It’s also a good example of how to reason about random systems and how they can tend to stable states. You may need to see this in a web-browser to see the simulations run.

The model is of a cell evolving in discrete time steps. The cell is a constrained volume of fluid, with a membrane that is meant to be the cell wall. Through diffusion, molecules called monomers can go in or out. These monomers are meant to be nutrients.

Within a cell there are N sites. Each monomer can attach to one of the N sites, or the site can be empty. The model evolves in discrete time steps. At every time step, a site can remain empty. If it has a monomer attached to it, the monomer can stay there, or it can unbind. The sites are meant to represent placeholders for monomers binding into polymers, but I’ll get into the biochem in a different post.

There are n+1 species of monomers that can bind to the sites. n of these are considered ‘inactive’, and one type is ‘active’. If a site has an active monomer, then the likelihood that other sites absorb an active monomer increases. A monomer being active is meant to represent a monomer being at the ‘right place at the right time’ that lets other active monomers bind to a cite. This is meant to capture ‘auto-catalysis’ — the binding of an active monomer makes it more likely that other active monomers will bind.

So each site has three possible states — empty, inactive monomer, or active monomer. Per unit time, the probabilities are modelled as follows:

If a site is filled by either an active or inactive monomer, then it has a probability pq of being emptied, or ‘desorbed’, where q is a constant.

On the other hand, if a site is empty,

• the probability of an inactive monomer being absorbed is np, where n is the number of types of inactive monomers.

• the probability of an active monomer being absorbed is ψ(x) · p, where x is the fraction of sites already occupied by active monomers.

  • ψ(x) is a monotonically increasing function of the fraction of sites that are already active.

  • This means that the more active sites there are, the more likely it is that active monomers will be absorbed.

This function psi is meant to encode the increased chance that an active monomer will be absorbed if there are active ones present already. This is shown in the video below.

For each site, the transition between each state at each time step is given by the following chain:

So if we let a simulation run, let’s see what happens to the fraction of empty, inactive, and active sites. If we start with a large number of inactive sites, the number of active sites doesn’t really seem to take off — and so the cell remains filled with inactive sites. In this case the cell is just filled with mostly ‘inactive’ monomers, and remains ‘dead’.

This is shown in the video below, where the blue monomers are inactive, and the green monomers are active.

On the other hand, if we initialise the system with a larger amount of active monomers, it seems to stay that way.

Now, if we increase the number of monomers and the time of the simulation rapidly, we see that the cell converges to one of two states. The green trajectory shows the convergence to an alive state in which there is a constant fraction of active monomers. And the blue one shows convergence to a dead state where most states are inactive monomers. This is pretty cool!

So we have got an apparently random, auto-catalytic system and have found that it has converged to two stable states. How do we know which initial conditions lead to which state, and what fraction there is of active monomers in each state? I will discuss this in my next post.

1. Dyson, Freeman J. Origins of Life. 2nd ed. Cambridge: Cambridge University Press, 1999.↩︎

The most celebrated result is the Maxwell-Boltzmann distribution. Given the temperature of an ideal gas, this result gives the probability distribution of the speeds of the particles in the gas. It’s not obvious to link these two together. The temperature is the macro variable, and the speeds of the particles are near impossible to observe in isolation.

Explicitly, the probability density for a particle of mass m to have speed v at temperature T is

where k_B is Boltzmann’s constant. The whole curve is set by the single macro number T — increasing T shifts the peak to higher speeds and broadens the tail.

Why is this important? Well, it’s not easy to observe the speeds of particles in isolation. But having a model for the speed distribution of particles then gives us the ability to compute other interesting things, like heat capacity, diffusion constants, and more. And these are observable.

There is one legendary result that uses the distribution to relate the viscosity, heat conductivity, and heat capacity all in one!

Elementary kinetic theory gives two macroscopic variables, viscosity and heat conductivity, in terms of microscopic quantities — the number density n, the mean speed ⟨v⟩, and the mean free path λ:

where η is the viscosity, κ is the heat conductivity, m is the mass per particle, and c_v is the heat capacity per particle. When you take the ratio — all the messy microscopic quantities (n, ⟨v⟩, λ) drop out and you are left with

where c_V = c_v / m is the specific heat capacity per unit mass. So if you measure any two of {η, κ, c_V}, the third is determined — a non-trivial macroscopic prediction that falls out of the micro-level distribution.

What would an analogous result in biology be?

Last time I used David Jordan’s bioreactor to observe the state variables of algae. The macro variables here are temperature, CO2 concentration, O2 concentration, and so on. It would be cool to link these variables to genetics, which are the micro variables of biology.

In other words, what information does a macro state variable give us about the probability distribution of genetic states in the algae?

And if we know that distribution, can we relate it to any other macro variables that we can observe, like behaviour?

Thanks to David Jordan for extensive discussions.

References

Roberto Livi and Paolo Politi, Nonequilibrium Statistical Physics: A Modern Perspective (Cambridge University Press, 2017), Chapter 1. DOI: 10.1017/9781107278974. Chapter 1 gives a clean derivation of the kinetic-theory transport coefficients above.

Anyways, they look complicated and scary. Like seriously, the hell is this:

In the above, ρ is the density, p is the pressure, v is the velocity field, μ is the viscosity coefficient, and f is any external body force per unit volume — gravity, for instance. p is the pressure gradient (how pressure changes from point to point), and ∇² is the second derivative in space.

The second equation ∇ · v = 0 is the incompressibility condition, which just says the fluid isn’t being squeezed or stretched anywhere — whatever flows into a region flows out.

Confused? Yep, it’s hard to work with.

But here’s a more intuitive way to think about it. I will show you how you can derive and think about this formula in a more natural way. For now, I will ignore the incompressibility assumption and the stress tensor, and focus on the main meat.

Here goes. It all goes back to Newton.

From your high school days you learnt in Newtonian Mechanics, acceleration is equal to the force divided by the mass

And acceleration is the derivative of velocity. So Newton’s Laws can be rewritten as

So what’s the fluids version of this equation?

Well, a fluid isn’t a point mass like in Newtonian mechanics; it’s a field where every point in space has a value associated. At different points in the fluid, we have a vector for the direction of the flow, a number for the pressure, a number for the density and so on.

So the ‘velocity’ of a fluid is a velocity field, where every point in the fluid has a given velocity associated with the flow. The velocity at this point is the velocity that a particle would travel if you placed it there. This is what the velocity field of a river looks like.

So how do we think about a ‘derivative‘ of a field? If you wanted to measure the derivative of a quantity in a field, the most natural approach is to pretend you put some measuring equipment in the flow, let it travel, and then observe the change in the measuring equipment. Imagine putting a thermometer in a river and then reading the change in temperature as it flows.

The change in quantity over time as something moves is called the material derivative. It measures the change in some quantity F, which is a function of a point in space and time, taking into account that the point at which we are measuring changes as well

The change, over a small time δt, is

Since the point itself is moving with the flow, its displacement over δt is δx = v δt, where v = dx/dt is the velocity at that point. Taylor expanding F around (x(t), t),

Don’t worry if the upside down triangles confuse you, these are just the derivatives but in three dimensions. Subtracting F(x, t), dividing by δt, and taking δt → 0, the material derivative is

The first term is how F changes at a fixed point in space; the remaining three are how F changes because the probe is being carried along by the flow — one for each spatial direction, weighted by the corresponding component of the velocity.

So the ‘acceleration’ in fluids space is just the material derivative of the velocity field,

Now what about the other side of the equation? Well Newton said that

But we are dealing with a fluid, so mass doesn’t make sense. Instead we should be using the density ρ (mass per unit volume), and so we have the equation

But what is the force on a fluid? Well it can be whatever the physical situation calls for. Gravity, friction, whatever you need. But for fluids, one force that always appears is the pressure gradient. Differences in pressure cause particles to move — fluid gets pushed from high pressure to low pressure, so the force per unit volume is −∇p.

Putting that in,

And then we also have friction. In fluids, friction is known as viscosity. This is a bit more technical, but David Tong has an excellent explanation here about why viscosity takes the form

where μ is the viscosity coefficient and ∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z² is the Laplacian. And so we finally have the Navier-Stokes equations,

Or, multiplying through by ρ to match the form at the top,

Bingo bango. We’re done.

I read about this experiment when reading about Lorenz’s history in Chaos by Gleick. Lorenz initially derived his chaotic strange attractor when looking at simplified versions of fluid convection. I’ll write another post on the theory later. For now, I’ll write about how to do it and show my results.

I got silicone gel, mica powder, a hot plate, and a borosilicate petri dish. I used 50 cSt High Temp Silicone Fluid, like this:

Then I just got the regular mica powder from Amazon.

I then put the oil in a borosilicate petri dish, a pinch of the mica powder, and then mixed it pretty thoroughly to get it even.

Then I heated it the mix using my cheaper hot plate that I got from Alibaba. At first, some discrete cells started to form, both and the middle and the sides:

You can see the boundaries start to form:

And then after a while the cells start to get really organised, even after I cooled it down.

Whilst the experiment is simple I think there are endless interesting questions.

• Is it easy to predict or model the size of the cells when it settles?

• Do the number of cells depend on how quickly the temperature was increased?

• What temperature do the cells break down and turbulence arises?

• Are the number of cells dependent on the steepness of the temperature gradient?

I also wanted to figure out a better way to automate the process here. I was thinking of attaching a camera on bird’s eye view and then doing some image segmentation over a long time periods to see the evolution. I also wanted a way to program my hotplate to control the temperature gradients.

I tested David Jordan’s tabletop bioreactor in my last post. I have a day job so need to be able to run experiments whilst I’m out of the house. I also realised how painful it is to change parameters in re-runs of experiments. So, I wanted to reduce friction in the whole process and make it much easier to interact with the bioreactor.

After all, the less friction there is, the more likely I’ll do stuff.

So, we thought it would be useful to automate the bioreactor so that it can run experiments on its own, in an agentic fashion. And it would be cool if we could tell it what to do in plain English!

So instead of writing an experiment with ad hoc code that looks like this…

import time
import requests

BIOREACTOR = "http://raspberrypi.local:8000"

def set_temperature(celsius: float):
    requests.post(f"{BIOREACTOR}/heater", json={"target_c": celsius})

def read_od():
    return requests.get(f"{BIOREACTOR}/od").json()["od600"]

def set_stirrer(rpm: int):
    requests.post(f"{BIOREACTOR}/stirrer", json={"rpm": rpm})

def pump_media(ml: float):
    requests.post(f"{BIOREACTOR}/pump/in", json={"volume_ml": ml})

def drain(ml: float):
    requests.post(f"{BIOREACTOR}/pump/out", json={"volume_ml": ml})

# Algae growth experiment: hold at 25C, stir at 200rpm,
# sample OD every 10 minutes, dilute once OD crosses 0.6.
set_temperature(25.0)
set_stirrer(200)

log = []
start = time.time()
while time.time() - start < 24 * 3600:
    od = read_od()
    log.append((time.time() - start, od))
    if od > 0.6:
        drain(20)
        pump_media(20)
    time.sleep(600)

Which is HARD AND ANNOYING, wouldn’t it be cool if you could just ask it something like this?

‘Please loop at different temperatures and measure the photosynthesis rate in increments of 2, making sure that each temperature has enough time to test what you need to test. For each loop flicker the light on for 10 seconds on and off’.

And so I tried to build a rough version. And got it working. Have a look at the first version, which is me telling to change the colours of the LED light in the bioreactor that the organisms are exposed to.

Here’s the architecture.

The bioreactor is controlled by a Raspberry Pi. I enabled ssh on it so that I could access the Pi remotely from my laptop on my local network. I then installed Claude Code on the Pi.

David Jordan then built an API that could control each of the hardware elements, which consist of temperature and gas sensors, a lighting element, a heating element, and an optical sensor.

To get it agentic, I built a model context protocol server on the Raspberry Pi. The model context protocol lets Claude ping the api provided that the server is running.

And to communicate with the bioreactor, the easiest way was to run Claude Code in headless mode. At that point, Claude then converts my plain English commands to api commands that control the bioreactor.

Moving forward, the next step is to create a memory file and data log so that the agent can continually refer to the logs of data as it makes decisions that are in line with the human’s commands. But this should be easy.

I’m excited about this for multiple reasons. It allows people do do experiments much more easily, myself included. It also lets people who are less physically able do science do experiments. And, since I care about biosecurity, it fits as a great minimal model in discovering where we should protect ourselves.

But you can get much cheaper lab equipment by buying direct from China. You can also buy second hand or through secondary auctions, but I’ve found this harder.

I’ve been ordering a lot of lab equipment from Alibaba or AliExpress. When I first scanned through these websites the prices were so low that I thought that most of the stuff wouldn’t work, or be of low quality.

But for my lab equipment at least, I’ve been wrong.

Most of the stuff I’ve got so far works fine. To do this I try to find things that have legit looking reviews and are not obvious scams. To tell what is an obvious scam, I ask Claude if the price is reasonable given the cost of components, and I usually get an idea of what’s completely fake.

This has now become the default way I buy equipment. I’ve been so impressed that I now suspect that European companies buy from Alibaba, stick a tag on, and jack up the price.

For example, I wanted to buy a magnetic stirrer and hot plate for my chemistry experiments. But the main scientific providers price this at around 300 bucks.

But on AliExpress I looked around and found the same thing for 50 bucks and free shipping. The reviews looked decent. And it’s literally a fifth of the cost. A fifth.

It arrived in a couple of days and works fine. Shipping was just as fast as anything on Amazon. Take a look.

On Alibaba I can also request for custom parts and they arrive fine as well. Here, I’m bartering for a custom Far UVC bandpass filter which they quoted 40 bucks. This would usually be 200 bucks if I ordered it from an optics company in the UK. The customer service is better, and the reps usually speak perfect English.

It came fine.

Wild. Surely this is a no brainer.

I am not sponsored by, endorsing, or non-endorsing anyone in this post.

Set within the countryside, its main attraction are its independent booksellers.

My personal favourite is converted cinema bookshop. It’s a babel-like temple of second hand books of any niche topic you can find. I especially enjoyed the polar expeditions section, the philosophy section, and the history of evolution section. Books were one pound to four pounds each, which is incredible value for knowledge. Here’s what it looked like:

And all of this set within this stunning backdrop.

In case you’re wondering, here are the books I got.

1. Gleick, J., Chaos

2. Devaney, R. L., An Introduction to Chaotic Dynamical Systems

3. Grandy, R. E. and Warner, R., Philosophical Grounds of Rationality

4. Russell, C. A., Science and Social Change, 1700–1900

5. Keynes, J. M., A Tract on Monetary Reform

6. Le Guin, U. K., The Dispossessed

7. Lenin, V. I., Selected Works, Volume 3

8. Locke, J., An Essay Concerning Human Understanding

9. Wells, H. G., The War of the Worlds

10. Mountevans, Admiral Lord, The Desolate Antarctic

11. Rousseau, J.-J., The Discourses and Other Early Political Writings

12. Tolansky, S., Introduction to Atomic Physics

Thanks George for showing us around.

My friend David Jordan built me a modular bioreactor. It ended up being really easy to use and it works! On my desk top at home! Here are my notes on testing it.

A bioreactor is something that monitors and controls biological media. This allows us to do experiments in a controlled environment. Bioreactors traditionally are a complicated and expensive piece of equipment, but the awesome thing that David did was to make one that’s super easy to use that fits on my table.

Here it is, with my finger for scale…

David’s bioreactor has a compartment to put a vial of biological media inside it. Then there’s a gas sensor that screws on top to complete the seal. Once the vial is inside the compartment, there are sensors to measure things. And also pieces that control the environment, like light and heat. The sensors and the environment components are all connected to a Raspberry Pi which can be automated or operated remotely.

This particular bioreactor has a few sensors

• a CO2 concentration sensor

• an O2 concentration sensor

• a temperature sensor

• an optical density sensor to see how cloudy the water is to track population growth.

Then there are components that change the environment of the vial.

• There is a heating element to control the temperature.

• There is an LED to expose the vial inside to light.

The sensors, heating element and LED are connected to a Raspberry Pi computer. On the Pi, theres a python script that records the reading of each element. David wrote a script, but it’s fully customisable and you can ssh into it too like a regular server. Robot lab geeks take note.

For the first experiment, I put some algae in some growth media. To prevent contamination I used my cheaper laminar flow hood, which pumps out HEPA filtered air over the workspace. Here’s a picture of me getting the algae out.

Then I put the algae in growth media and then screwed the gas sensor on top:

Sliding the vial into the compartment:

Then I turned the bioreactor on by hooking up the raspberry pi to a monitor, keyboard and mouse. Overall, super easy to set up. You can see the measurements on the monitor.

The first thing to try was turn the light on and off in 12 hour cycles. This was done with a python script, which also was used to regulate the temperature. So I started the bioreactor, turned on the sensors, turned on the light cycle, and the heating element kept the system at constant temperature.

And what did I observe?

The algae did photosynthesis and respiration!

The graph shows the oxygen and CO2 concentration during the periods where the light was off and on. The dark region shows when it’s off. When it’s dark, in the beginning, you can see the CO2 rising and the oxygen concentration falling. And then it’s light, it flips. Pretty cool! The CO2 eventually goes to zero but you can see the oxygen concentrations fluctuate.

The optical density plot is a proxy for the density of the species, which is also recorded. And you can see also that the bioreactor kept the temperature constant pretty well.

Above all, this bioreactor was super easy to set up, and easy to use. It makes me feel excited about the possibilities of being able to do science more easily. I’m also more optimistic about the role of table top, safe experiments for independent scientists.

David and I have spoke about the modularity of this design. I think a bioreactor has a cool interpretation in that you’re viewing life forms not as themselves, but through the macro variables of the environment that they are in. I’ve written a lot about state variables and phase space, both in thermodynamics and biology, and think there are a whole host of interesting things to explore along these lines with a bioreactor on your table top.

Acknowledgements

Thank you again to David Jordan for his generosity,

For example, the measurable variables for a single particle are its position and velocity. If it’s moving in one dimension, the position is a single number, and the velocity is a single number, for a total of two variables.

Phase space is the space of where these state variables can live.

Take a pendulum. I can describe the system through the angle it makes with the vertical axis, and the velocity of this angle. If we plot the values against each other, and then let time pass, we can trace out the phase space of a pendulum and see some interesting shapes evolve.

In the case of a pendulum with no friction, we get a closed loop. The x axis is the angle, and the y axis is the angular velocity. The animation below shows how phase space is traced out as the system evolves.

It’s not all the time that we get closed loops though. If we add air resistance, we get a spiral since the pendulum converges to a stationary point.

There are already interesting questions about viewing mechanical systems through this lens. A lot of these are geometric. For example, is the phase space ‘bounded’ in its geometry? Does the phase space lie on some interesting surface or subset, and if so what’s the dimension of the subset? What do the ‘orbits’ of points look like in phase space? And is there a way I can tell if orbits overlap?

Trajectories in phase space have their own nomenclature. Attracting points in phase space are points that the system converges to if the initial conditions start in some ‘basin of attraction’. Chaotic systems often have a never ending trajectory that is bounded, but never overlaps on itself. These are called strange attractors.

The Van Der Pol oscillator is a system where, no matter where you start in phase space, the point converges to a fixed cycle geometry.

Then, we have the questions of dynamics. If I start on a point in phase space, what path is subsequently traced out?

In high school, we use Newton’s laws to calculate the equations of motion. To do this, we use Newton’s laws to figure out what the acceleration will be once you solve for the forces.

But there is another way to do this, called the Hamiltonian method. This method tells you where a point in phase space will go to as time evolves. To do this, we first need to calculate the Hamiltonian of a physical system, which is the sum of its kinetic and potential energy as a function of its phase space variables. If we label momentum as p, and position as q the Hamiltonian is given by

where T(p) is the kinetic energy and V(q) is the potential energy.

With this Hamiltonian, we can calculate how each state variable will evolve according to Hamilton’s equations. The equations relate the time derivative of each state variable as a function of the Hamiltonian, and crucially gives us an avenue to see how the state of a system evolves.

As an example, lets look at the motion of a spring. A spring’s Hamiltonian is

where m is the mass, k is the spring constant, q is the displacement, and $p$ is the momentum. In this formulation, we treat position and momentum as its own variables, and then see what pops out. Hamilton’s equations for the spring give us the following equations

Now if we treat both variables as a function of time, and then solve it, we get that the solution is

where omega = sqrt{k/m} is the natural frequency of the spring, and q_0 and p_0 are the initial displacement and momentum. Now, if we choose a point in phase space, we can let the equations play out and see what lines we trace. In the case of a spring, it traces out an ellipse.

There are also interesting modern treatments that use phase space to predict the dynamics of a system. One such way is the delayed embedding method, which is a way to ‘learn’ the phase space trajectory by taking many observations. You could also think about phase space in terms of any state you want. For example, in biological systems, these could be temperature, CO2 concentration, and oxygen concentration.

I’m not an expert on this topic, but I wanted to write some parts of the book that have stuck out. None of this is my original work of course.

I learned that the most important event for an embryo is gastrulation. Wolpert himself famously said that “it is not birth, marriage, or death, but gastrulation, which is truly the most important time in your life” [1, p. 12]. A newly fertilised egg soon becomes a blastula, which is a spherical ball of cells. Gastrulation is when this ball of cells develops a form, which begins with the formation of the gut. To do this, cells move inward and take a different type, moulding the blastula to look like this:

As a mathematician, this to me is a case of biological symmetry breaking. Initially the blastula has a spherical symmetry, but then this breaks and the embryo becomes cylindrically symmetric, approaching the topology of a torus. An exercise to the reader; what Hamiltonian, if any, exhibits symmetry breaking from a sphere to C2?

On the empirical side of things, I learned that the history of embryology started during the time of the Greeks. Aristotle was amongst the first to investigate the development of embryos by cracking open fertilised chicken embryos [3]. And there was a debate until the 19th century about whether embryos came pre-formed, or developed throughout time (epigenesis) [1, Ch. 1].

One useful thing I learned is about the ease of using sea urchin embryos as a model study. Wolpert says that these are popular amongst developmental biologists because they’re easily available [1, Ch. 3]. The embryos are transparent, so you can easily see the development through a microscope. I’ve been thinking a lot about how I could do this safely at home too. More on that in a future post.

References

[1] Wolpert, L. (1991). The Triumph of the Embryo. Illustrated by D. Skinner. Oxford University Press.

[2] Aristotle. Generation of Animals (De Generatione Animalium). Translated by A. L. Peck. Loeb Classical Library, Harvard University Press, 1942.

I’m trying to run the London Independent Science Society’s meet-ups with less work. The less work I have, the more meet-ups I can run.

But reducing work does not mean spending less time on it, nor caring about it less. I love thinking about how to curate a dinner, designing an agenda, and meeting people. And I want to spend more time on that stuff.

So to me, less work means reducing the friction involved with bullshit admin. This friction is like a cognitive hump that stops me from doing something easy, for no reason.

The usual way I’ve setup events is by first polling people’s availability on Discord and WhatsApp. And then setting up an event page in a tool called Luma that’s then shared around. This already involves three websites I need to click through (cognitive friction). And one bullshit admin task is making these event pages in Luma. It’s fiddly and involves me clicking buttons to set-up an event with its date and time.

I already spend time on WhatsApp a lot so polling availability is not a problem for me. So I wanted to make the Luma part smoother.

I realised Luma had an api. So I:

1. Set up a lightweight Hetzner server.

2. Got the Luma api.

3. Got Claude code to learn and test the Luma api by pointing it to the docs.

4. Created an model context protocol on my Hetzner server so that my Claude could interact with the Luma api via its ‘Customisation’ function.

   1. This model context protocol is basically a set of API endpoints that are invoked by Claude, which in this case, take the suggested time and place of my commands and shove it into the Luma api call.

5. Added OAuth to secure the end point to the server.

6. Done!

So now when I’m on Claude on my phone, on the bus or whatever, when I have a free moment all I have to do is ask my Claude app to make the event page for me! I can preload in the notes of the MCP my preferred time of day for coffee, as well as the geo-locations of the cafes I always pick. I can’t do automated bookings yet, but I can do this:

And it works… I can also tell it to update the description, reschedule it, customise the image and all that jazz. This is enough friction reduction for me to make scheduling a lot easier.

I’m thinking of making a website for the society too, and I can easily add to the MCP protocol list to add the events to the website. Easy!