## 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.

## Reply to Recent NYT Article: “God, Darwin, and My College Biology Class”

I recently came across the article/op-ed in the NYTimes, titled, “God, Darwin, and My College Biology Class”, http://www.nytimes.com/2014/09/28/opinion/sunday/god-darwin-and-my-college-biology-class.html?_r=0

In this article, it is stated: “Since Darwin, however, we have come to understand that an entirely natural and undirected process, namely random variation plus natural selection, contains all that is needed to generate extraordinary levels of non-randomness. Living things are indeed wonderfully complex, but altogether within the range of a statistically powerful, entirely mechanical phenomenon.”

When it is stated that “contains all that is needed to generate extraordinary levels of non-randomness” is factually not accurate, for one is making the mistake that many reductionists make by assuming that all complexity arises from bottom-up causation alone, and are completely ignoring the effects of top-down causation. The reason is as follows: Lower levels of complexity are necessarily governed by uncertainties due to quantum mechanics, it is not clear how these quantum uncertainties transition to a classical state. Mathematically, these uncertainties that are at the heart of the random variation that is cited are governed in a Hilbert, L^2 Lebesgue integrable space. Classical systems, determined by phase space manifolds have these probabilistic domains in the cotangent bundle of the manifold. The article is essentially saying that the cotangent bundle determines the phase space and not the other way around which is not correct. Further, there remains the unsolved issue of how the quantum fluctuations become classical (unless you follow the untestable many worlds route, which has major problems – see S D Hsu Modern Physics Letters A27: 1230114 (2012) for one interesting comment, and the writings of Sudarsky.

On the other side, top-down causation via cosmology and Einstein’s equations seed the correct conditions for dynamical Darwinian evolution to take place to begin with, for some reason, the author completely leaves this out.

Without a question, the author is an expert an evolutionary biology, but I am afraid he has looked through these issues through a very narrow lens, which does not do the issue a full and complete justification, and indeed, is responsible for much of the discomfort with evolutionary theory that is described so accurately and well in the article.

In the mean time, I would humbly suggest that the interested reader look at the following articles which describe Darwinian evolution in a more complete context as a function of emergence and complexity through physics which underlies biology. Also, one should see the work of Denis Noble, http://musicoflife.co.uk who advocates for a dynamical systems-based view of biological systems, which I personally believe to be correct, as it is much more mathematically and physically sound compared to standard evolutionary theory. This YouTube video of a lecture from the noted cosmologist GFR Ellis also sums up the problem with the reductionist view of evolutionary biology: http://youtu.be/nEhTkF3eG8Q

Title:
Laws, Causation and Dynamics at Different Levels
Authors:
Butterfield, Jeremy
Publication:
eprint arXiv:1406.4732
Publication Date:
06/2014
Origin:
ARXIV
Keywords:
Physics – History and Philosophy of Physics, Physics – Popular Physics
Comment:
29 pages, 3 figures; Interface Focus (Royal Society London), volume 2, 2012, pp. 101-114; doi:10.1098/rsfs.2011.0052
Bibliographic Code:
2014arXiv1406.4732B
Abstract

I have two main aims. The first is general, and more philosophical (Section 2). The second is specific, and more closely related to physics (Sections 3 and 4). The first aim is to state my general views about laws and causation at different ‘levels’. The main task is to understand how the higher levels sustain notions of law and causation that ‘ride free’ of reductions to the lower level or levels. I endeavour to relate my views to those of other symposiasts. The second aim is to give a framework for describing dynamics at different levels, emphasising how the various levels’ dynamics can mesh or fail to mesh. This framework is essentially that of elementary dynamical systems theory. The main idea will be, for simplicity, to work with just two levels, dubbed ‘micro’ and ‘macro’ which are related by coarse-graining. I use this framework to describe, in part, the first four of Ellis’ five types of top-down causation.

Title:
The arrow of time and the nature of spacetime
Authors:
Ellis, George F R
Publication:
eprint arXiv:1302.7291
Publication Date:
02/2013
Origin:
ARXIV
Keywords:
General Relativity and Quantum Cosmology, Physics – History and Philosophy of Physics
Comment:
56 pages, 7 figures
Bibliographic Code:
2013arXiv1302.7291E
Abstract

This paper extends the work of a previous paper [arXiv:1208.2611] on the flow of time, to consider the origin of the arrow of time. It proposes that a past condition’ cascades down from cosmological to micro scales, being realized in many microstructures and setting the arrow of time at the quantum level by top-down causation. This physics arrow of time then propagates up, through underlying emergence of higher level structures, to geology, astronomy, engineering, and biology. The appropriate space-time picture to view all this is an emergent block universe (EBU’), that recognizes the way the present is different from both the past and the future. This essential difference is the ultimate reason the arrow of time has to be the way it is.

Title:
Recognising Top-Down Causation
Authors:
Ellis, George F R
Publication:
eprint arXiv:1212.2275
Publication Date:
12/2012
Origin:
ARXIV
Keywords:
Physics – Classical Physics, Nonlinear Sciences – Adaptation and Self-Organizing Systems, Physics – History and Philosophy of Physics
Comment:
11 pages, 2 figures, 2 tables. 2nd prize in FQXI essay competition
Bibliographic Code:
2012arXiv1212.2275E
Abstract

One of the basic assumptions implicit in the way physics is usually done is that all causation flows in a bottom up fashion, from micro to macro scales. However this is wrong in many cases in biology, and in particular in the way the brain functions. Here I make the case that it is also wrong in the case of digital computers – the paradigm of mechanistic algorithmic causation – and in many cases in physics, ranging from the origin of the arrow of time to the process of state vector preparation. I consider some examples from classical physics, as well as the case of digital computers, and then explain why this is possible without contradicting the causal powers of the underlying microphysics. Understanding the emergence of genuine complexity out of the underlying physics depends on recognising this kind of causation.

Title:
Evolutionary Transitions and Top-Down Causation
Authors:
Imari Walker, Sara; Cisneros, Luis; Davies, Paul C. W.
Publication:
eprint arXiv:1207.4808
Publication Date:
07/2012
Origin:
ARXIV
Keywords:
Nonlinear Sciences – Adaptation and Self-Organizing Systems, Quantitative Biology – Other Quantitative Biology
Comment:
8 pages, 4 figures; Proceedings of Artificial Life XIII (2012) p. 283-290
Bibliographic Code:
2012arXiv1207.4808I
Abstract

Top-down causation has been suggested to occur at all scales of biological organization as a mechanism for explaining the hierarchy of structure and causation in living systems. Here we propose that a transition from bottom-up to top-down causation — mediated by a reversal in the flow of information from lower to higher levels of organization, to that from higher to lower levels of organization — is a driving force for most major evolutionary transitions. We suggest that many major evolutionary transitions might therefore be marked by a transition in causal structure. We use logistic growth as a toy model for demonstrating how such a transition can drive the emergence of collective behavior in replicative systems. We then outline how this scenario may have played out in those major evolutionary transitions in which new, higher levels of organization emerged, and propose possible methods via which our hypothesis might be tested.

Title:
On the limits of quantum theory: Contextuality and the quantum-classical cut
Authors:
Ellis, George F. R.
Affiliation:
AA(Mathematics Department, University of Cape Town, South Africa)
Publication:
Annals of Physics, Volume 327, Issue 7, p. 1890-1932.
Publication Date:
07/2012
Origin:
ELSEVIER
(c) 2012 Elsevier Inc.
DOI:
10.1016/j.aop.2012.05.002
Bibliographic Code:
2012AnPhy.327.1890E
Abstract

This paper is based on four assumptions: 1. Physical reality is made of linearly behaving components combined in non-linear ways. 2. Higher level behaviour emerges from this lower level structure. 3. The way the lower level elements behaves depends on the context in which they are embedded. 4. Quantum theory applies to the lower level entities. An implication is that higher level effective laws, based on the outcomes of non-linear combinations of lower level linear interactions, will generically not be unitary; hence the applicability of quantum theory at higher levels is strictly limited. This leads to the view that both state vector preparation and the quantum measurement process are crucially based on top-down causal effects, and helps provide criteria for the Heisenberg cut that challenge some views on Schrödinger’s cat.

Title:
Top-Down Causation and Autonomy in Complex Systems
Authors:
Juarrero, Alicia
Affiliation:
AA(Emeritus, Prince George’s Community College)
Publication:
Downward Causation and the Neurobiology of Free Will, Understanding Complex Systems. ISBN 978-3-642-03204-2. Springer Berlin Heidelberg, 2009, p. 83
Publication Date:
00/2009
Origin:
SPRINGER
Keywords:
Physics
(c) 2009: Springer Berlin Heidelberg
DOI:
10.1007/978-3-642-03205-9_5
Bibliographic Code:
2009dcnf.book…83J
Abstract

Evolutionary evidence shows that complex dynamical systems become increasingly self-directed and decoupled from merely energetic forces over time. In this paper I analyze these transformations, concentrating on changes in the type of top-down causation that characterizes such self-organized and autopoietic pro cesses. Specifically, I show that the top-down selection criteria of these systems makes some of them autonomous, and that because once evolution reaches humans the criteria according to which voluntary actions are selected are semantic and symbolic – and can be self-consciously chosen – human self-direction constitutes a form of strong autonomy that can arguably be considered “free will.”

Title:
Top-Down Causation and the Human Brain
Authors:
Ellis, George F. R.
Affiliation:
AA(Mathematics Department, University of Cape Town)
Publication:
Downward Causation and the Neurobiology of Free Will, Understanding Complex Systems. ISBN 978-3-642-03204-2. Springer Berlin Heidelberg, 2009, p. 63
Publication Date:
00/2009
Origin:
SPRINGER
Keywords:
Physics
(c) 2009: Springer Berlin Heidelberg
DOI:
10.1007/978-3-642-03205-9_4
Bibliographic Code:
2009dcnf.book…63E
Abstract

A reliable understanding of the nature of causation is the core feature of science. In this paper the concept of top-down causation in the hierarchy of structure and causation is examined in depth. Five different classes of top-down causation are identified and illustrated with real-world examples. They are (1) al gorithmic top-down causation; (2) top-down causation via nonadaptive information control; (3) top-down causation via adaptive selection; (4) top-down causation via adaptive information control; and (5) intelligent top-down causation (i.e., the effect of the human mind on the physical world). Recognizing these forms of causation implies that other kinds of causes than physical and chemical interactions are effective in the real world. Because of the existence of random processes at the bottom, there is sufficient causal slack at the physical level to allow all these kinds of causation to occur without violation of physical causation. That they do indeed occur is indicated by many kinds of evidence. Each such kind of causation takes place in particular in the human brain, as is indicated by specific examples.

Title:
Top-Down Causation by Information Control: From a Philosophical Problem to a Scientific Research Program
Authors:
Auletta, G.; Ellis, G. F. R.; Jaeger, L.
Publication:
eprint arXiv:0710.4235
Publication Date:
10/2007
Origin:
ARXIV
Keywords:
Quantitative Biology – Other Quantitative Biology
Comment:
Revised version to meet referee’s comments, and responding to a paper by Wegscheid et al that was not mentioned in the previous version. 23 pages, 9 figures
Bibliographic Code:
2007arXiv0710.4235A
Abstract

It has been claimed that different types of causes must be considered in biological systems, including top-down as well as same-level and bottom-up causation, thus enabling the top levels to be causally efficacious in their own right. To clarify this issue, important distinctions between information and signs are introduced here and the concepts of information control and functional equivalence classes in those systems are rigorously defined and used to characterise when top down causation by feedback control happens, in a way that is testable. The causally significant elements we consider are equivalence classes of lower level processes, realised in biological systems through different operations having the same outcome within the context of information control and networks.

Title:
Physics and the Real World
Authors:
Ellis, George F. R.
Affiliation:
AA(Mathematics Department, University of Cape Town)
Publication:
Foundations of Physics, Volume 36, Issue 2, pp.227-262
Publication Date:
02/2006
Origin:
CROSSREF; SPRINGER
Keywords:
Physics, emergence, causality
DOI:
10.1007/s10701-005-9016-x
Bibliographic Code:
2006FoPh…36..227E
Abstract

Physics and chemistry underlie the nature of all the world around us, including human brains. Consequently some suggest that in causal terms, physics is all there is. However, we live in an environment dominated by objects embodying the outcomes of intentional design (buildings, computers, teaspoons). The present day subject of physics has nothing to say about the intentionality resulting in existence of such objects, even though this intentionality is clearly causally effective. This paper examines the claim that the underlying physics uniquely causally determines what happens, even though we cannot predict the outcome. It suggests that what occurs is the contextual emergence of complexity: the higher levels in the hierarchy of complexity have autonomous causal powers, functionally independent of lower level processes. This is possible because top-down causation takes place as well as bottom-up action, with higher level contexts determining the outcome of lower level functioning and even modifying the nature of lower level constituents. Stored information plays a key role, resulting in non-linear dynamics that is non-local in space and time. Brain functioning is causally affected by abstractions such as the value of money and the theory of the laser. These are realised as brain states in individuals, but are not equivalent to them. Consequently physics per se cannot causally determine the outcome of human creativity, rather it creates the possibility space allowing human intelligence to function autonomously. The challenge to physics is to develop a realistic description of causality in truly complex hierarchical structures, with top-down causation and memory effects allowing autonomous higher levels of order to emerge with genuine causal powers.

Title:
Emergence and Dissolvence in the Self-organisation of Complex Systems
Authors:
Testa, Bernard; Kier, Lemont B.
Publication:
Entropy, vol. 2, Issue 1, p.1-25
Publication Date:
03/2000
Origin: