## A Series of Lectures on Fine-Tuning in Biology

A recent lecture and a series of interviews has been posted online where cosmologist George F.R. Ellis discusses the issue of fine-tuning in biology at considerable length and in considerable detail. Of course, the larger theme here is that to discuss and understand things like Darwinian evolution properly, one needs to have an understanding of the underlying physics, as it is laws of physics that allow life to emerge and for Darwinian evolution to occur in the first place. Here are the lectures:

## Mathematical Origins of Life

The purpose of this post is to demonstrate some very beautiful (I think!) mathematics that arises form Darwinian evolutionary theory. It is a real shame that most courses and discussions dealing with evolution never introduce any type of mathematical formalism which is very strange, since at the most fundamental levels, evolution must also be governed by quantum mechanics and electromagnetism, from which chemistry and biochemistry arise via top-down and bottom-up causation. See this article by George Ellis for more on the role of top-down causation in the universe and the hierarchy of physical matter. Indeed, my personal belief is that if some biologists and evolutionary biologists like Dawkins, Coyne, and others took the time to explain evolution with some modicum of mathematical formalism to properly describe the underlying mechanics instead of using it as an opportunity to attack religious people, the world would be a much better place, and the dialogue between science and religion would be much more smooth and intelligible.

In this post today, I will describe some formalism behind the phenomena of prebiotic evolution. It turns out that there has been a very good book by Claudius Gros and understanding evolution as a complex dynamical system (dynamical systems theory is my main area of research), and the interested reader should check out his book for more details on what follows below.

We can for simplicity consider a quasispecies as a system of macromolecules that have the ability to carry information, and consider the dynamics of the concentrations of the constituent molecules as the following dynamical system:

$\boxed{\dot{x}_{i} = W_{ii}x_{i} + \sum_{j \neq i}W_{ij}x_{j} - x_{i} \phi(t)}$,

where $x_{i}$ are the concentrations of $N$ molecules, $W_{ii}$ is the autocatalytic self-replication rate, and $W_{ij}$ are mutation rates.

From this, we can consider the following catalytic reaction equations:

$\boxed{\dot{x}_i = x_{i} \left(\lambda_{i} + k_i^j x_j - \phi \right)}$,

$\boxed{\phi = x^{k}\left(\lambda_{k} + \kappa_k^j x_j\right) }$,

$x_i$ are the concentrations, $\lambda_i$ are the autocatalytic growth rates, and $\kappa_{ij}$ are the transmolecular catalytic rates. We choose $\phi$ such that

$\boxed{\dot{C} = \sum_i \dot{x}_i = \sum_i x_i \left(\lambda_i + \sum_j \kappa_{ij}x_{j} \right) - C \phi = (1-C)\phi}$.

Clearly:

$\lim_{C \to 1} (1-C)\phi = 0$,

that is, this quick calculation shows that the total concentration $C$ remains constant.

Let us consider now the case of homogeneous interactions such that

$\kappa_{i \neq j} = \kappa$, $\kappa_{ii} = 0$, $\lambda_i = \alpha i$,

$\boxed{\dot{x}_{i} = x_{i} \left(\lambda_i + \kappa \sum_{j \neq i} x_{j} - \phi \right)}$,

which becomes

$\boxed{\dot{x}_i = x_i \left(\lambda_i + \kappa - \kappa x_i - \phi\right)}$.

This is a one-dimensional ODE with the following invariant submanifolds:

$\boxed{x_{i}^* = \frac{\lambda_i + \kappa - \phi}{\kappa}}$,

$\boxed{x_i^* = 0, \quad \lambda_i = N \alpha}$.

With homogeneous interactions, the concentrations with the largest growth rates will dominate, so there exists a $N^*$ such that $1 \leq N^* \leq N$ where

$\boxed{x_i^* = \frac{\lambda_i + \kappa - \phi}{\kappa}, \quad N^* \leq i \leq N}$,

$\boxed{0, \quad 1 \leq i < N^*}$.

The quantities $N^*$ and $\phi$ are determined via normalization conditions that give us a system of equations:

$\boxed{1 = \frac{\alpha}{2\kappa} \left[N(N+1) - N^*(N^* - 1)\right] + \left[\frac{\kappa - \phi}{\kappa}\right] \left(N + 1 - N^*\right)}$,

$\boxed{0 = \frac{\lambda_{N^*-1} + \kappa - \phi}{\kappa} = \frac{\alpha(N^* - 1)}{\kappa} + \frac{\kappa - \phi}{\kappa} }$.

For large $N, N^*$, we obtain the approximation

$\boxed{N - N^* \approx \sqrt{\frac{2 \kappa}{\alpha}}}$,

which is the number of surviving species.

Clearly, this is non-zero for a finite catalytic rate $\kappa$. This shows the formation of a hypercycle of molecules/quasispecies.

These computations clearly should be taken with a grain of salt. As pointed out in several sources, hypercycles describe closed systems, but, life exists in an open system driven by an energy flux. But, the interesting thing is, despite this, the very last calculation shows that there is clear division between molecules $i = N^*, \ldots N$ which can be considered as a type of primordial life-form separated by these molecules belonging to the environment.