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The Greatest Story Never Told — Solved by Mathematical Pre-biology?

By John Brindley and Rowena Ball

In which we apply the power of mathematical modelling and stochastic dynamical systems to the origin of life enigma, and find that the dice on the primordial Earth were loaded in favour of life.

Life famously resembles the elephant of the ancient Indian parable; it is difficult to define or even describe, but unmistakeable when you see it. Life on Earth (currently the only example we have) is ubiquitous, persisting in nearly all environments on or near the planet’s surface, including extremely arduous conditions of temperature, pressure, salinity, pH, etc. To persist it must maintain a dynamic state far from thermodynamic equilibrium —through bewilderingly complex and beautiful, essentially autocatalytic processes within and through selectively permeable membranes. In other words, life’s (literally vital) characteristic is its ability to locally defy the apparently relentless progress towards higher entropy exhibited by the physical universe, dictated by the second law of thermodynamics. But how did life begin? Despite vast efforts by many researchers, there is still no universally-accepted answer to the question of life’s original emergence from the non-living environment presumed to have existed on Earth in its early history.

We can trace all known life on Earth to a single last universal common ancestor (LUCA), suggestive of a lone origin (at some currently-unknown time and place) dependent on a DNA-RNA apparatus to carry out the processes of replication, information transfer, and development, which are the very essence of life itself. Recent research seeks to understand the pre-LUCA processes that transformed chemistry to biochemistry and biology, and attempts to describe the physical characteristics of the environment in which this transformation took place. There is broad agreement that this took place within the range of 3.8-4.2 billion years ago (bya), but much debate exists regarding the possible environment and processes that led to the emergence and dominance of incredibly complex RNA molecules and a pre- or proto-cellular “RNA World.” Recent reviews offer up-to-date critical surveys of current thinking and research in all of these areas [1-3].

RNA’s key property is its capacity for information storage and transmission through replication (life itself has been described as “information which copies itself”).  In contrast to DNA, which demands independent protein enzymes to catalyse its replication, RNA is able to act as its own catalytic enzyme. How a polymer of RNA’s complexity could have emerged, presumably via many sequential steps from appropriate monomers—which themselves were far removed from any “prebiotic soup” that might have existed in the (unknown) conditions of early Earth—remains a challenging question. Several putative answers are currently under active debate. 

Our own initial contribution to this debate [4] proposed a purely chemical provider for the cyclical ambient temperature desirable for effective replication of RNA, simultaneously delivering an energy source for the maintenance of the far-from-equilibrium state of all life forms (and presumably of pre-LUCA chemistry/biochemistry). Our model is that of a thermochemical oscillator, specifically the hydrogen peroxide-thiosulphate redox system (well-known and exploited in the chemical engineering world [5]) in a continuous stirred-tank reactor (CSTR) or chain of CSTRs. We conjecture that such an oscillator could have sourced its reagents in a flow through the porous structure of a “hydrothermal  vent,” a feature currently found near mid-oceanic ridges and likely to have been more widely present in the geological conditions of 4 bya [6], or even from a “natural nuclear reactor,” extinct examples of which have been studied [7].This reaction system also behaves as a pH oscillator, which is potentially of considerable importance for the RNA world; we named it the thiosulphate hydrogen peroxide (THP) oscillator, and discussed its influence in several recent papers [8-11]. Figure 1 shows the sustained temperature and pH oscillations produced by the THP redox reactions, where they drive replication of RNA.

Figure 1. pH and temperature cycling of the thiosulfate-hydrogen peroxide (THP) redox oscillator drive replication of XX’, double-stranded polyribonucleotides.

Hydrogen peroxide, whether biotically or non-biotically produced, is often regarded solely as a poison to current life forms — unjustly so, for there is mounting evidence of its multiple essential biological functions [12].

In our model of reacting flow through a hydrothermal vent matrix, the THP oscillations manifest as waves travelling at some speed relative to the mean flow; their speed relative to the matrix will in general be nonzero. As a typical monomer entering the matrix is carried by the flow, we envisage its experience in Figure 2. The monomer will expect to spend a varying time in each pore; during these intervals, transported and empowered by the oscillating fields of the travelling THP waves and aided by catalysed surface reactions, it will polymerize and ultimately have a presumably low probability of reaching a degree of complexity with the ability to replicate. The fluctuations in temperature and pH drive the process, which will be enhanced by the localised concentration of longer polymers — both by selective trapping in individual pores and accumulation near the stable and unstable manifolds of saddle points in the chaotic advection inherent to flows in porous media [13-14].

Figure 2. 2a. Sketch of the CSTRs-in-series setup, in which n units are coupled serially. THP reactants and monomers flow in at one end, while dotted outlines indicate porous walls through which THP reactants also enter the system in cross-flow. The cross-flow mean residence times may differ between units, which allows the model to account for the more general situation shown in 2b, where the pore volumes are nonuniform.

Central to our latest paper [11] is the demonstration of the (positive) effects of inevitable stochastic input fluctuations on biomolecules. With respect to an evolving, diversifying pre-biotic chemical system, temperature and pH fluctuations allow a more extensive exploration of parameter space than would a regular periodicity. Indeed, the extent and saturation of parameter space explored by an initial distribution of monomers, each of which has a unique space/time trajectory through the matrix, offers a reasonable possibility of even an extremely rare event. Interestingly, recent results on the response of a closed system near to equilibrium and subject to a Gaussian stochastic input [15] have rigorously shown that the output fluctuations are non-Gaussian but have a gamma-type probability distribution, which is antithetical to the origin and persistence of life. However, our model is a nonlinear, open system, operating far from equilibrium, and the output of a Gaussian input is neither Gaussian nor gamma (see Figure 3). It demonstrates negative skewness and a positive median; in other words; it is weighted towards the higher temperature perturbations, favouring high activation energy reactions over lower ones. A negatively-skewed, right-weighted distribution of temperature fluctuations, themselves forced by Gaussian noise, enable the precious, high-energy synthetic reactions that the prebiotic non-enzymatic world must access to polymerise, diversify, and replicate. Thus the “Goldilocks” distribution, in explicit form, effectively becomes “the fundamental equation of life!”

Figure 3. The contrast between gamma and “Goldilocks” output distributions is striking. “Goldilocks” is obtained as the Bezier-smoothed envelope of the output temperature fluctuation histogram (bottom figure).

Where next? Though our results are interesting and exciting, we are far from establishing a fully convincing story of the origin of life, which almost certainly involved many collaborative physical processes. The extreme difficulty, or even impossibility, of obtaining direct physical evidence of the first life on Earth means that the modeller—physical, numerical, or mathematical—has a virtually clean sheet for plausible hypotheses. But scientific progress demands predictions which are critically testable now, and this is the real challenge. We hope that we can persuade at least some of our readers to rise to it!

References
[1] Walker, S.I. (2017). Origins of life: a problem for physics, a key issues review. Reports on Progress in Physics, 80(9), 092601.
[2] Szostak, J.W. (2017). The Narrow Road to the Deep Past: In Search of the Chemistry of the Origin of Life. Angewandte Chemie, 56(37), 11037-11043.
[3] Sutherland, J.D. (2017). Opinion: Studies on the origin of life — the end of the beginning. Nature Reviews Chemistry, 1.
[4] Ball, R., & Brindley, J. (2014). Hydrogen peroxide thermochemical oscillator as driver for primordial RNA replication. Interface, 11(95). 
[5] Chang, M., & Schmitz, R.A. (1975). An experimental study of oscillatory states in a stirred reactor. Chemical Engineering Science, 30(1), 21-34. 
[6] Foustoukos, D.I., Houghton, J.L., Seyfried Jr., W.E., Sievert, S.M., & Cody, G.D. (2011). Kinetics of H2–O2–H2O redox equilibria and formation of metastable H2O2 under low temperature hydrothermal conditions. Geochimica et Cosmochimica Acta, 75(6), 1594-1607. 
[7] Adam, Z.R. (2015). Temperature oscillations near natural nuclear reactor cores and the potential for prebiotic oligomer synthesis. Origins of Life and Evolution of Biospheres, 46(2-3), 171-187. 
[8] Ball, R., & Brindley, J. (2015). The life story of hydrogen peroxide II: a periodic pH and thermochemical drive for the RNA world. Journal of the Royal Society Interface, 12(109).
[9] Ball, R., & Brindley, J. (2016). Thiosulfate-Hydrogen Peroxide Redox Oscillator as pH Driver for Ribozyme Activity in the RNA World. Origins of Life and Evolution of Biospheres, 46(1), 133-147. 
[10] Ball, R., & Brindley, J. (2016). The Life Story of Hydrogen Peroxide III: Chirality and Physical Effects at the Dawn of Life. Origins of Life and Evolution of Biospheres, 46(1), 81-93.
[11] Ball, R., & Brindley, J. (2017). Toy trains, loaded dice and the origin of life: dimerization on mineral surfaces under periodic drive with Gaussian inputs. Royal Society of Chemistry. 
[12] Sies, H. (2017). Hydrogen peroxide as a central redox signaling molecule in physiological oxidative stress: Oxidative eustress. Redox Biology, 11, 613-619.
[13] Tél, T., de Moura, A., Grebogi, C., & Károlyi, G. (2005). Chemical and biological activity in open flows: A dynamical system approach. Physics Reports, 413(2-3), 91-196.
[14] Lester, D.R., Metcalfe, G., & Trefry, M.G. (2013). Is Chaotic Advection Inherent to Porous Media Flow? Physical Review Letters, 111, 174101.
[15] Betrola, V., & Cafaro, E. (2015). Response of a thermodynamic system subject to stochastic thermal perturbations. Physics Letters A, 379(47-48), 3035-3036.


John Brindley (j.brindley@leeds.ac.uk) is from the School of Mathematics and Statistics at the University of Leeds, UK. Rowena Ball (Rowena.Ball@anu.edu.au) is from the Mathematical Sciences Institute and the Research School of Chemistry at the Australian National University. 

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