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The editors of this journal have an announcement:

We are happy to announce the founding editorial board of Compositionality, featuring established researchers working across logic, computer science, physics, linguistics, coalgebra, and pure category theory (see the full list below). Our steering board considered many strong applications to our initial open call for editors, and it was not easy narrowing down to the final list, but we think that the quality of this editorial board and the general response bodes well for our growing research community.

In the meantime, we hope you will consider submitting something to our first issue. Look out in the coming weeks for the journal’s official open-for-submissions announcement.

The editorial board of Compositionality:

• Corina Cristea, Alexandru Ioan Cuza University, Romania
• Ross Duncan, University of Strathclyde, UK
• Andrée Ehresmann, University of Picardie Jules Verne, France
• Tobias Fritz, Max Planck Institute, Germany
• Neil Ghani, University of Strathclyde, UK
• Dan Ghica, University of Birmingham, UK
• Jeremy Gibbons, University of Oxford, UK
• Nick Gurski, University of Sheffield, UK
• Helle Hvid Hansen, Delft University of Technology, Netherlands
• Chris Heunen, University of Edinburgh, UK
• Aleks Kissinger, Radboud University, Netherlands
• Joachim Kock, Universitat Autònoma de Barcelona, Spain
• Martha Lewis, University of Amsterdam, Netherlands
• Samuel Mimram, École Polytechnique, France
• Simona Paoli, University of Leicester, UK
• Dusko Pavlovic, University of Hawaii, USA
• Christian Retoré, Université de Montpellier, France
• Mehrnoosh Sadrzadeh, Queen Mary University, UK
• Peter Selinger, Dalhousie University, Canada
• Pawel Sobocinski, University of Southampton, UK
• David Spivak, MIT, USA
• Jamie Vicary, University of Birmingham, UK
• Simon Willerton, University of Sheffield, UK

Best,
Josh, Brendan, and Nina
Executive editors, Compositionality

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

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

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

Here are my lectures so far:

Lecture 55 – Chapter 4: Enriched Profunctors and Collaborative Design
Lecture 56 – Chapter 4: Feasibility Relations
Lecture 57 – Chapter 4: Feasibility Relations
Lecture 58 – Chapter 4: Composing Feasibility Relations
Lecture 59 – Chapter 4: Cost-Enriched Profunctors
Lecture 60 – Chapter 4: Closed Monoidal Preorders

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Azimuth by John Baez - 3d ago

In Random Points on a Sphere (Part 1), we learned an interesting fact. You can take the unit sphere in , randomly choose two points on it, and compute their distance. This gives a random variable, whose moments you can calculate.

And now the interesting part: when n = 1, 2 or 4, and seemingly in no other cases, all the even moments are integers.

These are the dimensions in which the spheres are groups. We can prove that the even moments are integers because they are differences of dimensions of certain representations of these groups. Rogier Brussee and Allen Knutson pointed out that if we want to broaden our line of investigation, we can look at other groups. So that’s what I’ll do today.

If we take a representation of a compact Lie group we get a map from group into a space of square matrices. Since there is a standard metric on any space of square matrices, this lets us define the distance between two points on the group. This is different than the distance defined using the shortest geodesic in the group: instead, we’re taking a straight-line path in the larger space of matrices.

If we randomly choose two points on the group, we get a random variable, namely the distance between them. We can compute the moments of this random variable, and today I’ll prove that the even moments are all integers.

So, we get a sequence of integers from any representation of any compact Lie group So far we’ve only studied groups that are spheres:

• The defining representation of on the real numbers gives the powers of 2.

• The defining representation of on the complex numbers gives the central binomial coefficients

• The defining representation of on the quaternions gives the Catalan numbers.

It could be fun to work out these sequences for other examples. Our proof that the even moments are integers will give a way to calculate these sequences, not by doing integrals over the group, but by counting certain ‘random walks in the Weyl chamber’ of the group. Unfortunately, we need to count walks in a certain weighted way that makes things a bit tricky for me.

But let’s see why the even moments are integers!

If our group representation is real or quaternionic, we can either turn it into a complex representation or adapt my argument below. So, let’s do the complex case.

Let be a compact Lie group with a unitary representation on This means we have a smooth map

where is the algebra of complex matrices, such that

and

where is the conjugate transpose of the matrix

To define a distance between points on we’ll give its metric

This clearly makes into a -dimensional Euclidean space. But a better way to think about this metric is that it comes from the norm

where is the trace, or sum of the diagonal entries. We have

I want to think about the distance between two randomly chosen points in the group, where ‘randomly chosen’ means with respect to normalized Haar measure: the unique translation-invariant probability Borel measure on the group. But because this measure and also the distance function are translation-invariant, we can equally well think about the distance between the identity 1 and one randomly chosen point in the group. So let’s work out this distance!

I really mean the distance between and so let’s compute that. Actually its square will be nicer, which is why we only consider even moments. We have

Now, any representation of has a character

defined by

and characters have many nice properties. So, we should rewrite the distance between and the identity using characters. We have our representation whose character can be seen lurking in the formula we saw:

But there’s another representation lurking here, the dual

given by

This is a fairly lowbrow way of defining the dual representation, good only for unitary representations on but it works well for us here, because it lets us instantly see

This is useful because it lets us write our distance squared

in terms of characters:

So, the distance squared is an integral linear combination of characters. (The constant function 1 is the character of the 1-dimensional trivial representation.)

And this does the job: it shows that all the even moments of our distance squared function are integers!

Why? Because of these two facts:

1) If you take an integral linear combination of characters, and raise it to a power, you get another integral linear combination of characters.

2) If you take an integral linear combination of characters, and integrate it over you get an integer.

I feel like explaining these facts a bit further, because they’re part of a very beautiful branch of math, called character theory, which every mathematician should know. So here’s a quick intro to character theory for beginners. It’s not as elegant as I could make it; it’s not as simple as I could make it: I’ll try to strike a balance here.

There’s an abelian group consisting of formal differences of isomorphism classes of representations of , mod the relation

Elements of are called virtual representations of Unlike actual representations we can subtract them. We can also add them, and the above formula relates addition in to direct sums of representations.

We can also multiply them, by saying

and decreeing that multiplication distributes over addition and subtraction. This makes into a ring, called the representation ring of

There’s a map

where is the ring of continuous complex-valued functions on This map sends each finite-dimensional representation to its character This map is one-to-one because we know a representation up to isomorphism if we know its character. This map is also a ring homomorphism, since

and

These facts are easy to check directly.

We can integrate continuous complex-valued functions on so we get a map

The first non-obvious fact in character theory is that we can compute inner products of characters as follows:

where the expression at right is the dimension of the space of ‘intertwining operators’, or morphisms of representations, between the representation and the representation

What matters most for us now is that this inner product is an integer. In particular, if is the character of any representation,

is an integer because we can take to be the trivial representation in the previous formula, giving

Thus, the map

actually takes values in

Now, our distance squared function

is actually the image under of an element of the representation ring, namely

So the same is true for any of its powers—and when we integrate any of these powers we get an integer!

This stuff may seem abstract, but if you’re good at tensoring representations of some group, like you should be able to use it to compute the even moments of the distance function on this group more efficiently than using the brute-force direct approach. Instead of complicated integrals we wind up doing combinatorics.

I would like to know what sequence of integers we get for A much easier, less thrilling but still interesting example is This is the 3-dimensional real projective space which we can think of as embedded in the 9-dimensional space of real matrices. It’s sort of cool that I could now work out the even moments of the distance function on this space by hand! But I haven’t done it yet.

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This is the tale of a mathematical adventure. Last time our hardy band of explorers discovered that if you randomly choose two points on the unit sphere in 1-, 2- or 4-dimensional space and look at the probability distribution of their distances, then the even moments of this probability distribution are always integers. I gave a proof using some group representation theory.

On the other hand, with the help of Mathematica, Greg Egan showed that we can work out these moments for a sphere in any dimension by actually doing the bloody integrals.

He looked at the nth moment of the distance for two randomly chosen points in the unit sphere in and he got

This looks pretty scary, but you can simplify it using the relation between the gamma function and factorials. Remember, for integers we have

We also need to know at half-integers, which we can get knowing

and

Using these we can express moment(d,n) in terms of factorials, but the details depend on whether d and n are even or odd.

I’m going to focus on the case where both the dimension d and the moment number n are even, so let

In this case we get

Here ‘we’ means that Greg Egan did all the hard work:

From this formula

you can show directly that the even moments in 4 dimensions are Catalan numbers:

while in 2 dimensions they are binomial coefficients:

More precisely, they are ‘central’ binomial cofficients, forming the middle column of Pascal’s triangle:



So, it seems that with some real work one can get vastly more informative results than with my argument using group representation theory. The only thing you don’t get, so far, is an appealing explanation of why the even moments are integral in dimensions 1, 2 and 4.

The computational approach also opens up a huge new realm of questions! For example, are there any dimensions other than 1, 2 and 4 where the even moments are all integral?

I was especially curious about dimension 8, where the octonions live. Remember, 1, 2 and 4 are the dimensions of the associative normed division algebras, but there’s also a nonassociative normed division algebra in dimension 8: the octonions.

The d = 8 row seemed to have a fairly high fraction of integer entries:


I wondered if there were only finitely many entries in the 8th row that weren’t integers. Greg Egan did a calculation and replied:

The d=8 moments don’t seem to become all integers permanently at any point, but the non-integers become increasingly sparse.

He also got evidence suggesting that for any even dimension d, a large fraction of the even moments are integers. After some further conversation he found the nice way to think about this. Recall that

If we let

then this moment is just

so the question becomes: when is this an integer?

It’s good to think about this naively a bit. We can cancel out a bunch of stuff in that ratio of binomial coefficents and write it like this:

So when is this an integer? Let’s do the 8th moment in 4 dimensions:

This is an integer, namely the Catalan number 42: the Answer to the Ultimate Question of Life, the Universe, and Everything.  But apparently we had to be a bit ‘lucky’ to get an integer. For example, we needed the 10 on top to deal with the 5 on the bottom.

It seems plausible that our chances of getting an integer increase as the moment gets big compared to the dimension. For example, try the 4th moment in dimension 10:

This not an integer, because we’re just not multiplying enough numbers to handle the prime 5 in the denominator. The 6th moment in dimension 10 is also not an integer. But if we try the 8th moment, we get lucky:

This is an integer! We’ve got enough in the numerator to handle everything in the denominator.

Greg posted a question about this on MathOverflow:

• Greg Egan, When does doubling the size of a set multiply the number of subsets by an integer?, 9 July 2018.

He got a very nice answer from a mysterious figure named Lucia, who pointed out relevant results from this interesting paper:

• Carl Pomerance, Divisors of the middle binomial coefficient, American Mathematical Monthly 122 (2015), 636–644.

Using these, Lucia proved a result that implies the following:

Theorem. If we fix a sphere of some even dimension, and look at the even moments of the probability distribution of distances between randomly chosen points on that sphere, from the 2nd moment to the (2m)th, the fraction of these that are integers approaches 1 as m → ∞.

On the other hand, Lucia also believes Pomerance’s techniques can be used to prove a result that would imply this:

Conjecture. If we fix a sphere of some even dimension > 4, and consider the even moments of the probability distribution of distances between randomly chosen points on that sphere, infinitely many of these are not integers.

In summary: we’re seeing a more or less typical rabbit-hole in mathematics. We started by trying to understand how noncommutative quaternions are on average. We figured that out, but we got sidetracked by thinking about how far points on a sphere are on average. We started calculating, we got interested in moments of the probability distribution of distances, we noticed that the Catalan numbers show up, and we got pulled into some representation theory and number theory!

I wouldn’t say our results are earth-shaking, but we definitely had fun and learned a thing or two. One thing at least is clear. In pure math, at least, it pays to follow the ideas wherever they lead. Math isn’t really divided into different branches—it’s all connected!

Afterword

Oh, and one more thing. Remember how this quest started with John D. Cook numerically computing the average of over unit quaternions? Well, he went on and numerically computed the average of over unit octonions!

• John D. Cook, How close is octonion multiplication to being associative?, 9 July 2018.

He showed the average is about 1.095, and he created this histogram:

Later, Greg Egan computed the exact value! It’s

On Twitter, Christopher D. Long, whose handle is @octonion, pointed out the hidden beauty of this answer—it equals

Nice! Here’s how Greg did this calculation:

• Greg Egan, The average associator, 12 July 2018.

Details

If you want more details on the proof of this:

Theorem. If we fix a sphere of some even dimension, and look at the even moments of the probability distribution of distances between randomly chosen points on that sphere, from the 2nd moment to the (2m)th, the fraction of these that are integers approaches 1 as m → ∞.

you should read Greg Egan’s question on Mathoverflow, Lucia’s reply, and Pomerance’s paper. Here is Greg’s question:

For natural numbers , consider the ratio of the number of subsets of size taken from a set of size to the number of subsets of the same size taken from a set of size :

For we have the central binomial coefficients, which of course are all integers:

For we have the Catalan numbers, which again are integers:


           

However, for any fixed , while seems to be mostly integral, it is not exclusively so. For example, with ranging from 0 to 20000, the number of times is an integer for 2,3,4,5 are 19583, 19485, 18566, and 18312 respectively.

I am seeking general criteria for to be an integer.

Edited to add:

We can write:

So the denominator is the product of consecutive numbers , while the numerator is the product of consecutive numbers . So there is a gap of between the last of the numbers in the denominator and the first of the numbers in the numerator.

Lucia replied:

Put , and then we can write more conveniently as

So the question essentially becomes one about which numbers for divide the middle binomial coefficient . Obviously when ,..

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John D. Cook, Greg Egan, Dan Piponi and I had a fun mathematical adventure on Twitter. It started when John Cook wrote a program to compute the probability distribution of distances where and were two randomly chosen unit quaternions:

• John D. Cook, How far is xy from yx on average for quaternions?, 5 July 2018.

Three things to note before we move on:

• Click the pictures to see the source and get more information—I made none of them!

• We’ll be ‘randomly choosing’ lots of points on spheres of various dimensions. Whenever we do this, I mean that they’re chosen independently, and uniformly with respect to the unique rotation-invariant Borel measure that’s a probability measure on the sphere. In other words: nothing sneaky, just the most obvious symmetrical thing!

• We’ll be talking about lots of distances between points on the unit sphere in dimensions. Whenever we do this, I mean the Euclidean distance in , not the length of the shortest path on the sphere connecting them.

Okay:

If you look at the histogram above, you’ll see the length is between 0 and 2. That’s good, since and are on the unit sphere in 4 dimensions. More interestingly, the mean looks bigger than 1. John Cook estimated it at 1.13.

Greg Egan went ahead and found that the mean is exactly

He did this by working out a formula for the probability distribution:

All this is great, but it made me wonder how surprised I should be. What’s the average distance between two points on the unit sphere in 4 dimensions, anyway?

Greg Egan worked this out too:



So, the mean distance for two randomly chosen unit quaternions is

The mean of is smaller than this. In retrospect this makes sense, since I know what quaternionic commutators are like: for example the points at the ‘north and south poles’ of the unit sphere commute with everybody. However, we can now say the mean of is exactly

times the mean of and there’s no way I could have guessed that.

While trying to get a better intuition for this, I realized that as you go to higher and higher dimensions, and you standing at the north pole of the unit sphere, the chance that a randomly chosen other point is quite near the equator gets higher and higher! That’s how high dimensions work. So, the mean value of should get closer and closer to And indeed, Greg showed that this is true:

The graphs here show the probability distributions of distances for randomly chosen pairs of points on spheres of various dimensions. As the dimension increases, the probability distribution gets more sharply peaked, and the mean gets closer to

Greg wrote:

Here’s the general formula for the distribution, with plots for n=2,…,10. The mean distance does tend to √2, and the mean of the squared distance is always exactly 2, so the variance tends to zero.

But now comes the surprising part.

Dan Piponi looked at the probability distribution of distances in the 4-dimensional case:

and somehow noticed that its moments

when n is even, are the Catalan numbers!

Now if you don’t know about moments of probability distributions you should go read about those, because they’re about a thousand times more important than anything you’ll learn here.

And if you don’t know about Catalan numbers, you should go read about those, because they’re about a thousand times more fun than anything you’ll learn here.

So, I’ll assume you know about those. How did Piponi notice that the Catalan numbers

were the even moments of this probability distribution? Maybe it’s because he recently managed to get ahold of Richard Stanley’s book on Amazon for just $11 instead of its normal price of $77.

(I don’t know how that happened. Some people write 7’s that look like 1’s, but….)

Anyway, you’ll notice that this strange phenomenon is all about points on the unit sphere in 4 dimensions. It doesn’t seem to involve quaternions anymore! So I asked if something similar happens in other dimensions, maybe giving us other interesting sequences of integers.

Greg Egan figured it out, and got some striking results:

Here d is the dimension of the Euclidean space containing our unit sphere, and Egan is tabulating the nth moment of the probability distribution of distances between two randomly chosen points on that sphere. The gnarly formula on top is a general expression for this moment in terms of the gamma function.

The obvious interesting feature of this table is that only for d = 2 and d = 4 rows are all the entries integers.

But Dan Piponi made another great observation: Greg left out the rather trivial d = 1 row, and that all the entries of this row would be integers too! Even better, d = 1, 2, and 4 are the dimensions of the associative normed division algebras: the real numbers, the complex numbers and the quaternions!

This made me eager to find a proof that all the even moments of the probability distribution of distances between points on the unit sphere in are integers when is an associative normed division algebra.

The first step is to notice the significance of even moments.

First, we don’t need to choose both points on the sphere randomly: we can fix one and let the other vary. So, we can think of the distance

as a function on the sphere, or more generally a function of And when we do this we instantly notice that the square root is rather obnoxious, but all the even powers of the function are polynomials on

Then, we notice that restricting polynomials from Euclidean space to the sphere is how we get spherical harmonics, so this problem is connected to spherical harmonics and ‘harmonic analysis’. The nth moment of the probability distribution of distances between points on the unit sphere in is

where we are integrating with respect to the rotation-invariant probability measure on the sphere. We can rewrite this as an inner product in namely

where 1 is the constant function equal to 1 on the whole sphere.

We’re looking at the even moments, so let n = 2m. Now, why should

be an integer when d = 1, 2 and 4? Well, these are the cases where the sphere is a group! For d = 1,

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

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

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

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

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There’s a new conference series, whose acronym is pronounced “psycho”. It’s part of the new trend toward the study of “compositionality” in many branches of thought, often but not always using category theory:

First Symposium on Compositional Structures (SYCO1), School of Computer Science, University of Birmingham, 20-21 September, 2018. Organized by Ross Duncan, Chris Heunen, Aleks Kissinger, Samuel Mimram, Simona Paoli, Mehrnoosh Sadrzadeh, Pawel Sobocinski and Jamie Vicary.

The Symposium on Compositional Structures is a new interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students.

Description

The Symposium on Compositional Structures is a new interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students.

Submission is easy, with no format requirements or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere.

While no list of topics could be exhaustive, SYCO welcomes submissions with a compositional focus related to any of the following areas, in
particular from the perspective of category theory:

• logical methods in computer science, including classical and
quantum programming, type theory, concurrency, natural language
processing and machine learning;
• graphical calculi, including string diagrams, Petri nets and
reaction networks;
• languages and frameworks, including process algebras, proof nets,
type theory and game semantics;
• abstract algebra and pure category theory, including monoidal
category theory, higher category theory, operads, polygraphs, and
relationships to homotopy theory;
• quantum algebra, including quantum computation and representation theory;
• tools and techniques, including rewriting, formal proofs and proof
assistants, and game theory;
• industrial applications, including case studies and real-world
problem descriptions.

This new series aims to bring together the communities behind many
previous successful events which have taken place over the last
decade, including “Categories, Logic and Physics”, “Categories, Logic
and Physics (Scotland)”, “Higher-Dimensional Rewriting and
Applications”, “String Diagrams in Computation, Logic and Physics”,
“Applied Category Theory”, “Simons Workshop on Compositionality”, and
the “Peripatetic Seminar in Sheaves and Logic”.

The steering committee hopes that SYCO will become a regular fixture
in the academic calendar, running regularly throughout the year, and
becoming over time a recognized venue for presentation and discussion
of results in an informal and friendly atmosphere. To help create this
community, in the event that more good-quality submissions are
received than can be accommodated in the timetable, we may choose to
defer some submissions to a future meeting, rather than reject them.
This would be done based on submission order, giving an incentive for
early submission, and avoiding any need to make difficult choices
between strong submissions. Deferred submissions would be accepted for
presentation at any future SYCO meeting without the need for peer
review. This will allow us to ensure that speakers have enough time to
present their ideas, without creating an unnecessarily competitive
atmosphere. Meetings would be held sufficiently frequently to avoid a
backlog of deferred papers.

Invited Speakers

• David Corfield, Department of Philosophy, University of Kent: “The ubiquity of modal type theory”.

• Jules Hedges, Department of Computer Science, University of Oxford: “Compositional game theory”

Important Dates

All times are anywhere-on-earth.

• Submission deadline: Sunday 5 August 2018
• Author notification: Monday 13 August 2018
• Travel support application deadline: Monday 20 August 2018
• Symposium dates: Thursday 20 September and Friday 21 September 2018

Submissions

Submission is by EasyChair, via the following link:

https://easychair.org/conferences/?conf=syco1

Submissions should present research results in sufficient detail to allow them to be properly considered by members of the programme committee, who will assess papers with regards to significance, clarity, correctness, and scope. We encourage the submission of work in progress, as well as mature results. There are no proceedings, so work can be submitted even if it has been previously published, or has been submitted for consideration elsewhere. There is no specific formatting requirement, and no page limit, although for long submissions authors should understand that reviewers may not be able to read the entire document in detail.

Funding

Some funding is available to cover travel and subsistence costs, with
a priority for PhD students and junior researchers. To apply for this
funding, please contact the local organizer Jamie Vicary at by the deadline given above, with a short statement of your travel costs and funding required.

Programme Committee

The symposium managed by the following people, who also serve as the
programme committee.

• Ross Duncan, University of Strathclyde
• Chris Heunen, University of Edinburgh
• Aleks Kissinger, Radboud University Nijmegen
• Samuel Mimram, École Polytechnique
• Simona Paoli, University of Leicester
• Mehrnoosh Sadrzadeh, Queen Mary, University of London
• Pawel Sobocinski, University of Southampton
• Jamie Vicary, University of Birmingham and University of Oxford
(local organizer)

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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:

or more precisely two: the forward reaction

with its rate constant and the reverse reaction

with its rate rate constant Then as I recently explained, these reactions are in a detailed balanced equilibrium when

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

Note: we have three variables, the concentrations and 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:

Now we have a detailed balance equilibrium when

Again, one monomial equals another monomial.

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

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

and

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 defined by a system of equations, each of which equates two monomials in the coordinates

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

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?

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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:

Here 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:

oxaloacetate,
citrate,
cis-aconitate,
isocitrate,
oxalosuccinate,
α-ketoglutarate,
succinyl-CoA,
succinate,
fumarate,
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 into , 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

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 and 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

arises from the fact that , and contain a single sulfur atom, while none of the other species involved contain sulfur.

Similarly, the conserved quantity

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:

, due to the conservation of adenosine.

, due to conservation of flavin adenine dinucleotide.

, due to conservation of nicotinamide adenine dinucleotide.

. This expresses the fact that in the citric acid cycle each species is transformed to the next, modulo 10.

. 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 is destroyed and one molecule of 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


which expresses the fact that in each turn of the citric acid cycle, one molecule of is destroyed and three of 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:

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 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.

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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 consists of five reactions:

Ammonia () and carbonate () 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


expresses conservation of phosphorus. The conserved quantity


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) , due to conservation of adenosine by all reactions in the urea cycle.

B) , since the only reaction in the urea cycle involving either or has as a reactant and as a product.

C) , since the only reaction involving either or has as a reactant and as a product.

D) , since the only reaction involving either or has as a reactant and as a product.

E) , since the only reaction involving either or has as a reactant and as a product.

F) , 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) , 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

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

violates conservation of quantities B), C) and F). (In these reactions we are neglecting 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 , conserved quantity D) prevents the destruction of without the production of an equal amount of , 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!

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