## Information Processing in Chemical Networks (Part 2)

13 June, 2017

I’m in Luxembourg, and I’ll be blogging a bit about this workshop:

Dynamics, Thermodynamics and Information Processing in Chemical Networks, 13-16 June 2017, Complex Systems and Statistical Mechanics Group, University of Luxembourg. Organized by Massimiliano Esposito and Matteo Polettini.

I’ll do it in the comments!

I explained the idea of this workshop here:

and now you can see the program here.

## The Mathematics of Open Reaction Networks

8 June, 2017

Next week, Blake Pollard and I will talk about our work on reaction networks. We’ll do this at Dynamics, Thermodynamics and Information Processing in Chemical Networks, a workshop at the University of Luxembourg organized by Massimiliano Esposito and Matteo Polettini. We’ll do it on Tuesday, 13 June 2017, from 11:00 to 13:00, in room BSC 3.03 of the Bâtiment des Sciences. If you’re around, please stop by and say hi!

Here are the slides for my talk:

Abstract. To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Feynman diagrams and the like. In principle all these different diagrams fit into a common framework: the mathematics of monoidal categories. This has been known for some time. However, the details are more challenging, and ultimately more rewarding, than this basic insight. Here we explain how various applications of reaction networks and Petri nets fit into this framework.

If you see typos or other problems please let me know now!

I hope to blog a bit about the workshop… it promises to be very interesting.

## Phosphorus Sulfides

5 May, 2017

I think of sulfur and phosphorus as clever chameleons of the periodic table: both come in many different forms, called allotropes. There’s white phosphorus, red phosphorus, violet phosphorus and black phosphorus:

and there are about two dozen allotropes of sulfur, with a phase diagram like this:

So I should have guessed that sulfur and phosphorus combine to make many different compounds. But I never thought about this until yesterday!

I’m a great fan of diamonds, not for their monetary value but for the math of their crystal structure:

In a diamond the carbon atoms do not form a lattice in the strict mathematical sense (which is more restrictive than the sense of this word in crystallography). The reason is that there aren’t translational symmetries carrying any atom to any other. Instead, there are two lattices of atoms, shown as red and blue in this picture by Greg Egan. Each atom has 4 nearest neighbors arranged at the vertices of a regular tetrahedron; the tetrahedra centered at the blue atoms are ‘right-side up’, while those centered at the red atoms are ‘upside down’.

Having thought about this a lot, I was happy to read about adamantane. It’s a compound with 10 carbons and 16 hydrogens. There are 4 carbons at the vertices of a regular tetrahedron, and 6 along the edges—but the edges bend out in such a way that the carbons form a tiny piece of a diamond crystal:

or more abstractly, focusing on the carbons and their bonds:

Yesterday I learned that phosphorus decasulfide, P4S10, follows the same pattern:

The angles deviate slightly from the value of

$\arccos (-1/3) \approx 109.4712^\circ$

that we’d have in a fragment of a mathematically ideal diamond crystal, but that’s to be expected.

It turns out there are lots of other phosphorus sulfides! Here are some of them:

Puzzle 1. Why do each of these compounds have exactly 4 phosphorus atoms?

I don’t know the answer! I can’t believe it’s impossible to form phosphorus–sulfur compounds with some other number of phosphorus atoms, but the Wikipedia article containing this chart says

All known molecular phosphorus sulfides contain a tetrahedral array of four phosphorus atoms. P4S2 is also known but is unstable above −30 °C.

All these phosphorus sulfides contain at most 10 sulfur atoms. If we remove one sulfur from phosphorus decasulfide we can get this:

This is the ‘alpha form’ of P4S9. There’s also a beta form, shown in the chart above.

Some of the phosphorus sulfides have pleasing symmetries, like the
alpha form of P4S4:

or the epsilon form of P4S6:

Others look awkward. The alpha form of P4S5 is an ungainly beast:

They all seem to have a few things in common:

• There are 4 phosphorus atoms.

• Each phosphorus atom is connected to 3 or 4 atoms, at most one of which is phosphorus.

• Each sulfur atom is connected to 1 or 2 atoms, which must all be phosphorus.

The pictures seem pretty consistent about showing a ‘double bond’ when a sulfur atom is connected to just 1 phosphorus. However, they don’t show a double bond when a phosphorus atom is connected to just 3 sulfurs.

Puzzle 2. Can you draw molecules obeying the 3 rules listed above that aren’t on the chart?

Of all the phosphorus sulfides, P4S10 is not only the biggest and most symmetrical, it’s also the most widely used. Humans make thousands of tons of the stuff! It’s used for producing organic sulfur compounds.

People also make P4S3: it’s used in strike-anywhere matches. This molecule is not on the chart I showed you, and it also violates one of the rules I made up:

Somewhat confusingly, P4S10 is not only called phosphorus decasulfide: it’s also called phosphorus pentasulfide. Similarly, P4S3 is called phosphorus sesquisulfide. Since the prefix ‘sesqui-’ means ‘one and a half’, there seems to be some kind of division by 2 going on here.

## Diamondoids

2 May, 2017

I have a new favorite molecule: adamantane. As you probably know, someone is said to be ‘adamant’ if they are unshakeable, immovable, inflexible, unwavering, uncompromising, resolute, resolved, determined, firm, rigid, or steadfast. But ‘adamant’ is also a legendary mineral, and the etymology is the same as that for ‘diamond’.

The molecule adamantane, shown above, features 10 carbon atoms arranged just like a small portion of a diamond crystal! It’s a bit easier to see this if you ignore the 16 hydrogen atoms and focus on the carbon atoms and bonds between those:

It’s a somewhat strange shape.

Puzzle 1. Give a clear, elegant description of this shape.

Puzzle 2. What is its symmetry group? This is really two questions: I’m asking about the symmetry group of this shape as an abstract graph, but also the symmetry group of this graph as embedded in 3d Euclidean space, counting both rotations and reflections.

Puzzle 3. How many ‘kinds’ of carbon atoms does adamantane have? In other words, when we let the symmetry group of this graph act on the set of vertices, how many orbits are there? (Again this is really two questions, depending on which symmetry group we use.)

Puzzle 4. How many kinds of bonds between carbon atoms does adamantane have? In other words, when we let the symmetry group of this graph act on the set of edges, how many orbits are there? (Again, this is really two questions.)

You can see the relation between adamantane and a diamond if you look carefully at a diamond crystal, as shown in this image by H. K. D. H. Bhadeshia:

or this one by Greg Egan:

Even with these pictures at hand, I find it a bit tough to see the adamantane pattern lurking in the diamond! Look again:

Adamantane has an interesting history. The possibility of its existence was first suggested by a chemist named Decker at a conference in 1924. Decker called this molecule ‘decaterpene’, and registered surprise that nobody had made it yet. After some failed attempts, it was first synthesized by the Croatian-Swiss chemist Vladimir Prelog in 1941. He later won the Nobel prize for his work on stereochemistry.

However, long before it was synthesized, adamantane was isolated from petroleum by the Czech chemists Landa, Machacek and Mzourek! They did it in 1932. They only managed to make a few milligrams of the stuff, but we now know that petroleum naturally contains between .0001% and 0.03% adamantane!

Adamantane can be crystallized:

but ironically, the crystals are rather soft. It’s all that hydrogen. It’s also amusing that adamantane has an odor: supposedly it smells like camphor!

Adamantane is just the simplest of the molecules called diamondoids.
These are a few:

2 is called diamantane.

3 is called triamantane.

4 is called isotetramantane, and it comes in two mirror-image forms.

Here are some better pictures of diamantane:

People have done lots of chemical reactions with diamondoids. Here are some things they’ve done with the next one, pentamantane:

Many different diamondoids occur naturally in petroleum. Though the carbon in diamonds is not biological in origin, the carbon in diamondoids found in petroleum is. This was shown by studying ratios of carbon isotopes.

Eric Drexler has proposed using diamondoids for nanotechnology, but he’s talking about larger molecules than those shown here.

For more fun along these lines, try:

Diamonds and triamonds, Azimuth, 11 April 2016.

## Periodic Patterns in Peptide Masses

6 April, 2017

Gheorghe Craciun is a mathematician at the University of Wisconsin who recently proved the Global Attractor Conjecture, which since 1974 was the most famous conjecture in mathematical chemistry. This week he visited U. C. Riverside and gave a talk on this subject. But he also told me about something else—something quite remarkable.

### The mystery

A peptide is basically a small protein: a chain of made of fewer than 50 amino acids. If you plot the number of peptides of different masses found in various organisms, you see peculiar oscillations:

These oscillations have a frequency of about 14 daltons, where a ‘dalton’ is roughly the mass of a hydrogen atom—or more precisely, 1/12 the mass of a carbon atom.

Biologists had noticed these oscillations in databases of peptide masses. But they didn’t understand them.

Can you figure out what causes these oscillations?

It’s a math puzzle, actually.

Next I’ll give you the answer, so stop looking if you want to think about it first.

### The solution

Almost all peptides are made of 20 different amino acids, which have different masses, which are almost integers. So, to a reasonably good approximation, the puzzle amounts to this: if you have 20 natural numbers $m_1, ... , m_{20},$ how many ways can you write any natural number $N$ as a finite ordered sum of these numbers? Call it $F(N)$ and graph it. It oscillates! Why?

(We count ordered sums because the amino acids are stuck together in a linear way to form a protein.)

There’s a well-known way to write down a formula for $F(N)$. It obeys a linear recurrence:

$F(N) = F(N - m_1) + \cdots + F(N - m_{20})$

and we can solve this using the ansatz

$F(N) = x^N$

Then the recurrence relation will hold if

$x^N = x^{N - m_1} + x^{N - m_2} + \dots + x^{N - m_{20}}$

for all $N.$ But this is fairly easy to achieve! If $m_{20}$ is the biggest mass, we just need this polynomial equation to hold:

$x^{m_{20}} = x^{m_{20} - m_1} + x^{m_{20} - m_2} + \dots + 1$

There will be a bunch of solutions, about $m_{20}$ of them. (If there are repeated roots things get a bit more subtle, but let’s not worry about.) To get the actual formula for $F(N)$ we need to find the right linear combination of functions $x^N$ where $x$ ranges over all the roots. That takes some work. Craciun and his collaborator Shane Hubler did that work.

But we can get a pretty good understanding with a lot less work. In particular, the root $x$ with the largest magnitude will make $x^N$ grow the fastest.

If you haven’t thought about this sort of recurrence relation it’s good to look at the simplest case, where we just have two masses $m_1 = 1, m_2 = 2.$ Then the numbers $F(N)$ are the Fibonacci numbers. I hope you know this: the $N$th Fibonacci number is the number of ways to write $N$ as the sum of an ordered list of 1’s and 2’s!

1

1+1,   2

1+1+1,   1+2,   2+1

1+1+1+1,   1+1+2,   1+2+1,   2+1+1,   2+2

If I drew edges between these sums in the right way, forming a ‘family tree’, you’d see the connection to Fibonacci’s original rabbit puzzle.

In this example the recurrence gives the polynomial equation

$x^2 = x + 1$

and the root with largest magnitude is the golden ratio:

$\Phi = 1.6180339...$

The other root is

$1 - \Phi = -0.6180339...$

With a little more work you get an explicit formula for the Fibonacci numbers in terms of the golden ratio:

$\displaystyle{ F(N) = \frac{1}{\sqrt{5}} \left( \Phi^{N+1} - (1-\Phi)^{N+1} \right) }$

But right now I’m more interested in the qualitative aspects! In this example both roots are real. The example from biology is different.

Puzzle 1. For which lists of natural numbers $m_1 < \cdots < m_k$ are all the roots of

$x^{m_k} = x^{m_k - m_1} + x^{m_k - m_2} + \cdots + 1$

real?

I don’t know the answer. But apparently this kind of polynomial equation always one root with the largest possible magnitude, which is real and has multiplicity one. I think it turns out that $F(N)$ is asymptotically proportional to $x^N$ where $x$ is this root.

But in the case that’s relevant to biology, there’s also a pair of roots with the second largest magnitude, which are not real: they’re complex conjugates of each other. And these give rise to the oscillations!

For the masses of the 20 amino acids most common in life, the roots look like this:

The aqua root at right has the largest magnitude and gives the dominant contribution to the exponential growth of $F(N).$ The red roots have the second largest magnitude. These give the main oscillations in $F(N),$ which have period 14.28.

For the full story, read this:

• Shane Hubler and Gheorghe Craciun, Periodic patterns in distributions of peptide masses, BioSystems 109 (2012), 179–185.

Most of the pictures here are from this paper.

My main question is this:

Puzzle 2. Suppose we take many lists of natural numbers $m_1 < \cdots < m_k$ and draw all the roots of the equations

$x^{m_k} = x^{m_k - m_1} + x^{m_k - m_2} + \cdots + 1$

What pattern do we get in the complex plane?

I suspect that this picture is an approximation to the answer you’d get to Puzzle 2:

If you stare carefully at this picture, you’ll see some patterns, and I’m guessing those are hints of something very beautiful.

Earlier on this blog we looked at roots of polynomials whose coefficients are all 1 or -1:

The pattern is very nice, and it repays deep mathematical study. Here it is, drawn by Sam Derbyshire:

But now we’re looking at polynomials where the leading coefficient is 1 and all the rest are -1 or 0. How does that change things? A lot, it seems!

By the way, the 20 amino acids we commonly see in biology have masses ranging between 57 and 186. It’s not really true that all their masses are different. Here are their masses:

57, 71, 87, 97, 99, 101, 103, 113, 113, 114, 115, 128, 128, 129, 131, 137, 147, 156, 163, 186

I pretended that none of the masses $m_i$ are equal in Puzzle 2, and I left out the fact that only about 1/9th of the coefficients of our polynomial are nonzero. This may affect the picture you get!

## Information Processing in Chemical Networks (Part 1)

4 January, 2017

There’s a workshop this summer:

Dynamics, Thermodynamics and Information Processing in Chemical Networks, 13-16 June 2017, Complex Systems and Statistical Mechanics Group, University of Luxembourg. Organized by Massimiliano Esposito and Matteo Polettini.

They write, “The idea of the workshop is to bring in contact a small number of high-profile research groups working at the frontier between physics and biochemistry, with particular emphasis on the role of Chemical Networks.”

The speakers may include John Baez, Sophie de Buyl, Massimiliano Esposito, Arren Bar-Even, Christoff Flamm, Ronan Fleming, Christian Gaspard, Daniel Merkle, Philippe Nge, Thomas Ouldridge, Luca Peliti, Matteo Polettini, Hong Qian, Stefan Schuster, Alexander Skupin, Pieter Rein ten Wolde. I believe attendance is by invitation only, so I’ll endeavor to make some of the ideas presented available here at this blog.

### Some of the people involved

I’m looking forward to this, in part because there will be a mix of speakers I’ve met, speakers I know but haven’t met, and speakers I don’t know yet. I feel like reminiscing a bit, and I hope you’ll forgive me these reminiscences, since if you try the links you’ll get an introduction to the interface between computation and chemical reaction networks.

In part 25 of the network theory series here, I imagined an arbitrary chemical reaction network and said:

We could try to use these reactions to build a ‘chemical computer’. But how powerful can such a computer be? I don’t know the answer.

Luca Cardelli answered my question in part 26. This was just my first introduction to the wonderful world of chemical computing. Erik Winfree has a DNA and Natural Algorithms Group at Caltech, practically next door to Riverside, and the people there do a lot of great work on this subject. David Soloveichik, now at U. T. Austin, is an alumnus of this group.

In 2014 I met all three of these folks, and many other cool people working on these theme, at a workshop I tried to summarize here:

Programming with chemical reaction networks, Azimuth, 23 March 2014.

The computational power of chemical reaction networks, 10 June 2014.

Chemical reaction network talks, 26 June 2014.

I met Matteo Polettini about a year later, at a really big workshop on chemical reaction networks run by Elisenda Feliu and Carsten Wiuf:

Trends in reaction network theory (part 1), Azimuth, 27 January 2015.

Trends in reaction network theory (part 2), Azimuth, 1 July 2015.

Polettini has his own blog, very much worth visiting. For example, you can see his view of the same workshop here:

• Matteo Polettini, Mathematical trends in reaction network theory: part 1 and part 2, Out of Equilibrium, 1 July 2015.

Finally, I met Massimiliano Esposito and Christoph Flamm recently at the Santa Fe Institute, at a workshop summarized here:

Information processing and biology, Azimuth, 7 November 2016.

So, I’ve gradually become educated in this area, and I hope that by June I’ll be ready to say something interesting about the semantics of chemical reaction networks. Blake Pollard and I are writing a paper about this now.

## Compositionality in Network Theory

29 November, 2016

I gave a talk at the workshop on compositionality at the Simons Institute for the Theory of Computing next week. I spoke about some new work with Blake Pollard. You can see the slides here:

• John Baez, Compositionality in network theory, 6 December 2016.

and a video here:

Abstract. To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, Petri nets, electrical circuit diagrams, signal-flow graphs, chemical reaction networks, Feynman diagrams and the like. In principle all these different diagrams fit into a common framework: the mathematics of symmetric monoidal categories. This has been known for some time. However, the details are more challenging, and ultimately more rewarding, than this basic insight. Two complementary approaches are presentations of symmetric monoidal categories using generators and relations (which are more algebraic in flavor) and decorated cospan categories (which are more geometrical). In this talk we focus on the latter.

This talk assumes considerable familiarity with category theory. For a much gentler talk on the same theme, see: