’ because it affects the sign: for example, the work done by the gas is minus the work done to the gas. Work done to the gas increases its internal energy, while work done by it reduces its internal energy. Similarly for heat.

But what is this ‘infinitesimal’ stuff, and these weird \delta symbols?

In a minute I’m going to express everything in terms of P and V. So, T, N and U will be functions on the plane with coordinates P and V. dU will be a 1-form on this plane: it’s the differential of the function U.

But \delta W and \delta Q are not differentials of functions W and Q. There are no functions on the plane called W and Q. You can not take a box of gas and measure its work, or heat! There are just 1-forms called \delta W and \delta Q describing the change in work or heat. These are not exact 1-forms: that is, they’re not differentials of functions.

Fourth and finally:

\delta W = - P dV

This should be intuitive. The work done by the gas on the outside world by changing its volume a little equals the pressure times the change in volume. So, the work done to the gas is minus the pressure times the change in volume.

One nice feature of the 1-form \delta W = -P d V is this: as we integrate it around a simple closed curve going counterclockwise, we get the area enclosed by that curve. So, the area of this region:


is the work done by our container of gas during the Carnot cycle. (There are a lot of minus signs to worry about here, but don’t worry, I’ve got them under control. Our curve is going clockwise, so the work done to our container of gas is negative, and it’s minus the area in the region.)

Okay, now that we have our four basic equations, we can play with them and derive consequences. Let’s suppose the number N of atoms in our container of gas is fixed—a constant. Then we think of everything as a function of two variables: P and V.

First, since PV = NkT we have

\displaystyle{ T = \frac{PV}{Nk} }

So temperature is proportional to pressure times volume.

Second, since PV = NkT and U = \frac{n}{2}NkT we have

U = \frac{n}{2} P V

So, like the temperature, the internal energy of the gas is proportional to pressure times volume—but it depends on the dimension of space!

From this we get

dU = \frac{n}{2} d(PV) = \frac{n}{2}( V dP + P dV)

From this and our formulas dU = \delta W + \delta Q, \delta W = -PdV we get

\begin{array}{ccl}  \delta Q &=& dU - \delta W \\  \\  &=& \frac{n}{2}( V dP + P dV) + P dV \\ \\  &=& \frac{n}{2} V dP + \frac{n+2}{2} P dV   \end{array}

That’s basically it!

But now we know how to figure out everything about the Carnot cycle. I won’t do it all here, but I’ll work out formulas for the curves in this cycle:


The isothermal curves are easy, since we’ve seen temperature is proportional to pressure times volume:

\displaystyle{ T = \frac{PV}{Nk} }

So, an isothermal curve is any curve with

P \propto V^{-1}

The adiabatic reversible curves, or ‘adiabats’ for short, are a lot more interesting. A curve C in the P  V plane is an adiabat if when the container of gas changes pressure and volume while moving along this curve, no heat gets transferred to or from the gas. That is:

\delta Q \Big|_C = 0

where the funny symbol means I’m restricting a 1-form to the curve and getting a 1-form on that curve (which happens to be zero).

Let’s figure out what an adiabat looks like! By our formula for Q we have

(\frac{n}{2} V dP + \frac{n+2}{2} P dV) \Big|_C = 0

or

\frac{n}{2} V dP \Big|_C = -\frac{n+2}{2} P dV \Big|_C

or

\frac{dP}{P} \Big|_C = - \frac{n+2}{n} \frac{dV}{V}\Big|_C

Now, we can integrate both sides along a portion of the curve C and get

\ln P = - \frac{n+2}{n} \ln V + \mathrm{constant}

or

P \propto V^{-(n+2)/n}

So in 3-dimensional space, as you let a gas expand adiabatically—say by putting it in an insulated cylinder so heat can’t get in or out—its pressure drops as its volume increases. But for a monatomic gas it drops in this peculiar specific way: the pressure goes like the volume to the -5/3 power.

In any dimension, the pressure of the monatomic gas drops more steeply when the container expands adiabatically than when it expands at constant temperature. Why? Because V^{-(n+2)/n} drops more rapidly than V^{-1} since

\frac{n+2}{n} > 1

But as n \to \infty,

\frac{n+2}{n} \to 1

so the adiabats become closer and and closer to the isothermal curves in high dimensions. This is not important for understanding the conceptually significant features of the Carnot cycle! But it’s curious, and I’d like to improve my understanding by thinking about it until it seems obvious. It doesn’t yet.


Nonequilibrium Thermodynamics in Biology (Part 2)

16 June, 2021

Larry Li, Bill Cannon and I ran a session on non-equilibrium thermodynamics in biology at SMB2021, the annual meeting of the Society for Mathematical Biology. You can see talk slides here!

Here’s the basic idea:

Since Lotka, physical scientists have argued that living things belong to a class of complex and orderly systems that exist not despite the second law of thermodynamics, but because of it. Life and evolution, through natural selection of dissipative structures, are based on non-equilibrium thermodynamics. The challenge is to develop an understanding of what the respective physical laws can tell us about flows of energy and matter in living systems, and about growth, death and selection. This session addresses current challenges including understanding emergence, regulation and control across scales, and entropy production, from metabolism in microbes to evolving ecosystems.

Click on the links to see slides for most of the talks:

Persistence, permanence, and global stability in reaction network models: some results inspired by thermodynamic principles
Gheorghe Craciun, University of Wisconsin–Madison

The standard mathematical model for the dynamics of concentrations in biochemical networks is called mass-action kinetics. We describe mass-action kinetics and discuss the connection between special classes of mass-action systems (such as detailed balanced and complex balanced systems) and the Boltzmann equation. We also discuss the connection between the ‘global attractor conjecture’ for complex balanced mass-action systems and Boltzmann’s H-theorem. We also describe some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.

The principle of maximum caliber of nonequilibria
Ken Dill, Stony Brook University

Maximum Caliber is a principle for inferring pathways and rate distributions of kinetic processes. The structure and foundations of MaxCal are much like those of Maximum Entropy for static distributions. We have explored how MaxCal may serve as a general variational principle for nonequilibrium statistical physics—giving well-known results, such as the Green-Kubo relations, Onsager’s reciprocal relations and Prigogine’s Minimum Entropy Production principle near equilibrium, but is also applicable far from equilibrium. I will also discuss some applications, such as finding reaction coordinates in molecular simulations non-linear dynamics in gene circuits, power-law-tail distributions in ‘social-physics’ networks, and others.

Nonequilibrium biomolecular information processes
Pierre Gaspard, Université libre de Bruxelles

Nearly 70 years have passed since the discovery of DNA structure and its role in coding genetic information. Yet, the kinetics and thermodynamics of genetic information processing in DNA replication, transcription, and translation remain poorly understood. These template-directed copolymerization processes are running away from equilibrium, being powered by extracellular energy sources. Recent advances show that their kinetic equations can be exactly solved in terms of so-called iterated function systems. Remarkably, iterated function systems can determine the effects of genome sequence on replication errors, up to a million times faster than kinetic Monte Carlo algorithms. With these new methods, fundamental links can be established between molecular information processing and the second law of thermodynamics, shedding a new light on genetic drift, mutations, and evolution.

Nonequilibrium dynamics of disturbed ecosystems
John Harte, University of California, Berkeley

The Maximum Entropy Theory of Ecology (METE) predicts the shapes of macroecological metrics in relatively static ecosystems, across spatial scales, taxonomic categories, and habitats, using constraints imposed by static state variables. In disturbed ecosystems, however, with time-varying state variables, its predictions often fail. We extend macroecological theory from static to dynamic, by combining the MaxEnt inference procedure with explicit mechanisms governing disturbance. In the static limit, the resulting theory, DynaMETE, reduces to METE but also predicts a new scaling relationship among static state variables. Under disturbances, expressed as shifts in demographic, ontogenic growth, or migration rates, DynaMETE predicts the time trajectories of the state variables as well as the time-varying shapes of macroecological metrics such as the species abundance distribution and the distribution of metabolic rates over
individuals. An iterative procedure for solving the dynamic theory is presented. Characteristic signatures of the deviation from static predictions of macroecological patterns are shown to result from different kinds of disturbance. By combining MaxEnt inference with explicit dynamical mechanisms of disturbance, DynaMETE is a candidate theory of macroecology for ecosystems responding to anthropogenic or natural disturbances.

Stochastic chemical reaction networks
Supriya Krishnamurthy, Stockholm University

The study of chemical reaction networks (CRN’s) is a very active field. Earlier well-known results (Feinberg Chem. Enc. Sci. 42 2229 (1987), Anderson et al Bull. Math. Biol. 72 1947 (2010)) identify a topological quantity called deficiency, easy to compute for CRNs of any size, which, when exactly equal to zero, leads to a unique factorized (non-equilibrium) steady-state for these networks. No general results exist however for the steady states of non-zero-deficiency networks. In recent work, we show how to write the full moment-hierarchy for any non-zero-deficiency CRN obeying mass-action kinetics, in terms of equations for the factorial moments. Using these, we can recursively predict values for lower moments from higher moments, reversing the procedure usually used to solve moment hierarchies. We show, for non-trivial examples, that in this manner we can predict any moment of interest, for CRN’s with non-zero deficiency and non-factorizable steady states. It is however an open question how scalable these techniques are for large networks.

Heat flows adjust local ion concentrations in favor of prebiotic chemistry
Christof Mast, Ludwig-Maximilians-Universität München

Prebiotic reactions often require certain initial concentrations of ions. For example, the activity of RNA enzymes requires a lot of divalent magnesium salt, whereas too much monovalent sodium salt leads to a reduction in enzyme function. However, it is known from leaching experiments that prebiotically relevant geomaterial such as basalt releases mainly a lot of sodium and only little magnesium. A natural solution to this problem is heat fluxes through thin rock fractures, through which magnesium is actively enriched and sodium is depleted by thermogravitational convection and thermophoresis. This process establishes suitable conditions for ribozyme function from a basaltic leach. It can take place in a spatially distributed system of rock cracks and is therefore particularly stable to natural fluctuations and disturbances.

Deficiency of chemical reaction networks and thermodynamics
Matteo Polettini, University of Luxembourg

Deficiency is a topological property of a Chemical Reaction Network linked to important dynamical features, in particular of deterministic fixed points and of stochastic stationary states. Here we link it to thermodynamics: in particular we discuss the validity of a strong vs. weak zeroth law, the existence of time-reversed mass-action kinetics, and the possibility to formulate marginal fluctuation relations. Finally we illustrate some subtleties of the Python module we created for MCMC stochastic simulation of CRNs, soon to be made public.

Large deviations theory and emergent landscapes in biological dynamics
Hong Qian, University of Washington

The mathematical theory of large deviations provides a nonequilibrium thermodynamic description of complex biological systems that consist of heterogeneous individuals. In terms of the notions of stochastic elementary reactions and pure kinetic species, the continuous-time, integer-valued Markov process dictates a thermodynamic structure that generalizes (i) Gibbs’ microscopic chemical thermodynamics of equilibrium matters to nonequilibrium small systems such as living cells and tissues; and (ii) Gibbs’ potential function to the landscapes for biological dynamics, such as that of C. H. Waddington and S. Wright.

Using the maximum entropy production principle to understand and predict microbial biogeochemistry
Joseph Vallino, Marine Biological Laboratory, Woods Hole

Natural microbial communities contain billions of individuals per liter and can exceed a trillion cells per liter in sediments, as well as harbor thousands of species in the same volume. The high species diversity contributes to extensive metabolic functional capabilities to extract chemical energy from the environment, such as methanogenesis, sulfate reduction, anaerobic photosynthesis, chemoautotrophy, and many others, most of which are only expressed by bacteria and archaea. Reductionist modeling of natural communities is problematic, as we lack knowledge on growth kinetics for most organisms and have even less understanding on the mechanisms governing predation, viral lysis, and predator avoidance in these systems. As a result, existing models that describe microbial communities contain dozens to hundreds of parameters, and state variables are extensively aggregated. Overall, the models are little more than non-linear parameter fitting exercises that have limited, to no, extrapolation potential, as there are few principles governing organization and function of complex self-assembling systems. Over the last decade, we have been developing a systems approach that models microbial communities as a distributed metabolic network that focuses on metabolic function rather than describing individuals or species. We use an optimization approach to determine which metabolic functions in the network should be up regulated versus those that should be down regulated based on the non-equilibrium thermodynamics principle of maximum entropy production (MEP). Derived from statistical mechanics, MEP proposes that steady state systems will likely organize to maximize free energy dissipation rate. We have extended this conjecture to apply to non-steady state systems and have proposed that living systems maximize entropy production integrated over time and space, while non-living systems maximize instantaneous entropy production. Our presentation will provide a brief overview of the theory and approach, as well as present several examples of applying MEP to describe the biogeochemistry of microbial systems in laboratory experiments and natural ecosystems.

Reduction and the quasi-steady state approximation
Carsten Wiuf, University of Copenhagen

Chemical reactions often occur at different time-scales. In applications of chemical reaction network theory it is often desirable to reduce a reaction network to a smaller reaction network by elimination of fast species or fast reactions. There exist various techniques for doing so, e.g. the Quasi-Steady-State Approximation or the Rapid Equilibrium Approximation. However, these methods are not always mathematically justifiable. Here, a method is presented for which (so-called) non-interacting species are eliminated by means of QSSA. It is argued that this method is mathematically sound. Various examples are given (Michaelis-Menten mechanism, two-substrate mechanism, …) and older related techniques from the 50s and 60s are briefly discussed.


Non-Equilibrium Thermodynamics in Biology (Part 1)

11 May, 2021

Together with William Cannon and Larry Li, I’m helping run a minisymposium as part of SMB2021, the annual meeting of the Society for Mathematical Biology:

• Non-equilibrium Thermodynamics in Biology: from Chemical Reaction Networks to Natural Selection, Monday June 14, 2021, beginning 9:30 am Pacific Time.

You can register for free here before May 31st, 11:59 pm Pacific Time. You need to register to watch the talks live on Zoom. I think the talks will be recorded.

Here’s the idea:

Abstract: Since Lotka, physical scientists have argued that living things belong to a class of complex and orderly systems that exist not despite the second law of thermodynamics, but because of it. Life and evolution, through natural selection of dissipative structures, are based on non-equilibrium thermodynamics. The challenge is to develop an understanding of what the respective physical laws can tell us about flows of energy and matter in living systems, and about growth, death and selection. This session will address current challenges including understanding emergence, regulation and control across scales, and entropy production, from metabolism in microbes to evolving ecosystems.

It’s exciting to me because I want to get back into work on thermodynamics and reaction networks, and we’ll have some excellent speakers on these topics. I think the talks will be in this order… later I will learn the exact schedule.

Christof Mast, Ludwig-Maximilians-Universität München

Coauthors: T. Matreux, K. LeVay, A. Schmid, P. Aikkila, L. Belohlavek, Z. Caliskanoglu, E. Salibi, A. Kühnlein, C. Springsklee, B. Scheu, D. B. Dingwell, D. Braun, H. Mutschler.

Title: Heat flows adjust local ion concentrations in favor of prebiotic chemistry

Abstract: Prebiotic reactions often require certain initial concentrations of ions. For example, the activity of RNA enzymes requires a lot of divalent magnesium salt, whereas too much monovalent sodium salt leads to a reduction in enzyme function. However, it is known from leaching experiments that prebiotically relevant geomaterial such as basalt releases mainly a lot of sodium and only little magnesium. A natural solution to this problem is heat fluxes through thin rock fractures, through which magnesium is actively enriched and sodium is depleted by thermogravitational convection and thermophoresis. This process establishes suitable conditions for ribozyme function from a basaltic leach. It can take place in a spatially distributed system of rock cracks and is therefore particularly stable to natural fluctuations and disturbances.

Supriya Krishnamurthy, Stockholm University

Coauthors: Eric Smith

Title: Stochastic chemical reaction networks

Abstract: The study of chemical reaction networks (CRNs) is a very active field. Earlier well-known results (Feinberg Chem. Enc. Sci. 42 2229 (1987), Anderson et al Bull. Math. Biol. 72 1947 (2010)) identify a topological quantity called deficiency, easy to compute for CRNs of any size, which, when exactly equal to zero, leads to a unique factorized (non-equilibrium) steady-state for these networks. No general results exist however for the steady states of non-zero-deficiency networks. In recent work, we show how to write the full moment-hierarchy for any non-zero-deficiency CRN obeying mass-action kinetics, in terms of equations for the factorial moments. Using these, we can recursively predict values for lower moments from higher moments, reversing the procedure usually used to solve moment hierarchies. We show, for non-trivial examples, that in this manner we can predict any moment of interest, for CRNs with non-zero deficiency and non-factorizable steady states. It is however an open question how scalable these techniques are for large networks.

Pierre Gaspard, Université libre de Bruxelles

Title: Nonequilibrium biomolecular information processes

Abstract: Nearly 70 years have passed since the discovery of DNA structure and its role in coding genetic information. Yet, the kinetics and thermodynamics of genetic information processing in DNA replication, transcription, and translation remain poorly understood. These template-directed copolymerization processes are running away from equilibrium, being powered by extracellular energy sources. Recent advances show that their kinetic equations can be exactly solved in terms of so-called iterated function systems. Remarkably, iterated function systems can determine the effects of genome sequence on replication errors, up to a million times faster than kinetic Monte Carlo algorithms. With these new methods, fundamental links can be established between molecular information processing and the second law of thermodynamics, shedding a new light on genetic drift, mutations, and evolution.

Carsten Wiuf, University of Copenhagen

Coauthors: Elisenda Feliu, Sebastian Walcher, Meritxell Sáez

Title: Reduction and the Quasi-Steady State Approximation

Abstract: Chemical reactions often occur at different time-scales. In applications of chemical reaction network theory it is often desirable to reduce a reaction network to a smaller reaction network by elimination of fast species or fast reactions. There exist various techniques for doing so, e.g. the Quasi-Steady-State Approximation or the Rapid Equilibrium Approximation. However, these methods are not always mathematically justifiable. Here, a method is presented for which (so-called) non-interacting species are eliminated by means of QSSA. It is argued that this method is mathematically sound. Various examples are given (Michaelis-Menten mechanism, two-substrate mechanism, …) and older related techniques from the 50-60ies are briefly discussed.

Matteo Polettini, University of Luxembourg

Coauthor: Tobias Fishback

Title: Deficiency of chemical reaction networks and thermodynamics

Abstract: Deficiency is a topological property of a Chemical Reaction Network linked to important dynamical features, in particular of deterministic fixed points and of stochastic stationary states. Here we link it to thermodynamics: in particular we discuss the validity of a strong vs. weak zeroth law, the existence of time-reversed mass-action kinetics, and the possibility to formulate marginal fluctuation relations. Finally we illustrate some subtleties of the Python module we created for MCMC stochastic simulation of CRNs, soon to be made public.

Ken Dill, Stony Brook University

Title: The principle of maximum caliber of nonequilibria

Abstract: Maximum Caliber is a principle for inferring pathways and rate distributions of kinetic processes. The structure and foundations of MaxCal are much like those of Maximum Entropy for static distributions. We have explored how MaxCal may serve as a general variational principle for nonequilibrium statistical physics – giving well-known results, such as the Green-Kubo relations, Onsager’s reciprocal relations and Prigogine’s Minimum Entropy Production principle near equilibrium, but is also applicable far from equilibrium. I will also discuss some applications, such as finding reaction coordinates in molecular simulations non-linear dynamics in gene circuits, power-law-tail distributions in “social-physics” networks, and others.

Joseph Vallino, Marine Biological Laboratory, Woods Hole

Coauthors: Ioannis Tsakalakis, Julie A. Huber

Title: Using the maximum entropy production principle to understand and predict microbial biogeochemistry

Abstract: Natural microbial communities contain billions of individuals per liter and can exceed a trillion cells per liter in sediments, as well as harbor thousands of species in the same volume. The high species diversity contributes to extensive metabolic functional capabilities to extract chemical energy from the environment, such as methanogenesis, sulfate reduction, anaerobic photosynthesis, chemoautotrophy, and many others, most of which are only expressed by bacteria and archaea. Reductionist modeling of natural communities is problematic, as we lack knowledge on growth kinetics for most organisms and have even less understanding on the mechanisms governing predation, viral lysis, and predator avoidance in these systems. As a result, existing models that describe microbial communities contain dozens to hundreds of parameters, and state variables are extensively aggregated. Overall, the models are little more than non-linear parameter fitting exercises that have limited, to no, extrapolation potential, as there are few principles governing organization and function of complex self-assembling systems. Over the last decade, we have been developing a systems approach that models microbial communities as a distributed metabolic network that focuses on metabolic function rather than describing individuals or species. We use an optimization approach to determine which metabolic functions in the network should be up regulated versus those that should be down regulated based on the non-equilibrium thermodynamics principle of maximum entropy production (MEP). Derived from statistical mechanics, MEP proposes that steady state systems will likely organize to maximize free energy dissipation rate. We have extended this conjecture to apply to non-steady state systems and have proposed that living systems maximize entropy production integrated over time and space, while non-living systems maximize instantaneous entropy production. Our presentation will provide a brief overview of the theory and approach, as well as present several examples of applying MEP to describe the biogeochemistry of microbial systems in laboratory experiments and natural ecosystems.

Gheorge Craciun, University of Wisconsin-Madison

Title: Persistence, permanence, and global stability in reaction network models: some results inspired by thermodynamic principles

Abstract: The standard mathematical model for the dynamics of concentrations in biochemical networks is called mass-action kinetics. We describe mass-action kinetics and discuss the connection between special classes of mass-action systems (such as detailed balanced and complex balanced systems) and the Boltzmann equation. We also discuss the connection between the “global attractor conjecture” for complex balanced mass-action systems and Boltzmann’s H-theorem. We also describe some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.

Hong Qian, University of Washington

Title: Large deviations theory and emergent landscapes in biological dynamics

Abstract: The mathematical theory of large deviations provides a nonequilibrium thermodynamic description of complex biological systems that consist of heterogeneous individuals. In terms of the notions of stochastic elementary reactions and pure kinetic species, the continuous-time, integer-valued Markov process dictates a thermodynamic structure that generalizes (i) Gibbs’ macroscopic chemical thermodynamics of equilibrium matters to nonequilibrium small systems such as living cells and tissues; and (ii) Gibbs’ potential function to the landscapes for biological dynamics, such as that of C. H. Waddington’s and S. Wright’s.

John Harte, University of Berkeley

Coauthors: Micah Brush, Kaito Umemura

Title: Nonequilibrium dynamics of disturbed ecosystems

Abstract: The Maximum Entropy Theory of Ecology (METE) predicts the shapes of macroecological metrics in relatively static ecosystems, across spatial scales, taxonomic categories, and habitats, using constraints imposed by static state variables. In disturbed ecosystems, however, with time-varying state variables, its predictions often fail. We extend macroecological theory from static to dynamic, by combining the MaxEnt inference procedure with explicit mechanisms governing disturbance. In the static limit, the resulting theory, DynaMETE, reduces to METE but also predicts a new scaling relationship among static state variables. Under disturbances, expressed as shifts in demographic, ontogenic growth, or migration rates, DynaMETE predicts the time trajectories of the state variables as well as the time-varying shapes of macroecological metrics such as the species abundance distribution and the distribution of metabolic rates over individuals. An iterative procedure for solving the dynamic theory is presented. Characteristic signatures of the deviation from static predictions of macroecological patterns are shown to result from different kinds of disturbance. By combining MaxEnt inference with explicit dynamical mechanisms of disturbance, DynaMETE is a candidate theory of macroecology for ecosystems responding to anthropogenic or natural disturbances.


Talc

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:

Ice.

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’.

Huh?

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}]

and

\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.