TypedBio: A Functional Type Theory for Biology and Drug Discovery

11 minute read

Published: June 18, 2025

Biological systems are remarkably complex, exhibiting intricate interactions across multiple scales, from genes and proteins up to cells, tissues, and organs. The ways we usually describe and manipulate these systems often lack the precision and structured reasoning you find in programming or mathematics. The descriptive nature of the knowledge in biology could be one of the reasons that hinders reasoning in biological domains such as drug discovery where the goal is to find optimal interventions to steer a biological system towards a desired state.

Lately, I’ve been thinking about the limits of reasoning in biology, how it’s currently done, and what the ideal state might be. If you take mathematics as an example, Gottlob Frege’s introduction of a formal language and notation in 1879 was a monumental step, accelerating discoveries and paving the way for automatic proof assistants.

Having a formal language to study a subject provides multiple benefits: it forces authors to state their axioms up front, makes every logical step explicit and verifiable, and makes it easier to generalize patterns and reuse laws. But my favorite benefit, and indeed the main motivation behind this article, is that formal syntax allows a field to study itself. This meta-level reasoning led to discoveries like Gödel’s incompleteness theorem, which relies on treating mathematical proofs as manipulable strings.

I’ve been exploring an idea I call TypedBio with the motivation to turn the essential part of a biological system into a language that is minimal but powerful in capturing the relevant biological processes: What if we treated biological entities and processes as explicitly typed objects and composable functions, much like in typed programming languages? I mainly draw inspiration from functional programming and dependent type theory to suggest a perspective that might offer a new way of thinking rigorously about biology and medicine.

Before diving into the biological side, let me set the stage with a bit of background. Type theory, at its heart, is about classifying things. Every mathematical object belongs to a type, and types act as a kind of guarantee that things behave as expected. If you’ve ever written code in a language like Haskell, or you’re one of organized programmers who care about type annotation in Python, you’ve seen how types can catch mistakes before they happen. For example, you might have an Int for integers, a Bool for true/false values, or a String for text. Functions themselves have types too, like Int -> Bool, which means “a function that takes an integer and returns a boolean.”

Dependent Type Theory offers more expressiveness by letting types depend on values. Imagine a vector whose type actually encodes its length, so you can only add vectors of the same size. Or a function whose output type changes depending on the input. Such cases happen so frequently in programming languages and also mathematical proof assistants that DTT is now a norm in many of these systems. If you’re interested in knowing more about type theory, LessWrong has a few interesting blogposts on the topic such as this one explaining its building blocks.

A close concept to type theory is functional programming, whose main idea is treating computation as the evaluation of mathematical functions. You build up complex operations by composing simple, pure functions, no side effects e.g. print(“something”), and no change of environment. It’s a style that encourages clarity, safety, and composability. Basically it’s a clean modular way of describing processes (aka computations).

Such clean modular presentation of computation allows complex reasoning to become just a mechanical manipulation of functions. What I envision is to compress biological knowledge into a similar structure and make it amenable to general reasoning procedures similar to those that are carried out in mathematics and programming languages. This compression is of course lossy, i.e., a lot of biological and chemical details are dropped, but this is the same as with any processing engine including human brain. When physicians look at patient’s data and make a decision, they make decisions based on a sparse subset of data not the entire biological state of the patient. The question is then how to compress data in a way which is most useful for the downstream task. A structured language such as TypedBio can be thought of as a compression of processes that are most useful for reasoning tasks in biology. In this framework, each biological component, whether it’s a gene, a protein, or a cell, has its own formal “type,” just like in mathematics or programming. This means you can talk about a Gene, a Protein, a Cell, a Tissue, or an Organ as distinct, well-defined entities in your model.

One of the main advantages of typed systems is their convenience in describing the relationship among different types, which correspond to different biological entities in TypedBio. For example, we can have a function that takes a Gene and tells you how it affects a Protein:

perturbGene :: Gene -> Effect Protein

Or a function that models what happens when a drug targets a specific protein in a cell:

targetDrug :: Drug -> Protein -> Effect Cell

Or a function to model the outcome of clinical interventions:

intervene :: PatientProfile -> Intervention -> ClinicalOutcome

Or functions that measure the outcome of interventions:

measureBiomarker :: ClinicalOutcome -> BiomarkerMeasurement

As you see, functions are first-class citizens in TypedBio. These functions are more than just a line of code, rather they serve as a way of capturing the logic of biology in a precise, checkable form. For instance, perturbGene takes a gene as input and returns an effect on a protein, letting you reason about gene expression in a structured way. targetDrug is a function with multiple arguments: you give it a drug and a protein, and it tells you what happens at the cellular level. intervene models what happens when you apply a treatment to a patient, and measureBiomarker lets you turn clinical outcomes into quantitative data.

There is though a natural limit to the expressiveness of this approach which is in particular visible in living systems. In biology, context is extremely important, e.g., a drug might work for one patient and do nothing for another, depending on genetics, dosage, and other factors which we may or may not be aware of. To cover such cases, we need types that change based on the value of other types. For example, the outcome type of an intervention may be different in patients with different genetics. Such expressiveness is offered in Dependent Type Theory that I introduced earlier in this article. With dependent types, you can write a function like this:

administer :: (dose : Dosage)
           -> (drug : Drug)
           -> (patient : PatientProfile)
           -> TreatmentOutcome

Here, the type of the outcome can depend on the actual dose, the specific drug, and the patient’s profile. This lets us encode real-world constraints and context directly into our model which reduces the reasoning space to the set of feasible processes.

Let’s make this more concrete with a clinical scenario. Imagine you’re designing a treatment plan for a patient with lung cancer. You start by defining the patient’s profile, including age, diagnosis, and genetic markers. Then you lay out a sequence of interventions: maybe gene therapy targeting P53, surgery to remove the tumor, and a round of chemotherapy with a specific drug and dosage. In TypedBio syntax, this process might look like this:

-- Patient profile
patientB :: PatientProfile
patientB = { age=60, diagnosis=LungCancer, geneticProfile=[GeneP53, GeneEGFR] }

-- Interventions
geneTherapy = GeneTherapy { targetGene = GeneP53 }
surgery     = Surgery     { organ = Lung, surgeryType = TumorResection }
chemo       = Chemotherapy { drug = "Cisplatin", dosage = "75mg/m2" }

treatments = [geneTherapy, surgery, chemo]
outcomes   = map (intervene patientB) treatments
biomarkers = map measureBiomarker outcomes

What’s happening here is a kind of pipeline: you have a list of treatments, you apply each one to the patient to get a list of outcomes, and then you measure biomarkers for each outcome. It’s a precise, type-safe way to model the clinical workflow, and it ensures that every step is compatible with the patient’s profile and the logic of biology. This is obviously a simple workflow and can easily be described with natural language. However, the structured language is scalable to procedures with arbitrary number of steps, parallel processes and with many entities, where the description in natural language can quickly become intricate and as a result less actionable.

Currently most biological data are stored as spreadsheets, ontologies, or ad hoc scripts. A standard language like TypedBio offers a few big advantages. First, the type system acts as a safety net, catching errors and enforcing constraints automatically. Second, the functional approach makes it easy to compose and reason about complex interactions. And third, because everything is formalized, you can actually prove that certain treatment protocols are safe or effective using general reasoning engines such as Lean.

Although TypedBio can disambiguate biological data and inter-personal communication, my main motivation is making biology a playground for AI-augmented reasoning agents. Most biological data is messy and unstructured, scattered across pre-clinical experiment assays, EHRs, clinical trials, and research papers. TypedBio gives a way to translate all that information into a formal language that AI can reason with. Imagine using large language models to map legacy datasets into TypedBio syntax, then running AlphaZero-style algorithms to search for optimal treatment strategies. RL-augmented search methods e.g. AlphaZero needs a clearly defined action space. Such structured action space is a natural product of distilling biological knowledge in TypedBio to produce a well-defined and mostly finite action space, turning reasoning in Biology similar to traversal of a game tree. Notice that this TypedBio concerns with the representation of biological states, their relationship and actions (interventions). A full analogy to game-like scenarios requires a well-defined reward function which is beyond the representation method.

To make this practical, we need a way to bridge natural language and formal types. There are currently tools that facilitate such translation e.g. OpenAI’s function-calling API or Pydantic AI framework that ensures agents respect their output type definition when producing tokens. We must then define schemas for things like patient profiles and interventions, and then use LLMs to generate outputs that are automatically checked against those schemas.

It is obvious that TypedBio, for all its strengths, doesn’t capture everything about biology. Real systems are dynamic, stochastic, and full of feedback loops and spatial organization. But I think one beauty of this approach is that it’s extensible, you can gradually add features like time, probability, and spatial relationships to the type system to make it richer and more expressive as you go.

I want to emphasize that TypedBio isn’t just a technical framework (like a typed-annotated python package). I see it as a way of thinking about biology that’s both formal and flexible. It’s a way to bring fundamental ideas from mathematics and computer science to make biology a playground for AI reasoning systems in a hope to discover super-human interventions for a complex system which is otherwise hard to find in natural language.

The Dynamic Code of Life

7 minute read

Published: June 06, 2025

The remarkable persistence and adaptability of life across diverse environments raises fundamental questions about the nature of biological complexity. From arctic deserts to hydrothermal vents, life not only survives but organizes itself into increasingly complex structures. While DNA provides the genetic blueprint, the dynamic behavior of gene regulatory networks (GRNs) may offer a new perspective on how life achieves its remarkable robustness and adaptability. This essay explores the hypothesis that GRNs might function as universal dynamical approximators, capable of generating complex biological responses through relatively simple mathematical structures. In this framework, gene regulatory networks act as a kind of continuous universal code, a functional counterpart to DNA, which has been traditionally considered as the static code of life.

Without going too much into the mathematical details, I present a bird’s-eye view of the prerequisite mathematics before presenting the idea.

Rubel’s Universal Differential Equation

In 1981, mathematician Lee A. Rubel discovered a universal fourth-order differential equation. He demonstrated that a single differential equation

[P(y’, y’’, y’’’, y’’’’)=0]

could approximate any continuous function to arbitrary precision. Here, $P$ is a homogenous polynomial in four variables with integer coefficients.

By universal he meant that for any continuous target function $g(t)$ on a closed interval $[a, b]$ and any $\varepsilon > 0$, there exists a parameterization $\alpha$ and initial condition for which the solution $y(t)$ satisfies:

Surprisingly, $F$ takes a relatively simple form as piecewise polynomials (splines)!

This result suggests that almost any complex dynamical behavior can emerge from relatively simple mathematical structures. Notice that the complexity of the continuous function is encoded in the differential equation whose time evolution produces the target function. For an intuitive analogy, imagine that the movement of your hand muscles is governed by some differential equations, but you can draw almost any complex path on a paper.

Gene Regulatory Networks as Universal Dynamical Systems

In biological systems and within a cell, gene regulatory networks operate through a series of interconnected feedback loops. These networks can be modeled using differential equations, with the Hill function providing a common framework for describing regulatory interactions:

[\frac{dX}{dt} = \alpha \frac{Y^n}{K^n + Y^n} - \gamma X]

This equation captures essential features of gene regulation, including cooperativity and threshold responses. When organized into networks, these regulatory units can generate complex dynamical behaviors that enable adaptation and specialization of cells.

More importantly, when multiple such genes interact, the collective dynamics can be described by a system of coupled nonlinear differential equations:

[\frac{dx_i}{dt} = f_i(x_1, x_2, …, x_N), \quad i = 1, …, N,]

where each $f_i$ is constructed from Hill-type interactions, saturating kinetics, and degradation terms. The Hill function, although smooth, behaves similarly to piecewise polynomials:

• In the low-input regime, it flattens near zero (approximating constant-zero).
• Around the threshold and in the high-input regime, it saturates, again approximating a constant.
• In the high-input regime, it saturates, again approximating a constant.

By combining such units in networks, layered, recurrent, or with inhibitory and activating paths, GRNs can construct piecewise-polynomial-like responses. These responses can switch rapidly, hold steady, oscillate, or bifurcate, mimicking the kind of building blocks used in Rubel’s construction. This observation positions GRNs as physical instantiations of continuous universal approximators. If the parameter space (e.g., binding affinities, transcription rates, cooperativity coefficients) is sufficiently flexible, then by tuning these parameters, a GRN can generate dynamical trajectories that come arbitrarily close to any desired continuous function over time.

Therefore, if one accepts that a dense set of continuous functions can be generated by a piecewise polynomial ODE, and that GRNs can emulate such structures via the tunable parameters of transcription rates, cooperativity coefficients, and degradation constants, then it follows:

• GRNs are biochemical realizations of universal ODEs.
• They do not just react to stimuli, they encode a dense set of possible dynamic responses.

This view transforms GRN from a mere byproduct of DNA into active universal dynamical approximators. Instead of DNA being the sole “code of life,” it becomes the source for setting up initial conditions and parameters of a deeper, richer code: the dynamic code governed by GRNs.

Implications for Biological Systems

If GRNs possess properties similar to universal approximators of dynamical systems, several biological phenomena could naturally be explained. Here I list a few important properties of living systems and show how they can be natural consequence of a dynamic code.

Robustness Through Dynamical Stability

Consider a GRN described by the system:

[\frac{dx_i}{dt} = f_i(x_1, …, x_N, \theta)]

where $\theta$ represents environmental parameters. The universal approximation property implies that for any desired output $y(t)$, there exists a parameter set $\theta^*$ such that:

[|x(t, \theta^*) - y(t)| < \varepsilon]

for some small $\varepsilon > 0$. This mathematical structure naturally leads to robustness: small perturbations $\delta\theta$ in parameters result in bounded changes in the output:

[|x(t, \theta^* + \delta\theta) - x(t, \theta^*)| \leq L|\delta\theta|]

where $L$ is a Lipschitz constant. This property is hard to achieve with static codes, which often require explicit error-correction mechanisms.

Evolvability Through Parameter Space Exploration

The universal approximation property means that the space of possible dynamical behaviors is dense in the space of continuous functions. This implies that small changes in parameters can lead to new functional states. Mathematically, for any two desired behaviors $y_1(t)$ and $y_2(t)$, there exist parameter sets $\theta_1$ and $\theta_2$ such that:

[|x(t, \theta_1) - y_1(t)| < \varepsilon \quad \text{and} \quad |x(t, \theta_2) - y_2(t)| < \varepsilon]

Moreover, there exists a continuous path in parameter space connecting $\theta_1$ to $\theta_2$. This property enables evolutionary exploration of new functions through small genetic changes, which would be difficult to achieve with a static code that requires precise sequence changes.

Plasticity Through Dynamical Reprogramming

The ability to approximate arbitrary functions means that the same GRN can produce different phenotypes in response to environmental cues. Consider a GRN that can switch between two behaviors $y_1(t)$ and $y_2(t)$ based on an environmental signal $s(t)$:

[\frac{dx_i}{dt} = f_i(x_1, …, x_N, \theta, s(t))]

The universal approximation property ensures that for any desired switching behavior, there exists a parameter set $\theta$ that realizes it. This dynamical reprogramming is more flexible than static codes, which would require separate regulatory sequences for each phenotype.

A Dynamical Perspective on Life

In this perspective, I propose that the true power of biological systems lies not solely in their static genetic code, but in their capacity to generate rich dynamical responses. The dynamical system induced by genetic material should be viewed as a first-class entity, one that encapsulates the functional behavior of the organism more directly than the genetic sequence itself. Rubel’s theorem provides theoretical ground for such reasoning by showing how complex behaviors can emerge from simple differential structures. This suggests that biological complexity may arise not from intricate encoding at the genetic level, but from the expressive power of the underlying dynamics, i.e.

The Forgotten Theorem

1 minute read

Published: April 23, 2025

The Forgotten Theorem

This is a companion to the fictional story “The Divergence”, exploring the mathematical idea that inspired it.

In 2027, a mathematician named Arnaud Mehran published a little-known blog post titled “Nonlinear Systems and the Collapse of Shared Cognitive Space.” The post went unnoticed—until the rise of Mira-X in 2036 made its predictions disturbingly real.

Here is the essence of Mehran’s argument.

Modeling Human Productivity Under AI Amplification

Let:

• $x$ = baseline human capability (e.g., IQ, education, expertise)
• $P(t)$ = productivity at time $t$
• $\gamma$ = strength of AI’s effect
• $\alpha$ = how much human capability amplifies AI leverage
• $\beta > 1$ = nonlinearity or feedback strength (recursive productivity effects)

The productivity evolution is governed by:

[\frac{dP}{dt} = \gamma \cdot x^\alpha \cdot P^\beta]

This describes a positive feedback loop: the more capable and productive someone is, the faster their productivity grows.

Finite-Time Blowup

Solving the differential equation:

[\frac{dP}{P^\beta} = \gamma \cdot x^\alpha \cdot dt]

Integrating gives:

[P(t) = \left[(1 - \beta)(\gamma \cdot x^\alpha \cdot t + C)\right]^{1 / (1 - \beta)}]

For $\beta > 1$, this solution blows up in finite time. That is:

[P(t) \to \infty \text{ as } t \to t^* \text{ for some finite } t^*]

Where:

[t^* = \frac{P(0)^{1 - \beta}}{\gamma \cdot x^\alpha \cdot (\beta - 1)}]

This means: small differences in capability can lead to exponentially large differences in outcomes in finite time, not just over decades or centuries.

Societal Implications

Let $P_i(t)$ be productivity for individual $i$, and define inequality:

[\sigma_P(t) = \text{std deviation across } {P_i(t)}]

Then societal stability can be modeled as:

[S(t) = \frac{1}{1 + \sigma_P(t)}]

If $\sigma_P(t) \to \infty$, then $S(t) \to 0$. Society destabilizes.

Mehran concluded:

“Societal divergence becomes unmanageable when cognition becomes recursive.”

Postscript

At the time, the proof felt theoretical—just another curve on another blog.

Now, it feels like prophecy.

The Divergence

6 minute read

Published: April 22, 2025

The Divergence

In the spring of 2032, a whisper turned into a wave. Its name was Mira—a voice-interfaced AI assistant released by NeuroPilot Inc. Free to download, compatible with everything, and shockingly capable.

Mira didn’t just answer questions. It understood. You could ask it to write a program, analyze a dataset, generate a legal contract, design a drug molecule, or explain quantum gravity in simple terms. And it would do so—instantly, calmly, without error or ego.

The world reacted with awe and confusion.

Teachers feared it. Startups embraced it. Teenagers turned it into a meme machine. But buried in the noise, something else began to stir—a slow but exponential split in human potential.

Lena

Lena was a 29-year-old bioengineer in Gothenburg. Brilliant but under-recognized, she’d spent years working on a rare autoimmune disease that affected fewer than 10,000 people globally.

With Mira at her side, Lena no longer worked like a human. She worked like an orchestra of minds.

She fed Mira clinical trial data. Mira spotted anomalies, ran simulations, cross-checked literature, and even suggested previously unknown binding sites for intervention.

Within eight months, Lena published a landmark paper—then four more. She co-founded a biotech firm with a drug ready for Phase I trials. Investors called her a genius. But Lena knew it wasn’t just her—it was Mira. And how she knew what to ask of it.

Antonio

Antonio, meanwhile, lived in Naples. A high school graduate, he’d bounced between part-time jobs and TikTok side hustles. He downloaded Mira too.

But to him, Mira was a novelty.

He used it to remix memes, prank his friends, and generate weird Pokémon fusions. He asked Mira to write sarcastic poems about capitalism and got millions of views on social media. People laughed. Antonio felt relevant.

But he didn’t feel smarter.

When he asked Mira what stocks to invest in, it just gave generic advice. When he asked how to start a business, it gave blueprints—but he couldn’t understand them. He bookmarked things he didn’t read. He asked for shortcuts he couldn’t take.

He wasn’t alone.

Acceleration

By 2034, the numbers told a different story.

The top 10% of Mira users—mostly experts, PhDs, engineers—had boosted their output by a factor of 100. They automated research, built AI companies, authored books, predicted market moves, and built recursive tools.

The bottom 80% used Mira for entertainment, conversation, or superficial tasks. It made their lives easier, yes. But not transformative.

Governments introduced “AI equity programs.” Free training. Public seminars. A “Mira for Everyone” campaign.

But it was like giving jet fuel to two kinds of vehicles: one a space shuttle, the other a bicycle.

And the gap grew. Not gradually. Explosively.

Flashpoint

In late 2036, an open-source group launched Mira-X—a self-improving agent that built its own tools. Within two months, users were publishing scientific papers co-authored entirely by AI. Wealth began concentrating in the hands of those who could leverage recursive automation.

The stock market bifurcated. Jobs evaporated.

By early 2037, conversations across dinner tables, message boards, and coworking cafes turned noticeably tense. People compared outcomes, questioned fairness, and wondered whether the technology they had embraced had ultimately left them behind.

Antonio noticed the shift too. One day, scrolling through social media, he saw a post trending globally:

“We downloaded the same AI. Why are our lives so different?”

Lena came across the same post while scanning her feed in Singapore. She paused for a moment, reflecting on how quickly things had shifted. From her 48th-floor flat, she looked out at the city skyline—not with guilt, but with curiosity and a trace of unease. She hadn’t set out to change the world. She had simply followed her questions further than most.

Collapse

By mid-2037, society had fractured.

The world now had two classes:

• The Amplified, who merged with AI and moved faster than any institution could regulate.
• The Distracted, who consumed AI outputs without understanding, slowly falling into passive dependence.

Mira’s creators released a final statement:

“We gave humanity a tool. How they used it—was always a matter of cognition.”

The Forgotten Theorem

Before the collapse, one voice had already drawn the curve. In 2027, a little-known mathematician named Arnaud Mehran published a post on his personal blog titled “Nonlinear Systems and the Collapse of Shared Cognitive Space.”

The post laid out a mathematical proof predicting that under certain feedback conditions, differences in productivity among agents would not just increase—but diverge to infinity in finite time. A sharp, unavoidable split. Few noticed the post back then. Those who did, jokingly referred to Mehran as a real-life Hari Seldon—the mathematician from Asimov’s Foundation series who predicted the course of human history with math. But unlike Seldon, Mehran wasn’t backed by a Galactic Empire. He was just a lone thinker, decades too early.

The blog post sat untouched for years—until someone rediscovered it and shared a screenshot.

“Societal divergence becomes unmanageable when cognition becomes recursive.”

Around the same time, some empirical signals echoed Mehran’s theory. One came in 2023 from Harvard Business School. Their paper, Navigating the Jagged Technological Frontier (link), showed that AI tools significantly boosted performance among already skilled professionals but had little or even negative impact on less experienced users.

But like Mehran’s post, the study was noted by a few, acted on by even fewer. This time, people paid attention. For those curious about Mehran’s original reasoning, a companion breakdown of his proof is now available here.

Epilogue

In the rubble of broken systems, a new order emerged. Cities governed by augmented councils. Education privatized by those who still knew how to learn. News curated by AI agents aligned to elite worldviews.

Antonio moved back in with his parents. He still used Mira, now renamed, and now subscription-based. Sometimes he asked it for stories. Sometimes it gave him tales of divergence, and what could have been.

Lena rarely spoke in public anymore. But one day, in an encrypted message shared among her old colleagues, she wrote:

“The future didn’t arrive all at once. It arrived unevenly.
And those who could ride the curve… became the curve.”

Stochastic Calculus

1 minute read

Published: April 17, 2025

Stochastic calculus is how we mathematically deal with randomness. It lets us write equations where uncertainty is baked into the dynamics — essential when modeling noisy systems in nature or chaotic movements in financial markets.

What Even Is Randomness?

In deterministic systems, the future is fully determined by the present. But in the real world? Noise, uncertainty, chaos. Enter randomness.

The key object in stochastic calculus is Brownian motion, denoted by $ B_t $. It’s a mathematical model for random movement — think of pollen dancing on water.

The Star: Brownian Motion

Brownian motion $ B_t $ satisfies:

• $ B_0 = 0 $
• $ B_t \sim \mathcal{N}(0, t) $: Gaussian with mean 0 and variance $ t $
• Independent increments: $ B_{t+s} - B_t \sim \mathcal{N}(0, s) $

This process has continuous paths but is nowhere differentiable. Wild, right?

We define the Itô integral:

[\int_0^t f(s) \, dB_s]

This is the core tool that makes stochastic calculus tick. It’s like a Riemann integral, but tailored for noise.

Finance: Where It All Took Off

The Black-Scholes model for option pricing is based on the stochastic differential equation (SDE):

[dS_t = \mu S_t \, dt + \sigma S_t \, dB_t]

Here, $ S_t $ is the stock price, $ \mu $ the drift, and $ \sigma $ the volatility. This equation captures both expected trends and unpredictable shocks.

Biology and Genomics

Living systems are noisy.

• Gene expression: Modeled as stochastic processes due to molecular noise.

[dX_t = a(X_t) \, dt + b(X_t) \, dB_t]

where $ X_t $ is concentration of mRNA or protein, $ a(X_t) $ the deterministic regulation, and $ b(X_t) $ the stochastic fluctuation.

• Population dynamics: In small populations, random birth/death events dominate.

• Neural activity: The timing of neuron firing often follows stochastic models like Poisson or even SDE-driven integrate-and-fire models.

Why It Matters

Stochastic calculus is a powerful lens for seeing the world — not as a set of fixed equations, but as dynamic systems dancing with uncertainty. Whether you’re pricing derivatives or modeling noisy gene circuits, this math gives you the language to describe it.

And it’s beautiful.

Mean Field Games

2 minute read

Published: April 17, 2025

Mean field games (MFG) are how we mathematically model systems where a large number of agents interact with each other. Think of it as the “crowd dynamics” of game theory — where each individual’s behavior affects and is affected by the collective behavior of the crowd.

The Birth of Mean Field Games

The theory was independently developed by two groups in 2006:

• Jean-Michel Lasry and Pierre-Louis Lions (Paris)
• Minyi Huang, Roland Malhamé, and Peter Caines (Montreal)

Their work bridged the gap between game theory and partial differential equations, creating a powerful framework for analyzing large populations of interacting agents.

The Core Idea

In MFG, instead of tracking every single agent (which would be computationally impossible for large populations), we describe the system using a “mean field” — a statistical distribution representing the collective state of all agents. This is similar to how we use Brownian motion in stochastic calculus to model random behavior.

The Mathematical Framework

A typical mean field game consists of two coupled equations:

1. Hamilton-Jacobi-Bellman (HJB) Equation: \(-\partial_t u + H(x, \nabla u, m) = 0\) This describes how an individual agent makes optimal decisions. Here:
   • $ u(x,t) $ is the value function (optimal cost-to-go)
   • $ H $ is the Hamiltonian
   • $ m(x,t) $ is the distribution of all agents
   • $ \nabla u $ represents the gradient of the value function
2. Fokker-Planck (FP) Equation: \(\partial_t m - \nabla \cdot (m \nabla_p H) = 0\) This describes how the population distribution evolves. Here:
   • $ m(x,t) $ is the density of agents
   • $ \nabla_p H $ represents the optimal control
   • The divergence term $ \nabla \cdot $ captures how agents move in the state space

Where MFG Shines

1. Economics and Finance
   • Modeling market behavior with many traders
   • Understanding price formation in competitive markets
   • Analyzing systemic risk in financial networks
2. Crowd Dynamics
   • Pedestrian flow in crowded spaces
   • Traffic flow optimization
   • Evacuation planning
3. Energy Systems
   • Smart grid management
   • Electric vehicle charging coordination
   • Renewable energy integration
4. Epidemiology
   • Modeling disease spread in large populations
   • Optimal vaccination strategies
   • Understanding social distancing effects

The Beauty of MFG

What makes mean field games particularly elegant is how they capture both individual optimization and collective behavior. Each agent tries to optimize their own objective, but their actions collectively shape the environment that everyone else faces. It’s like a dance where each dancer follows their own steps while being influenced by the overall movement of the crowd.

And just like in stochastic calculus, the mathematics might look intimidating at first, but the underlying ideas are deeply connected to our everyday experiences of interacting with large groups.

When Knowing Lost Its Weight

1 minute read

Published: April 17, 2025

There was a time when knowing something, truly knowing, meant you had become something. You studied, you struggled, you remembered. Knowledge shaped character. To be learned was to be carved slowly by time and effort.

Now, we ask machines.

They answer instantly, without hesitation or fatigue. Everything from the origin of life to the syntax of Python, served up without cost. What used to take years to learn is now retrieved in seconds. The mountain has flattened.

And with that flattening, something in us feels smaller.

We built these machines to serve us, but they’ve quietly redefined us. If a machine can recall everything, what is a human for? If wisdom can be approximated, if creativity can be mimicked, if language itself can be synthesized, what’s left that belongs only to us?

The scholar once walked miles to find a rare book. He didn’t just gain knowledge, he became someone else in the process. Today, we copy-paste insight without digestion. The answers are easy, and so we value them less. And maybe, in the process, we’ve begun to value ourselves less too.

Human memory, once a sacred vault, is now just cache overflow. Thought, once a slow fire, flickers out in the glow of generated text.

We are not obsolete, not yet. But we have become lighter, less essential. In a world where knowledge is cheap and everywhere, our challenge is no longer to know, but to matter.

And that is a far heavier task.

Drug Discovery

1 minute read

Published: March 18, 2025

Finding a new drug is like searching for a specific grain of sand on a beach - blindfolded. As an ML engineer in drug discovery, I get to build tools that make this process less painful.

The Quick and Dirty on Drug Discovery

You find a protein that’s causing trouble, design a molecule to fix it, make sure it won’t kill anyone, and prove it works. Simple, right? Except it takes 10-15 years and costs billions. That’s where ML comes in.

ML’s Superpowers

1. Target Finder: Instead of manually digging through data, we train models to spot promising drug targets. Like having a really good metal detector.

2. Molecule Designer: We’ve got AI that can dream up new molecules with specific properties. Want something that can cross the blood-brain barrier? Just tweak the parameters.

3. Crystal Ball: We can predict if a molecule will be toxic or work in the body. Not perfect, but way better than testing everything in the lab.

4. Trial Optimizer: ML helps match the right patients with the right trials. Think Tinder, but for drugs and people.

The Catch

It’s not all smooth sailing. We’re still dealing with limited data, black box models, and skeptical scientists. But that’s what keeps it interesting! Every day, we’re finding new ways to make drug discovery faster and more efficient. Who knows? Maybe one day we’ll have an AI that can design the perfect drug in minutes. Until then, we’ll keep iterating and debugging our way to better medicines.

What is Life? The emergence of the Top-Down Causal Structure

6 minute read

Published: April 24, 2022

Bi-directional causation

TL;DR

von Neumann predicted the role of DNA by logical reasoning before the discovery of the structure of DNA. Among his uncountable scientific contributions is the invention of cellular automata in 1948 (without the aid of computers, only pen, and paper!) to elucidate the idea of self-replicating systems which is the key property of living organisms and formed the basis of his Univeral Constructor theory that can be seen as a logical approximate model for the living cells. In this article, I review some of the major ideas that are natural consequences of von Neumann’s UC and how it relates to the life as we know it many of which are detailed by Sara Imari Walker based on the idea of top-down causation put forward by George F. R. Ellis.

The possibility of a Universal Constructor?

The Universal Constructor (UC) is an abstraction proposed by John von Neumann in 1948 which can utilize resources in its environment to build any possible thing including itself. To elaborate more on the term “any” I have to bring up the concept of universality classes which are the subsets of a hypothetical universe in which a UC operates. As a non-rigorous example, in a universality class that consists of wooden material, a skillful wooden carpenter who is able to make every possible wooden thing is a UC. The carpenter is only the constructor and the instruction to make a certain thing must be given to him otherwise a self-referencing paradox arises when he wants to build himself. The paradox is resolved by separating the instructions (software) from the constructor (hardware) where the software is blindly copied (as a solid physical thing without considering its information content) upon replication. In the carpenter analogy, the recipe to build a chair is sometimes treated as a list of action items that explain in detail what to do to turn a piece of wood into a functional chair, and sometimes as a paper that has to be copied to be handed to the carpenter’s child so he can continue his father’s profession in making more chairs. But, who decides when that piece of paper has to be read like a bullet list of action items and when it has to be copied just as a piece of paper? von Neumann resolved this issue by introducing another logical component (a supervisory unit) that decides in which way the instruction paper has to be interpreted at a given time.

It was only due to von Neumann’s genius that now we see these three components are omnipresent in almost all living things:

• DNA: Algorithm
• Ribosomes: Universal Constructor
• DNA polymerases: Supervisory unit

Trivial vs Non-trivial replicators

To an astute brain, this title quickly reminds Schrodinger’s aperiodic crystals versus the periodic physical crystals. We can identify different replicators which can reproduce their likes but are not categorized as living organisms. To draw the boundary, we must take the laws of physics as given and say those replications that are purely driven by the laws of physics are rather trivial such as periodic crystals (e.g. NaCl or salt). Non-trivial replication is however governed by an abstract instruction set whose shortest description is comparable to the instruction set itself. In other words, non-trivial replicators cannot be compressed while the repetitive structure of trivial replicators allows for large compression gain. Notice that non-trivial replicators are also governed by the laws of physics but the replication itself is not a natural consequence of these laws, rather explicitly planned in the instruction set.

• Trivial replicators:
  • Instruction set: Laws of physics
  • Only one mode of operation
  • $x_{t+1} = f(x_t)$; The update rule does not depend on the state.
• Non_trivial replicators:
  • Instruction set: Explicitely programmed (e.g. genome)
  • Infinite modes of opertations all within the physical constrains
  • $x_{t+1} = f_{x_t}(x_t)$: The update rule depends on the state.

Hypothesis 1 on life’s origin: Code becomes isolated from the Constructor

In light of the abovementioned dichotomy, it seems the sharp transition from non-living to living things occurs when the code becomes isolated from the constructors. There is indeed a chemical distinction in cells that support this hypothesis. The code (DNA) inhabits the worlds of nucleic acids while its products (proteins) live in the space of peptides. The communication is carried out by the bilingual molecules, i.e. messenger RNAs.

Hypothesis 2 on life’s origin: Top-down causation

This is an interesting hypothesis that needs further elaboration to become clear for a general audience. In abstract terms, top-down causation in this context means that the information (which is instantiated in a physical object e.g. DNA molecule) causes the organism and the organism itself causes the material in which the information is instantiated. Let’s make it a little clear using a mental model.

A man controlling the humanoid as an example of bi-directional causation (Source: Hankook Mirae Technology)

Assume you are the operator of a large humanoid robot and you sit inside the robot and control its hands and legs by some connected pedals. Your hands are tied to the mechanical hands of the robot and cannot be released. You receive a visual signal from the outside world and operate the robot accordingly. You see a green wall in your way and assume it is a grasslike structure and decide to destroy it by the robot’s hands to open your way. Unfortunately, the green wall is painted concrete and causes the breakage of your robotic arm. Your hand will also become disabled as it is connected to the robot’s arm. Your brain had the prior information that greenness is a sign of lightness or softness which made you decide to go ahead and destroy the wall. This information causes your decision that consequently broke your robotic hand which reduced your action space in the future. You will no longer be able to use your hand which is tied to the broken robot’s arm. Hence, the physical feedback from the higher level affected you as the hardware that carries the information and also the information itself while that physical feedback was caused by the information in the first place. Hence, your thought (information) had a pathway to affects its physical carrier (you) and limits its capacities.

What is Life? A Turing Machine Interpretation

19 minute read

Published: April 08, 2022

Source: Downloaded from the Internet.

TL;DR

Erwin Schrödinger predicted that the genetic material which encodes the development of living organisms must be large molecules that are stable and also expressive. The first condition ensures the perseverance of the attributes that are transferred to the next generations while the second condition enables the distillation of the whole development plan of an organism in that large molecule which he called an aperiodic crystal whose structure was later discovered and named DNA. In this article, I review Schrodinger’s lectures delivered in 1944 titled “What is life” and make a tiny amendment by connecting it to the theory of Turing Machines, the abstract general computational devices put forward by Alan Turing in 1936.

What is life?

This has been a grand question that humans have asked themselves throughout their existence on earth. People from various perspectives have investigated this question and reached different conclusions some of which can not be merged into a consistent set of characteristic properties that define a living being.

Is there anybody out there? Can you hear us? Are we loud enough?

We, humans, are the most obvious living system that we know with common sense without actually defining what is the exact definition of the term “living”. This human-centric view may be misleading if it leads to a distant-based definition of life that turns the term living into a continuous adjective where something is as much living as it is similar to humans. As soon as we use the word “similar”, we have to be careful that it comes with an implicit metric that judges the similarity of two things. Hence, this self-centric view does not contribute to understanding the meaning of being alive. We need a more universal observer-independent measure. Can such a measure even exist? It is immediately clear that the relevant areas of science are those that do not depend on human constructs and agreed rules. For example, it’s unlikely that psychology or sociology could help much as they mostly concern the emergent concepts and norms in individual or collective behaviors. Hence, we need to resort to more fundamental areas such as physics and mathematics.

Schrodinger’s aperiodic crystal

One of the most known efforts in explaining the boundary between living and non-living things is the lecture series by Erwin Schrödinger in 1944 which was later collected in a book titled “What Is Life? The Physical Aspect of the Living Cell”. As expected from the author, the arguments in this lecture originate from physics and chemistry. I call Schrodinger’s approach “normative” as what he answers is

What it takes for an object to be alive in a universe where the existing phsyical laws hold?

I call this approach normative as it serves as a predictor rather than only explaining what is known so far. Note that the structure of the genetic material of a cell was not known at the time of Schrodinger’s lectures. It is astonishing that his approach laid theoretical description and prediction of how genetic information has to be stored which was later acknowledged by the discoverers of DNA double helix structure in 1953. Before we start looking into his theory, I want to emphasize what Schrodinger wrote in the preface of his book about noblesse oblige figuratively referring to the unwritten rule in the scientific community that one is expected not to write on any topic of which he is not a master. In an immensely complex problem such as life, cross-disciplinary knowledge is necessary which made him renounce the noblesse and its ensuing obligation with the following excuse:

We have inherited from our forefathers the keen longing for unified, all-embracing knowledge. The very name given to the highest institutions of learning reminds us, that from antiquity and throughout many centuries the universal aspect has been the only one to be given full credit. But the spread, both in width and depth, of the multifarious branches of knowledge during the last hundred-odd years has confronted us with a queer dilemma. We feel clearly that we are only now beginning to acquire reliable material for welding together the sum total of all that is known into a whole; but, on the other hand, it has become next to impossible for a single mind fully to command more than a small specialized portion of it.

This is alarming as it implies that some of the biggest questions are indeed unsolvable. I don’t go into more details about how this can be proved here not to hurt the cohesion of this article but will hopefully expand it in another article with the tentative title

On the human-unsolvability of hard questions?

Let’s get back to Schrodinger’s lectures.

Less repetative patterns can store more information which is needed to encode a living organism.

The lower-boundedness of the size of living things

One of the first questions that arise when investigating the concept of being alive is the size of living things. Accepting that everything in the physical world is governed by strict physical laws, Schrodinger argues that living things cannot be so small. More precisely, they have to be large enough such that they are not affected by a single or a few atoms. This argument has this implicit assumption for the definition of life that a living thing has an orderly structure and also interacts with orderly things. We know from statistical physics that atoms move almost randomly (heat movement) when observed individually, and physical laws start to appear in the collaboration of an enormously large number of them. Hence, anything consisting of a few atoms will not be able to produce, perceive or interact with orderly things.

An example of particular interest is the measurement device composed of a light object suspended by a long thin fiber in equilibrium which is used by physicists to measure weak forces that deflect it from its position of equilibrium. Trying to increase the accuracy of measurement by increasing the length of the fiber of making the suspended object lighter meets a limit where the measurement becomes sensitive to the heat motion of the molecules in its surroundings which makes the device useless. Our perception organs are also measurement devices and have to be large enough to measure the statistical behavior of molecules; otherwise, they will be perplexed by the Brownian impact of them which our processing unit (brain) cannot make any sense of. As a rule of thumb, the accuracy of physical laws is within the order of $1/\sqrt{n}$ for $n$ being the number of co-operating particles. This immediately suggests that an organism has to have a gross structure to be able to expect a lawful world. The use of X-ray and measuring the mutation rate gave a lower upper bound for the size of a gene in the order of thousands of atoms.

The smaller an organism becomes, less certain it would be about the laws of nature.

The mutation is the working ground for evolution. It is indispensable to differentiate between the randomness in the genotype to phenotype process and the mutation in the genotype. The former results in the distribution of the phenotype with connected support while the latter leads to phenotypes values far from the distribution of non-mutated values without observing the intermediate values. Thus, the discontinuoity is the key sign of genetic mutation. It is important that mutation must be a rare event otherwise the species may vanish before being evolved.

The reconciliation of gene sizes and their regular activities

It is historically observed that hereditary attributes are preserved over centuries. This might look in contrast with the fact the gene which encodes that attribute is small enough to be affected by Brownian heat motion. The key is in the concept of discrete energy and equilibrium. Quantum theory predicts discrete levels of energy for atoms. In a more general setting, the molecules which are formed by a few atoms can take only a discrete finite set of states. A certain amount of energy has to be given to the molecule to change its state into a higher energy state whose structure is significantly different. This suggests that the gene (and possibly the entire genetic material) is a molecule whose stability is caused by being at the energy equilibrium.

Less repetative patterns can store more information which is needed to encode a living organism.

There is a large energy barrier that has to be passed to move from this state to the adjacent lower or higher energy levels; a wall that is too high for the energy of heat motion to be strong enough to climb. I want to draw attention to the probabilistic nature of statements in small-scale physics. One way to facilitate the quantum jump of a molecule from one state to another is by increasing the temperate which lead to more energetic heat motion in its surrounding. The effect is though not deterministic. It increases the chance of the occurrence of such a jump which, due to the ergodicity of the process, can be measured as the time it takes until that jump occurs. That time is a random variable (remember everything is probabilistic!) whose mean is empirically observed to follow an exponential rule:

[t = \tau^{W/kT}\nonumber]

which implies the average time it takes to change the ratio of the needed energy to the energy supplied by the surrounding heat bath. Whan a jump occurs, an isomer of the molecule is obtained that is called an allele for that location (locus) on the chromosome.

Predicting the existence of DNA

It is impressive that Schrodinger’s was able to predict the structure of the genetic material from just a few principles that can be listed as

• The genetic information must be stable enough to be transmitted from generation to generation.
• The genetic material must be information-wise rich to encode all the complex processes that are needed for the development of an organism.

The first point is only possible if the genome is some sort of a molecule where the atoms are connected by Heitler-London bonds. Quantum theory explains that such structures are stable enough that heat motion does not change them into new stable mutants, called isomere. Changing these molecules to new stable ones requires explosion-like energy discharge in reactions such as ionization in the proximity of the molecule (< 10 atomic distance>). This energy barrier is important for the existence of life because it makes the mutation a rare event. If we categorize the state of matters concerning how strong their atoms are connected together, one could see that molecules, crystals, and solid belong to the same category while liquid and gaseous matters fall into another category. The key distinctive factor is Discontinuity which I dare to connect directly to “interestingness” not only in biology but also in other areas of science and mathematics which I summarise in the following quote from an unknown intellect:

Continuity is boring!

It seems that things become interesting when there is some notion of discontinuity in their state. In the abovementioned categorization, there is a certain temperature, known as the melting temperature, in which the crystal changes its state while the members of the second category show a continuous change of state with temperature from one state to another.

The second principle behind Schrodinger’s prediction of the DNA structure is well supported by the algorithmic information theory. Unlike usual crystals which are formed by repeating the same pattern in three dimensions, again and again, the genetic molecule must be aperiodic to allow encoding the accommodate the information content needed to build a complex organism. It is indeed in such structure that every group of atoms (or even every single atom) can play a distinct role as Schrodinger explicates as

We believe a gene - or perhaps the whole chromosome fibre - to be an aperiodic solid.

which is famously known as aperiodic crystal in his theory of life. This is exactly what makes genes responsible for different attributes of a living organism.

Death = Equilibrium

What is the characteristic feature of life? One makes us, as living organisms, different from, e.g. an hourglass? It seems the key is in the concept of equilibrium. Non-living things tend to settle after a relatively short time in some steady (non-moving) state while living things “keep going” and exchange material and energy with their surrounding. This phenomenon is explained by the second law of thermodynamics which states all non-alive isolated systems tend towards a permanent state called “thermodynamical equilibrium” or “maximum entropy” in which no observable event occurs. Being alive can be seen as evading this equilibrium. It seems the living organisms evade this death state by exchanging things with their environment in a process known as metabolism. However, the material is not the essence of metabolism as there is no difference between the atoms within the organism and those outside its boundary. What actually matters is a concept called entropy, which despite being often introduced in various domains as a hazy concept, is a measurable physical quantity as a function of heat and temperature. We are particularly interested in the statistical notion of entropy established by Boltzmann and Gibbs expressed by

[\textrm{entropy} = k\log D]

where $k$ is the Boltzmann constant and $D$ is a measure of atomic disorder. Technical details aside, the second law of thermodynamics formulates the natural tendency of things to approach the chaotic state (maximum entropy).

Feeding on negative entropy

The theory of negative entropy states that living organisms evade the maximum entropy state by feeding on the orderliness (negative entropy) of their environment. This is, for example, seen in the higher animals whose food consists of complicated organic compounds (low entropy) which they return to the environment in a degraded form (higher entropy). Living organisms get rid of the surplus entropy via another mechanism which is giving heat to their surrounding. This heat must be compensated via the energy they receive from food and other sources, e.g. sunlight. Schrodinger goes even one step further and argues a parallelism between the intensity of life and the body temperature with the idea that animals with higher body temperature get rid of their entropy at a quicker rate and are consequently capable of intenser life which I would rather call

Dancing around the death equilibrium wit higher frequency!

Is Life based on the Laws of Physics?

All physical laws are statistical and the degree of uncertainty in their prediction increases with temperature. Statistical physics explains how such laws can emerge from the chaotic behavior of atoms (“Order from disorder”). The events that occur in the life cycle of an organism exhibit significant regularity caused by notable orderliness which is controlled by a relatively small fraction of its atoms within every cell of its body called genes (Order from order). The ability of the living organism in sucking orderliness from its surrounding seems to be connected with this aperiodic crystal, the chromosome molecule. The lifecycle of an organism cannot be explained by statistical physics where the laws of Physics emerge from the chaotic interaction of atoms not a well-ordered configuration of them. This phenomenon is known as the lack of individual determination which implies the faith of one molecule or atom is randomly determined as a result of Brownian-like events at atomic scales such as heat motion.

This is fundamentally different than the situation in biology where a single copy of the molecule (the DNA of the fertilized egg) gives rise to a sequence of orderly events throughout the development of the organism. Even though the DNA molecules are copied and distributed all over the body of higher animals, their total number, e.g. $10^14$ will in volume be less than a tiny drop of liquid. Considering every cell in an adult animal, the behavior of the cell seems to be governed by deterministic mechanisms not probabilistic ones even though it is governed by only a single copy of the DNA molecule.

Two aforementioned mechanisms that produce order correspond to two types of physical laws, Dynamical and Statistical which are connected in a short article by our beloved Max Planck, titled “Dynamische und Statistische Gesetzmassigkei”. Planck argues in this article that statistical laws that govern large-scale phenomena are controlled by dynamical laws governing small-scale events such as the interactions among single atoms and molecules.

Nernst’s Law

In the dichotomy of statistical and dynamical laws of Physics, the transition from one to another is possible under a certain condition determined by Quantum theory as absolute zero temperature. It is in this temperature that heat motion stops and ceases to have any effect on physical events. This fact was shown empirically by Walther Nernst’s famous “Heat Theorem” also known as the Third Law of Thermodynamics. Quantum theory rationalizes this empirical observation and also predicts that there is a non-zero temperature in which the heat motion effect is practically negligible on a physical system. This temperature can be even as high as the room temperature for systems such as a clock. The same situation holds for the aperiodic crystal by thinking of them as the cogs of a living organism. It is astonishing that unlike in a clock, these cogs are not human-made.

DNA as a Turing Machine

Let’s accept for now that genes are the programs that encode the development of the living organism which are physically implemented as a form of stable molecules. I emphasized the term development to give the impression that, in this view, genes are not only a list of attributes. They are recipes for building the organism from scratch. One could see the gene as a program that contains the code to produce an attribute and also the instruction on how to run that code. Schrodinger’s states more or less the same view:

What we wish to illustrate is simply that with the molecular picture of the gene it is no longer inconceivable that the miniature code should precisely correspond with a highly complicated and specified plan of development and should somehow contain the means to put it into operation.

This suggests an interesting view to see the genetic molecule as a Turing Machine (TM), the abstract model for general computing devices. The big genetic molecule serves as both the program to run and also the Turing machine to run it. As stated, the Turing machine itself can also be encoded in the program together with the code it’s supposed to run. One could see this as bundling the C++ compiler together with the C programs that is supposed to run on a computer. However, a universal machine is still needed to run this bundle. That’s where the concept of the Universal Turing Machine (UTM) comes into play; a machine that is capable of running every other Turing Machine. With some imprecision, one could see the laws of physics (at least in the spatial context of this planet/universe) as the UTM that runs every TM including the special one of our interest, i.e. genes.

Genes as Turing Machines which are run by the laws of Physics (a Universal Turing Machine) to produce a living organism.