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Thermodynamics and the Structure of Living Systems

Nathaniel Virgo

Submitted for the degree of D.Phil.

University of Sussex

July, 2011

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Declaration

I hereby declare that this thesis has not been submitted, either in the same or different form, to this
or any other university for a degree.

Signature:

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Chapter 4. Entropy Production in Ecosystems 65

4.3 Organisms as Engines: an Evolutionary Model

In the previous section we defined a model in which the population metabolic rate was a parameter.
In effect we defined only the boundary conditions of the system, with the single parameter M
representing the overall effect of all the processes that take place within the system.

In this section we will develop a specific model of the population, which allows us to calculate
the value of M and to study how M changes over evolutionary time. We will find that in this very
simple model there is always evolutionary pressure towards an increase in M, regardless of whether
M is greater or less than MMEP, meaning that in general, in this model there is no dynamical trend
towards maximising the entropy production. A steady state in which M = MMEP is optimal for
the population as a whole (it represents not only a maximum in the rate at which chemical work
is extracted but also a maximum in the population itself) but evolutionary competition drives the
system towards a faster, less optimal, population metabolic rate.

There are many factors that can limit the increase of M in natural systems, including physical
constraints on the efficiency of individual metabolisms, altruistic restraint (which could evolve in
a variety of ways) predation and spatial effects. We will discuss some of these possibilities in
Section 4.4.

4.3.1 A Heat Engine Metaphor

In order to motivate this model it is instructive to consider an analogy in terms of heat engines.
Imagine that we have two heat baths A and B, with temperatures TA > TB. We could position a
(not necessarily reversible) heat engine between the two heat baths, taking heat from A at a rate
Qin and adding heat to B at a rate Qout, producing work at a rate W = Qin−Qout.

If this heat engine is to be in place for a long time then its parts will suffer wear and tear
and need replacing. Let us assume that the appropriate raw materials needed to achieve this are
readily available. Work will need to be done at a certain rate, Wrepair (constant over a long enough
time scale) to transform these raw materials into replacement parts for the heat engine. But if
W ≥Wrepair then we can simply use some of the engine’s output to perform this repair process.
The worn-out parts are returned to the pool of raw materials. The actual value of Wrepair depends
on many factors, including the nature of the heat engine and of the available raw materials, and on
the way in which the repairs are carried out.

Heat is produced as an effect of producing the spare parts. We assume this heat is returned to
the cold reservoir B. If the total energy content of the raw material pool is not changing over time
then by the conservation of energy this heat must be produced at a rate Qwaste = Wrepair.

This leaves a rate Wexcess = W −Wrepair of work to be put to other uses. One use that this
work could be put to is building another heat engine, similar to the original. This is analogous
to biological reproduction, while the use of work to repair the engine is similar to biological
metabolism2 (it could be thought of as a literal interpretation of Kauffman’s (2000) idea of a

2An important difference is that in this metaphor I have described the repair and reproduction as being done by
us, whereas in a biological situation it is achieved by the dynamics of the engines/organisms’ structures. This doesn’t
make any difference for the energetics of the present model, but it is important for a proper understanding of biological

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Chapter 4. Entropy Production in Ecosystems 66

“work-constraint cycle”).
We may now consider a population of n such engines. If the temperatures of A and B are fixed

then each engine can produce work at the same rate, regardless of the size of n, and the population
can grow without limit, unless the available space or the size of the raw material pool becomes a
factor.

However, we can consider the heat baths’ temperatures to be functions of the rate at which
heat is added or removed from them, in a similar manner to the equatorial and polar regions in the
two-box atmosphere model, or to the chemical potentials in the ecosystem model above. In line
with both these examples, we assume that the difference between the two temperatures narrows as
the overall rate of heat flow from one bath to the other increases.

If W > Wrepair then Wexcess > 0 and it is possible for new engines to be built; if Wexcess < 0
then then not enough work is being produced to maintain all the engines and some will have to be
decommissioned. But W depends on the difference between the two temperatures. If we assume
that each engine removes heat from A at the same rate, regardless of the temperature difference,
then a greater population means a greater total heat flow between the two heat baths, and hence
a lower difference in temperatures. The second law implies a maximum possible value for W of
Qin(1−TB/TA), which decreases as the population increases and the values of TA and TB become
closer. A balance is therefore reached, where Wexcess = 0 and the population neither grows nor
shrinks.

By reasoning along these lines we can create a model of the population dynamics of these
engines. We can then go on to consider the effects of evolution, in the sense that a new, slightly
more efficient or faster-running type of engine might be introduced to compete with the older
model. Before we do this I will complete the metaphor by translating the argument into the
domain of chemical rather than heat engines.

4.3.2 Organisms as Chemical Engines

For the most part the translation is straightforward. As explained in detail in Chapter 2, the quantity
λU = ∂S/∂U = 1/T has a close formal relationship to λNX = ∂S/∂NX = µ/T . Rather than thinking
about engines that transport heat from A to B we can switch to thinking about engines that convert
a reactant or set of reactants X into products Y , and most of the formal details will stay the same.

The one thing that needs a little consideration when making this switch is the concept of
work. In the heat engine analogy we were using the classical thermodynamic idea of work as
energy concentrated into a single macroscopic degree of freedom, so that it doesn’t contribute to
the calculation of the entropy. Effectively, work in this sense can be thought of as energy with an
infinite temperature. It is not clear whether work in this form plays any role in a biological systems
(though there are certainly energy quantities in biology that can be thought of as having extremely
high temperatures; see Jaynes (1989) for an example). It is therefore better to use the concept of
entropy reduction in our model rather than work.

Like heat engines, organisms have a low entropy structure. Entropy is produced within this

phenomena. This concept will be more fully explored in Chapter 5.

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