A New Kind of Law Beyond Entailing Law: The Ensemble Approach : 13.7: Cosmos And Culture The reductionist view of the universe is inadequate. We live beyond the watershed of life and beyond entailing law.
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A New Kind of Law Beyond Entailing Law: The Ensemble Approach

Two posts ago, in "The End Of A Physics Worldview: Heraclitus And The Watershed of Life," I agued, I believe correctly, that no law entails the detailed evolution of the biosphere, or human life. At stake is our entire framework of understanding the universe, for reductionism claims that there shall be a "theory of everything" that entails all that happens in the universe, evolution, life, social systems, and history.

If I, with Giuseppe Longo, am right, the reductionist view is inadequate. We live beyond the watershed of life and beyond entailing law.

In this post I wish to speak to a new issue: If life is beyond entailing law, what forms of law can we have where life is concerned, if any? I believe we can. One approach is based on what I call the "ensemble approach."

I begin with the insights of Longo and Bailly in Mathematics and the Natural Sciences: The Physical Singularity of Life (Imperial College Press, 2011). Longo and Bailly establish convincingly that the modern foundations of physics, classical and quantum, have the following two essential properties: i. The phase space of the system in question, i.e., the space of its "possible behaviors," is pre-stated and pre-fixed. ii. The behavior of the system is, however, always a unique trajectory that optimizes some criterion. For a ball rolling down a set of interconnected bowls, the optimum is a path that minimizes an action, the principle of least action. In general terms, they claim, the actual trajectory follows a shortest path or geodesic on some surface, or manifold which may be curved.

Bailly and Longo, and I in Reinventing the Sacred, believe that the phase space of life is ever changing and unprestatable. Life follows myriad pathways, not geodesics, without entailing law.

The Ensemble Approach And A Form Of Self Organization

The ensemble approach may prove useful. I will give four examples where it has been applied: genetic regulatory networks, the origin of life, that statistical features of rugged fitness landscapes, and in physics, spin glasses. I discuss the first in detail.

First, as a young man I was thinking about cell differentiation. How could different cells in you, all having the same genes, be different: liver, kidney etc.?

It was known that in different cells types, different genes were active making specific and different sets of proteins. In 1961 and 1963, French microbiologists, F. Jacob and J. Monod, cracked the problem when they showed that, in E. coli bacteria, one gene could make a protein, say A, that bound to a DNA region next to another gene, say the B gene, and turn on or turn off the B gene's formation of its own B protein. In a seminal 1963 paper they argued that if two genes, A and B, each repressed, or turned off, the other gene, the little two gene circuit had two dynamical steady states: A on B off; A off B on. Hence they said, the same set of genes could express different proteins corresponding to two cell types.

Their work led to the central question of systems biology: What is the genetic regulatory network among 23,000 human genes, of which about 2,200 regulate one another's activities and regulate the rest of the 23,000?

Here we need to know which genes regulate which genes, and by what rules. Then we need to integrate the equations of motion of such a network to discover its integrated behavior.

Just as Newton's laws for billiard balls yield, upon integration and given initial and boundary conditions, the trajectory of the balls, for a classical physics genetic network, the behavior of the system has a trajectory from each initial state. These typically end up on small subsets of states, called attractors, each of which drains a basin of attraction. Cell types probably correspond to attractors, differentiation to noise or signal induced flow among attractors.

Here is the ensemble approach: I wondered if natural selection had to struggle to create very specific, engineered networks to achieve controlled differentiation from the fertilized egg, or zygote, called ontogeny, or, I hoped, some broad class, or ensemble of networks would all have good enough dynamical behavior to underlie ontogeny with just some tuning by natural selection.

To ask this question I idealized the behavior of a gene as an on-off device, a light bulb, and asked was there a class of large genetic networks that yielded orderly behavior. To ask this question is inherently to take the ensemble approach. It asks whether there are typical (i.e., generic) behaviors in different classes or ensembles of networks.

In my case I imagined N genes, each with K inputs. There are vastly many networks, an entire ensemble of networks, with N = 23,000 genes, and K = say, 2 inputs per gene. To study the typical properties of this ensemble, one approach is to sample at random from this ensemble. Thus, I chose the K = 2 inputs to each gene at random from among the N, and for each I assigned at random one of the 16 possible logical, or Boolean, functions prescribing the behavior of the regulated gene at the next-time moment, given the on or off states of its two inputs at the current moment. The and function is such a Boolean function. It says the regulated gene will be on at the next moment only if both its inputs are on at the present moment.

To summarize many years of work by many on Random Boolean Networks, it turns out that they behave in three regimes: ordered, chaotic, and a critical edge of chaos regime which is a phase transition between order and chaos. K = 2 networks turn out to be critical for the ensemble of networks with randomly chosen Boolean functions. Critical networks can have other values of K greater than 2 by using non-random choices of Boolean function of K inputs.

Now three essential facts: i. critical and ordered networks exhibit very ordered, and also multiple, attractors, hence the generic behaviors of these networks exhibits a new form of self organization — generic order in an ensemble of systems. These ordered attractors are so ordered that the different attractors could explain the order of the different cell types in an organism. ii. It is becoming clear that differentiated cell types are almost certainly attractors. iii. More amazingly, cells appear to be critical, to live on the edge of chaos.

Note two essential features of the ensemble approach: i. There is a vast ensemble of NK Random Boolean Networks, or more realistic models of genetic networks, all of which are dynamically critical. In short, criticality is a feature of an entire ensemble of networks, not just of one. ii. Critical networks are a subset of all Random Boolean Networks, those at the edge of chaos. If cells are critical, natural selection must hold networks at the edge of chaos for adaptive reasons — here is the mixture of ensemble self-organization and natural selection.

The Ensemble Approach Can Yield Statistical Laws Beyond Entailing Laws

As noted at the start of this post, no law entails the detailed evolution of the biosphere, including the evolution of genetic regulatory networks. This means we cannot deduce ab initio what those networks are. But the ensemble approach allows statistical laws about the typical features and behaviors of the entire ensemble of critical networks.

More profoundly evolution does not follow geodesics. Thus evolution is not entailed. It follows myriad pathways mixing quantum random indeterminate mutations and non-random natural selection. The ensemble approach is the natural way to seek statistical laws about the behaviors of genetic regulatory networks, without needing to know the details of any specific genetic regulatory network. As we learn more about real networks, we can refine the specifications of the ensemble, hence the generic behaviors of the refined ensemble, for better statistical predictions.

In short, the ensemble approach marries to the lack of entailing law for evolution, to yield a viable approach to statistical laws beyond entailing law.

Finally, the ensemble approach has been used by physicists with respect to spin glasses, and by me and others with respect to models of the generic origin of self-reproducing, collectively autocatalytic sets of organic molecules, where it appears that there is a phase transition in complex reaction networks to molecular self reproduction.

I hope life is generic.

The ensemble approach has also been used by me and others to study the typical statistical properties of tunably rugged fitness (or payoff) landscapes, of use in biology, and even economics, (See my Origins of Order or At Home in the Universe, Oxford University Press 1993, 1995).