Degrading a Gradient

If physico-chemical processes can be summarized by the dictum “nature abhors a gradient”, how then is that gradient reduced or degraded?

 

I've been puzzling over this, and have sketched out a preliminary picture, but I'd like to get into the weeds. Let’s begin with a cliff. A ball falling over the edge of the cliff goes down its (potential energy) gradient (A). When it hits the bottom, equilibrium has been reached. Nothing else happens. There is no longer any gradient where the ball is concerned.

 

Let’s now imagine a waterfall with the same cliff. As long as there is water (perhaps from rain) at the top, it will keep falling down along its gradient to the bottom. If it’s a hard cliff that cannot be altered by the flowing water, this picture remains unchanged, as long as the water keeps flowing (B). This is a steady state situation. The potential energy of the water continues to be dissipated as it falls. It might be possible to extract work from this continuous falling water. Unlike the single ball that has hit rock bottom.

 

If it’s not a hard cliff, but consists of softer dirt, the cliff can be eroded. The extent of erosion would depend on both the force of the flowing water and how easily the dirt can be moved in little bits and pieces. This could reshape the top of the cliff, but also the bottom of the cliff as slit accumulates (C). The flow has now changed, albeit very slightly. If the flow is of very low force, but continuously removes dirt from the top and deposits it at the bottom, the cliff might eventually become a slope (D) and perhaps reduce in steepness (E), although the overall gradient for water flow (from top to bottom) remains the same. Eventually when you run out of cliff, that overall gradient may finally begin to reduce.

 

But what if you had a variable flow? You might imagine a different situation where silt is carried some distance but then starts to build up a small mound (F). This creates a small pond at the bottom of the waterfall pent up by a natural dam. The pond may overflow when water is abundant. The natural dam may break, depending on how sturdy it is and how well it can hold together against the force of water. The water may find some other way out beyond the single dimension represented by these simple figures. Imagine a mountain lake constantly fed by rains, its water finding its way down through many streams and rivers, the flows changing, and their paths evolving over time as natural dams are built up and broken down. The landscape might look quite complicated as nature finds its way to degrade the gradient.

 

On planet Earth, there are three source gradients that could power the evolution of complex structures that we call life. These are (1) the sun (providing both a thermal gradient and energy in the form of photons), (2) the gradient in chemical redox potential between the minerals of the lithosphere and the hydrosphere/atmosphere, and (3) geothermal energy from radioactivity. I’ve also ranked them by importance in extant life today. Most of life today essentially harnesses its energy from the sun. But deep down in hydrothermal vents, other (albeit) weaker sources of energy can power life.

 

How would we map chemistry on this landscape? Kinetically stable compounds are pools that can build up until they overflow their banks. Increasing energy is akin to increasing the flow of water. “Low barrier” pools with low banks are easily overflowed as energy increases. “High barrier” steep banks are less easily overcome, but a weakness in the bank may cause a breach may result in an alternate pathway for water to escape – this is the role played by chemical catalysts that lower the energy barrier. A high enough flow can result in water sometimes traveling uphill, but only if there’s an inexorable downhill flow at a later point. The need to reduce the thermodynamic gradient reigns supreme. Our vision is limited by our discomfort imagining higher dimensions, but this is what we need to imagine in a truly complex chemical landscape. I don’t know how to describe the picture, nor am I sure I can grasp it in my little feeble brain.

 

Let’s therefore return to one or two dimensions to explore if there are other conceptual gains in our simple picture. In 1900, Henri Benard “created” hexagonal cell-like structures by heating sperm whale oil; order arising from disorder spontaneously.  Intriguingly, in a paper titled “Life as a Manifestation of the Second Law of Thermodynamics” (Math. Comput. Modelling, 1994, 19, 25-48), Schneider and Kay present a scheme analyzing these Benard cells that looks similar to the waterfall cliffs (B, D, C, respectively). In (a), Q represents heat flow in an “ideal” isothermal system. In these experiments, initially heat flows by conduction and shows a linear slope across the temperature gradient in (b). At a critical point, conduction transitions to convection as shown in (c).

 

While the zig-zag flow across the gradient represents a “longer” path compared to the straight-line slope, it is more efficient at dissipating heat. Below is a plot showing both the heat dissipation rate and entropy production along the gradient, emphasizing the shift from conduction to convection. This critical point is where order seemingly arises from disorder. Increase the heat flow further eventually the hexagonal structures collapse once again into chaos.

 

I’ve been puzzling over the meaning of “length” and “time”, which in scientific terms are two independent and fundamental variables. But when we think about energy flow in non-equilibrium thermodynamics, the two categories might get fuzzy. For one thing, I recently saw a similar picture proposed for the evolution of “global structure of metabolic pathways” using a toy model (Abedpour, N.; Kollman, M. BMC Syst. Biol. 2015, 9:88). Their analogy is that there’s a trade-off between metabolic “path length” and “maintenance costs” and depending on the particular situation, you may observe the switch.

 

And I happen to simultaneously be reading an academic history book (The Art of Not Being Governed by James C. Scott) about the various “tribal” peoples in the highlands that border Burma, Thailand, Vietnam, Laos, and China. When asked how far it is from location A to location B, the locals don’t tell you the distance in length units, but rather in time units. Because that’s what counts. And therefore that’s what is counted. Fascinatingly, useful politico-economic history “boundaries” can be drawn when one makes map based on time-travel rather than distance-travel. What seems like a longer distance to us non-locals unfamiliar with the local terrain turns out to be the practical “shortest” time. The simple one-dimensional pictures I’ve drawn are misleading in a sense. In a multi-dimensional landscape, the most efficient way to degrade a gradient is the one that life figures out evolutionarily. How it does so… ah, that’s the fun of research.