Is There A Maximum Power Efficiency Principle In Cells?

In this blog I wish to explore a new issue: Might there be a maximum power efficiency per unit fuel in a growing biological cell?

The answer may be yes. It appears to be a new idea.

Cells are non-equilibrium open thermodynamic systems, taking in food, excreting waste, and typically dividing, as in a colony of the bacterium E. coli.

We lack an optimization principle for such growth.  Indeed, we have two famous steps toward a thermodynamics of non-equilibrium processes. The physicist Lars Onsager formulated his famous linear theory of flows and forces for systems just slightly displaced from thermodynamic systems and open to their worlds. It has been very useful and Onsager won the Nobel Prize for his work.

More recently, Ilya Prigogine has explored what he calls dissipative “structures,” such that chemical systems like the Bielosov - Zhabatinzki reaction are able to oscillated or in a petri plate, send out concentric circles of oxidation reduction oscillations, or even form beautiful spiral waves, with implications for cardiac arythmias such as sudden cardiac death due to such spiral waves of activities in the ventricle.

Prigogine also won a Nobel Prize.  But he attempted to show that such dissipative structures maximize entropy production, a theorem disproven by others.

We lack a maximization principle for “how far from equilibrium” a system “ought” to be by some success criterion.

In this blog I will tell you about recent work with my Finnish colleagues, Tommi Aho and Olli yli-Harja at the Tampere University of Technology that hints at such a maximization principle: Maximize the power efficiency per unit fuel.  Cells may have evolved to achieve this.  In the next blog, I will take initial steps to tie this to a second order phase transition in water flowing down an incline plane, where the “food” is the gravitational potential.  For abiotic systems no “selection principle” yet seems available to achieve such systems, which, as we will see, are poised at a transition between order and chaos.

How Fast Should You Drive Your Car to Maximize “Miles per Gallon?"

 

Consider driving your car at 1 mile per hour, mph, 47 mph, or 2046 mph. Roughly where will you maximize miles per gallon?  Well at about 47 miles per hour.

Now in physics, work is force acting through a distance, for example accelerating a hockey puck with a hockey stick.  Power is “work per unit time”.  So “miles per hour” is a power measure, since work is done against friction of various sorts to maintain the speed of the car.  Then miles per gallon is a measure of a power efficiency per unit fuel, and as we guessed, is maximized at about 47 miles per hour.

What is the analogue for growing and dividing biological cells?  Well, it is cell division rate - a power term, divided by metabolic rate, the use of metabolic fuel.

Equivalently we can think of the above as “biomass production rate divided by glucose utilization rate in a very simple example.

Tommi, Olli and I have examined a “stoichiometric model of E. coli metabolism to relate biomass production rate per unit glucose utilization.

As I will show you in a moment, we find a maximum power efficiency per unit fuel.

But first I have to tell you about two types of enzymes catalyzing chemical reactions in metabolism.  The first are called “Michaelis-Menton kinetiics” enzymes.  Picture an X Y plot, on the X axis put down varying concentrations of substrates to the reaction, but a fixed concentration of the enzyme, called, “Z”. Now on the Y axis plot the rate of production of product, say little x converts to little y for fun, so plot dy/dt.  Then one obtains a curve in the X Y plot above showing for each concentration of substrate x, how fast product y is produced, dy/dt.  For Michaelis Menton enzymes, the plot goes up steeply at first as x increases, then slowly bends over to become horizontal.  That is, the second derivative of the rate is positive, so the curve monotonically gets flatter.

Why?

Simple, think of Z as a monomeric enzyme, one protein.  It acts to convert substrate, x, to y.  To do so, substrate x binds to the enzyme Z and is transformed to y.  At first as the concentration of substrate increases, the “saturation” of the enzyme X increases as fast, so the production of y, dy/dt, increases linearly as x increases in concentration. But eventually, there is so much substrate x around that the poor Z enzyme is fully saturated, or bound by x all the time, so the rate of conversion of x to y, dy/dt, cannot increase. That is why the curve dy/dt starts steep then flattens out as the concentration of substrate x increases.

OK, but there are a critical class of enzymes called “positively cooperative”. Hemoglobin, while not an enzyme, is an example. Four homoglobin proteins bind to one another in a “tetramer”.  Here is the trick.  Binding the first oxygen, which hemoglobin in your blood transports, to the FIRST of the four hemoglobin molecules in the tetramer is “hard” or a slow process.  BUT ONCE THE FIRST OXYGEN IS BOUND TO THE FIRST HEMOGLOBIN PROTEIN, IT IS EASIER FOR A SECOND OXYGEN TO BIND TO A SECOND HEMOGLOBIN PROTEIN. Then it is even easier for a third than a fourth  oxygen molecule to bind to the third and fourth hemoglobin molecule.  As oxygen concentration goes up and up, at first the oxygen per hemoglobin tetramer increases slowly, the increases faster and faster bending the curve of oxygen bound versus oxygen concentration, upwards.  So it is NOT a straight line, the line bends upwards. But eventually, all four hemoglobins have bound oxygen and further increases in oxygen concentration do NOT increase the fraction of the hemoglobin tetramer bound by oxygen. Thus, as oxygen concentration increases, we get a SIGMOIDAL binding curve. As oxygen increases, at first there is little hemoglobin bound oxygen, then the curve bends upward, until all four hemoglobins are bound by oxygen. After this, further increases in oxygen do no increase the oxygen bound to the tetramer of hemoglobin.

This sigmoidal curve is characteristic of positively cooperative kinetics and common in biology and metabolism.  Were we talking about a sigmoidal positive cooperative enzyme, Z, converting x substrate to y product, dy/dt, we would get the same sigmoidal behavior.

OK. So we have Michaelis -Menton enzymes in metabolism and positively cooperative enzymes in metabolism.

Imagine all the enzymes in E. coli were Michaelis Menton, which is NOT true.  Then at what x concentration would we get the fastest conversion of that x food molecule to biomass, y?  For food concentration x, just slightly above no influx of food at all.

What about if all the enzymes are sigmoidal positively cooperative enzymes.  Well, the sigmoidal curve is steepest in its rate of increase where the sigmoidal curve is steepest, at its inflection point. But this occurs for a finite concentration of substrate, or food, x, greater than 0!  But this means that the most rapid production of product y, or biomass, dy/dt, will occur at a finite input rate of food stuff, x.  

We are ready.  Tommi, Olli and I have shown, using the rate of biomass production as a function of glucose input rate, and most of the enzymes assumed to be sigmoidal, positively cooperative enzymes, that maximum rate of biomass production divided by glucose feeding rate, occurs for a finite rate of feeding in glucuse.

So what have we shown?  Biomass production rate is a power term.  Biomass production rate divided by glucose utiliziation rate is, like miles per gallon, a power efficency rate per fuel, glucose, intake rate.  So we have shown that E coli, if full of lots of sigmoidal enzymes, maximizes a power per unit fuel, biosmass production rate per glucose input rate, at a finite greater than 0 input rate of glucose.

In short, we have found the first evidence for the existence of an optimal power efficiency per unit food, biomass production rate divided by glucose utilization, for a finite optimal displacement from equilibrium, i.e. the optimal glucose input rate.

For the first time for cells that I know of, we have a optimization of a power per unit fuel at a finite measurable displacement from equilibrium - the glucose feed rate.

We do not yet know that E coli grows at this optimal rate. We are looking into it. The possibility is hopeful. Bacteria in colonies sense their local bacterial concentration by “quorum” sensing their numbers and speed up or slow down growth rate.  They may be optimizing our power efficiency, biosmass producution rate per unit fuel used.