The above diagram is taken from a paper[ref] published in 1995 by the Hungarian biochemist Gaspar Banfalvi in which he introduced circular graphs to map energy changes in metabolic cycles – in this particular case showing the relationship between the Gibbs free energy (dotted line) and the average carbon oxidation state (solid line) of intermediates in the Krebs cycle.

This post adopts Banfalvi’s innovative approach in order to further explore the redox behavior of Krebs cycle intermediates. But before we start, a brief section on calculating average carbon oxidation states.

Calculating Average Carbon Oxidation States

The overall oxidation state of a molecule is its charge magnitude taking sign into account:
Charge magnitude = sum of carbon oxidation states + sum of oxidation states of other atoms

The sum of oxidation states of other atoms is computed by assigning a value to each of these atoms in the molecule as appropriate e.g. H = +1, O = -2, N = -3 in amino group [-NH2], P = +5 in phosphate group [-O-PO3]^2-, and summing them. Subtracting this sum from the charge magnitude gives the sum of carbon oxidation states. Dividing this by the number of carbon atoms in the molecule gives the average carbon oxidation state.

Example 1: Citrate

Charge magnitude = -3
Sum of oxidation states of other atoms Hx5, Ox7 = +5 -14 = -9
Sum of carbon oxidation states = -3 – (-9) = +6
Number of carbon atoms in molecule = 6
Average carbon oxidation state = +6/6 = +1

Example 2: Urea

Charge magnitude = 0
Sum of oxidation states of other atoms Hx4, Ox1 , Nx2 = +4 -2 -6 = -4
Sum of carbon oxidation states = 0 – (-4) = +4
Number of carbon atoms in molecule = 1
Carbon oxidation state = +4/1 = +4

Example 3: 1,3-diphosphoglycerate

Charge magnitude = -4
Sum of oxidation states of other atoms Hx4, Ox10, Px2 = +4 – 20 + 10 = -6
Sum of carbon oxidation states = -4 – (-6) = +2
Number of carbon atoms in molecule = 3
Average carbon oxidation state = +2/3

Useful links for determining oxidation states
– Sulfur and Phosphorus
https://www2.chemistry.msu.edu/faculty/reusch/VirtTxtJml/special2.htm
– Nitrogen https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Supplemental_Modules_(Organic_Chemistry)/Amines/
Properties_of_Amines/Oxidation_States_of_Nitrogen

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Carbon Oxidation States and Krebs Cycle

OK so here’s a different way of illustrating the Krebs Cycle. This diagram shows the cycle in terms of the number of carbon atoms in the input, output and cycle intermediate molecules, together with their average oxidation states. The numbers inscribed in blue show the changes in oxidation state of the intermediates taking place around the cycle.

The diagram reveals a feature that is not easily discernible in conventional depictions of Krebs Cycle – namely that the cycle intermediates undergo a progressive oxidation state reduction of -1 from oxaloacetate (C4,+1½) to succinyl-CoA (C4,+½), followed by a progressive oxidation state increase of +1 from succinyl-CoA back to oxaloacetate. Since a lower carbon oxidation state reflects more energy residing in chemical bonds, these changes indicate that cycle intermediates store energy in the first half of the cycle and liberate it in the second half.

Within this overall movement of energy lies a more detailed redox picture. Consider the steps in the first half where carbon dioxide is released. Decarboxylation of isocitrate per se (C6H5O7^3-) → [C5H5O5^3-] equates to a reduction of carbon oxidation state from +1 to +2/5 in the theoretical residue shown in [ ]. But isocitrate also undergoes oxidation, which we can notate as the theoretical residue transferring charge and a hydrogen atom to the NAD+ cofactor [C5H5O5^3-] → (C5H4O5^2-) resulting in a lesser overall reduction of carbon oxidation state from +1 to +4/5. Isocitrate thus combines both reductive and oxidative roles.

Similarly, decarboxylation of α-ketoglutarate per se (C5H4O5^2-) → [C4H4O3^2-] equates to a reduction of carbon oxidation state from +4/5 to 0 in the theoretical residue. But α-ketoglutarate also undergoes two oxidative processes, which we can notate as the theoretical residue adding an -SCoA group to the molecule and transferring charge to the NAD+* cofactor [C4H4O3SCoA^2-] → (C4H4O3S-CoA^1-) resulting in a lesser overall reduction of carbon oxidation state from +4/5 to +½. Thus α-ketoglutarate also combines both reductive and oxidative roles.

Two intermediates combine reductive and oxidative roles in the second half of the cycle. Succinyl-CoA undergoes two reductive processes: the deesterification of succinyl-CoA per se and receiving charge from a phosphate ion (C4H4O3S-CoA^1-) → [C4H4O3^2-] equates to a reduction of carbon oxidation state from +½ to +0 in the theoretical residue. Succinyl-CoA also undergoes oxidation, which we can notate as the theoretical residue receiving an oxygen atom from the same phosphate ion [C4H4O3^2-] → (C4H4O4^2-) resulting in an increase of carbon oxidation state from 0 to +½. Thus succinyl-CoA exhibits equal reductive and oxidative capacity resulting in no change in the average oxidation state of the next intermediate succinate.

Finally oxaloacetate undergoes two reductive processes: the addition of an acetyl group and receiving charge from the hydrogen of a water molecule (C4H2O5^2-) → [C6H5O6^3-] equates to a reduction of carbon oxidation state from +1½ to +2/3 in the theoretical residue. Oxaloacetate also undergoes oxidation, which we can notate as the theoretical residue receiving an oxygen atom from the same water molecule [C6H5O6^3-] → (C6H5O7^3-) resulting in a lesser overall reduction of carbon oxidation state from +1½ to +1.

*note that the hydrogen atom transferred to NADH comes from CoASH not α-ketoglutarate

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Krebs Cycle net equation restricted to cycle intermediates

There are plenty of sources out here on the web which will give you the Net Equation of Krebs Cycle showing every single substance involved, and there are lots of them. What I want to do here is write a simple net equation which shows only the inputs and outputs that are directly incorporated into or sourced from the structure of the cycle intermediate molecules themselves. The equation is a concise statement of what much of the foregoing has been trying to say – that the cycle intermediates have both oxidative and reductive roles:

Question. In overall terms two molecules of water enter the Krebs cycle, so why is there only one molecule of water shown on the input side? The answer is that while the water molecule which hydrates fumarate to malate is fully incorporated into the cycle intermediate structure, the water molecule involved in the conversion of oxaloacetate to citrate is not. Only the oxygen atom of this water molecule is incorporated into the citrate structure; one hydrogen atom is used to regenerate CoA-SH from acetyl-CoA while the other donates charge (C4^2- → C6^3-) and is oxidized to H+.

One of the two input oxygen atoms shown in the equation is now accounted for, but what about the other? The answer reveals a fascinating detail of the Krebs Cycle: it comes from the phosphate ion HPO4^2- involved in GTP/ATP formation associated with the conversion of succinyl-CoA to succinate. A PO3^1- moiety is used in the conversion of GDP to GTP while the hydrogen atom regenerates CoA-SH from succinyl-CoA, leaving a single oxygen which is incorporated into the succinate molecule together with a transfer of charge (C4^1- → C4^2-).

In summary we can say that at each turn of the Krebs Cycle, an acetyl group, a water molecule and two oxygen atoms – one from another water molecule and one from a phosphate ion – are incorporated into the structures of the cycle intermediates to facilitate the oxidative formation of carbon dioxide for release and hydrogen-reduced cofactors for onward transmission to the electron transport chain and ATP production.

– – – –

Krebs Cycle Thermodynamics

All cells that generate ATP by metabolizing acetyl-CoA to carbon dioxide make use of the Krebs cycle, which drives energy production in a way that shares characteristics with other more familiar engines operating in cycles. Like them, the Krebs cycle executes energy conversion and in so doing produces heat.

As with all energy conversion processes, metabolism is not particularly efficient and only around 40% of the available energy is converted to ATP; the rest is dissipated as heat. There are two aspects to this heat production that are worthy of note. The first is the effect that heat has on reaction kinetics; even a modest increase in temperature over that of the surroundings can speed up chemical processes significantly. Warmer organisms react faster, which is vital for survival.

The second aspect is the scale of heat release; the Gibbs free energy changes in the Krebs cycle reactions are very modest so that heat production and conduction of heat out of the cell can occur while maintaining thermal stability. Even in situations of high ATP demand, the mitochondrion won’t get fried.

Step Reaction ΔG’° kJ/mol Keq
1 Acetyl-CoA + Oxaloacetate → Citrate -32.2 2.65 x 10^5
2 Citrate → Isocitrate 13 6.47 x 10^-3
3 Isocitrate → α-ketoglutarate -8.4 2.60 x 10^1
4 α-ketoglutarate → Succinyl-CoA -33.5 4.38 x 10^5
5 Succinyl-CoA → Succinate -2.9 3.08
6 Succinate → Fumarate 0 1
7 Fumarate → Malate -3.8 4.36
8 Malate → Oxaloacetate 29.7 9.96 x 10^-6

space
The equilibrium constants of the Krebs cycle reactions exhibit a mixture of reactant-favored and product-favored processes. The second half of the cycle (Steps 5-8) is essentially reversible, with Steps 1 and 4 providing the product-favored momentum. It should be noted that Steps 3 and 4 involve the release of carbon dioxide which continuously diffuses out of the cell to neighborhoods of lower concentration, thereby lowering the reaction quotient Q and enhancing product favorability.

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Structure and Nomenclature of Krebs Cycle Intermediates

The sequence of intermediates in the Krebs cycle was worked out long ago, and part of that history is preserved in their esoteric names. For example succinic acid used to be obtained from amber by distillation, which is why it is named after the Latin word for amber – succinum. All very interesting, but it doesn’t help to visualize what the molecule looks like. A better way is to learn the carbon chain shapes of the cycle intermediates, and the scientific names that describe them. The advantage of doing this is that one reinforces the other; once you have memorized these scientific names you can draw molecules directly from them.

Try this out for yourself. You know citric acid is a C6 molecule; learn to draw it with the longest carbon chain (shown in red) around three sides of a rectangle with the three -COOH groups pointing out to one side like this. Practice drawing this until you can see it in your sleep.


Now you’ve learned it, you can name it. The longest carbon chain is C5 so the scientific name is based on pentane (C5H12). The C5 chain has -COOH groups at each end, so the core molecule is pentanedioic acid, which has a carboxyl and hydroxyl group on the third carbon atom in the chain. Adding these gives 3-carboxy-3-hydroxypentanedioic acid. That’s the scientific name for citric acid.

If you can learn this name, you can draw citric acid from memory. Awesome. And it’s an easy progression from there to learn the structure of isocitric acid


The only difference is that the hydroxyl group drops down to the second carbon atom in the C5 chain. So the scientific name simply changes to 3-carboxy-2-hydroxypentanedioic acid. The molecule has one carboxyl group on a side chain (the one shown in black) and it’s this one which is released as carbon dioxide along with a hydrogen atom to reduce NAD+, the molecule rearranging itself so that an oxo- group replaces the hydroxyl group on C2


The pentanedioic acid core molecule now has only one substituent; its scientific name is 2-oxopentanedioic acid. Once learned, it’s easy to draw this C5 molecule from memory which is more than can be said for α-ketoglutaric acid. The lower carboxyl group now departs to form carbon dioxide along with a hydrogen atom to reduce NAD+, with a Coenzyme A group taking its place. The resulting molecule is easier to visualize with its C4 chain drawn around three sides of a square like this


Don’t bother with the scientific name of this one, it’s too long. Just call it succinyl-CoA. All the remaining steps of the Krebs cycle retain the C4 chain. Following deesterification the next intermediate, succinic acid, is the simplest structure of all the Krebs cycle intermediates


The scientific name is that of the core molecule, butanedioic acid. Learn this name and you can easily draw succinic acid from memory. The next steps are the removal of two hydrogen atoms from carbon atoms 2 and 3 to form fumaric acid (not shown) followed by hydration to malic acid


whose scientific name is 2-hydroxybutanedioic acid, before a final dehydrogenation to oxaloacetic acid

whose structure is much easier to draw if you can learn its scientific name, 2-oxobutanedioic acid. Here’s a summary of the Krebs cycle intermediate names, ancient and modern:

HISTORICAL NAME SCIENTIFIC NAME
Citric acid 3-carboxy-3-hydroxypentanedioic acid
Isocitric acid 3-carboxy-2-hydroxypentanedioic acid
α-ketoglutaric acid 2-oxopentanedioic acid
Succinyl-CoA
Succinic acid butanedioic acid
Fumaric acid (E)-butenedioic acid
Malic acid 2-hydroxybutanedioic acid
Oxaloacetic acid 2-oxobutanedioic acid

space
– – – –

Suggested further reading

Gaspar Banfalvi, Constructing Energy Maps of Metabolic Cycles (1995)
The pioneering paper whose graphical idea is adopted in this blogpost. Balfalvi’s paper was published in Biochemical Education, now known as Biochemistry and Molecular Biology Education.
Link: https://iubmb.onlinelibrary.wiley.com/doi/pdf/10.1016/0307-4412(94)00172-L

S.V. Eswaran, A New Look at the Citric Acid Molecule (1976)
This paper tells the story of Krebs’ belief that if radiolabeled CO2 was assimilated then both the CH2COOH groups in citric acid would be labeled since citric acid is a symmetrical molecule. He was wrong and this paper explains why. It’s worth reading about what was briefly known as the “Ogston Effect”* and the influence A.G. Ogston’s thinking about substrates and planar enzyme surfaces had on the subsequent development of prochirality.
Link: https://insa.nic.in/writereaddata/UpLoadedFiles/PINSA/Vol43A_1977_4_Art01.pdf

*named after Alexander George Ogston (1911-1996), a British physical chemist.

Nazaret, C et al., Mitochondrial energetic metabolism: A simplified model of TCA cycle with ATP production (2009)
If you’re into modeling the chemical kinetics of metabolic systems on the basis of the Mass Action Law, you might like this paper published in the Journal of Theoretical Biology. The authors describe their model as “very simple and reduced” but even so it’s knee-deep in differential equations. Enjoy.
Link: https://hal.archives-ouvertes.fr/hal-00554511/document

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P Mander February 2020

Credit: Carbon Engineering

Direct Air Capture (DAC) is the somewhat misleading term that has come into use for chemical fixation processes designed to extract carbon dioxide from the atmosphere. Chemical fixation of atmospheric CO2 is nothing new – organisms capable of photosynthesis are thought to have evolved billions of years ago, while limestone formation from CO2 taken up by the oceans has been occurring for hundreds of millions of years. But mankind hasn’t had a go at it until recently.

The driving force behind the development of DAC is the conviction based on the best available science that current (and rising) atmospheric CO2 levels constitute an existential threat due to global warming and climate change, as well as ocean acidification and marine ecosystem disruption. There is a desire for technologies to augment both naturally-occurring fixation processes and emission-reducing initiatives in order to accelerate the process of bringing the rise in atmospheric CO2 levels to a halt and subsequently to achieve drawdown.

To put this task in perspective, let’s put a few facts and figures into it. Carbon dioxide is a thermodynamically stable gas and the densest component of air, 1.977 kgm-3 at STP compared with 1.293 kgm-3 for air. This does not mean however that CO2 sinks in air and accumulates in the lower atmosphere. Like any gas, CO2 exhibits the phenomenon of diffusion, which is the tendency of a substance to spread uniformly throughout the space available to it. And this is where the challenge of DAC lies. Although there is a lot of CO2 – ca. 3210 gigatonnes in 2018 (ref) – up in the air, the atmosphere is a big place and the concentration of CO2 (currently around 410 ppm) is tiny in the context of extraction. In other words a large amount of air needs to be processed for a modest rate of capture. It is therefore relevant to consider the processes of capture and conversion from a carbon oxidation state perspective, which has a direct bearing on process thermodynamics and economics.

Note: If it proves possible to reduce the partial pressure of CO2 in the atmosphere through DAC, oceanic release of dissolved CO2 should take place according to Henry’s Law to restore equilibrium between oceanic and atmospheric partial pressures. If this were the case, the removal of CO2 from the atmosphere could not be achieved separately from removing CO2 from the vastly bigger ocean reservoir, and the task of stopping the rise of the Keeling curve could not be separated from the rise in oceanic acidification. This would add a whole new dimension to the task of mitigation.

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Oxidation states and conversion energy

Carbon can exist in nine oxidation states, which are conveniently if somewhat abstractly represented by dimensionless numbers ranging from +4 (most oxidized state) to –4 (most reduced state). The lower the number, the more energy is present in carbon’s chemical bonds. So the process of reducing a carbon compound requires energy, the amount of which increases with the degree of reduction.

As the table shows, carbon in carbon dioxide has the highest oxidation state (+4). In terms of conversion energy requirement, it is self-evident that the least demanding option is to capture CO2 without reduction for subsequent release, containment and onward supply to industry. Capture technologies such as aqueous alkanolamine absorption are well-known, but in terms of product use it should be noted that carbon dioxide applications such as synthetic fuel feedstock and carbonated beverages will lead to atmospheric re-release and therefore cannot contribute to CO2 drawdown.

Note that Mother Nature makes use of CO2 capture without reduction in the process of oceanic uptake. The oxidation state of carbon in the carbonate ion that eventually becomes chemically fixed in limestone is still +4.

Turning to carbon dioxide capture with conversion, the oxidation state table shows that there are eight levels of reduction, each possessing its own product outcomes and synthetic opportunities. The formulas shown in the table indicate two strands in the reduction process – the addition of hydrogen and the removal of oxygen (for simplicity I have restricted the examples to combinations solely of C, H and O).

If we ask ourselves the question – What conversion process can we apply such that the energy requirement for reduction of CO2 is minimized, the answer suggests itself: a process that reduces the carbon oxidation state by just one unit, from +4 to +3.

Simple arithmetic suggests this can be achieved by combining two atoms of hydrogen with two molecules of carbon dioxide. And this is borne out in practice, although as we shall see the reducing agent need not necessarily be hydrogen.

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Converting carbon dioxide to oxalic acid

Consider a DAC process in which two moles of carbon dioxide are captured from the atmosphere and converted into one mole of oxalic acid (involving the addition of a mole of hydrogen). Now think about density change. Carbon dioxide is a gas, two moles of which occupy 44.8 liters at STP. Oxalic acid is by contrast a solid, a mole of which occupies 0.0474 liters. So in this conversion process the carbon atoms get packed into a space which is 946 times smaller. Woo. And because oxalic acid is a solid there is no requirement for pressurized containment or its associated cost.

One method by which this capture and conversion process can be realized has been published by researchers at Louisiana State University (ref). The chemical capture unit is built out of four pyridyltriazole chelating units linked by two meta-xylylene groups; complexation with CuCl2 gives a dimeric macrocycle which following cation reduction to Cu+ by sodium ascorbate is able to selectively capture and convert two carbon dioxide molecules into an oxalate ion as shown here

The oxalate ion is released as oxalic acid when treated with dilute mineral acid, regenerating the original copper complex. The reaction conditions are mild, reflecting the minimal reduction of carbon oxidation state from +4 to +3.

As I understand it the essential reaction sequence is as follows:

The oxidation of the ascorbate ion to a radical cation would appear to provide the thermodynamic impulse for this electron transfer sequence. Obviously there are other anions involved here, but the paper didn’t detail them so nor can I. What is evident however is that the energy demands of this process are modest, as one would anticipate for a unit reduction in oxidation state.

This paper was published 5 years ago as exploratory work. The kinetics in particular needed improvement and no doubt this has been addressed in further studies. My point in highlighting this work is to show the advanced level of innovation in facilitating oxidation state reductions by which atmospheric CO2 can be converted to carbon compounds of significant synthetic potential.

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Electrochemical reduction

With an enthalpy of formation ΔHf = -393 kJmol-1, carbon dioxide is a very thermodynamically stable compound. This explains why electrochemical reduction routes using the power of cathodic electrons have been sought to produce a CO2 anion which can enter into protic or aprotic reaction yielding formate or oxalate ions.

In Europe the acronym-intensive SPIRE (Sustainable Process Industry through Resource and Energy Efficiency) Association representing innovative process industries has a project running called OCEAN, which stands for Oxalic acid from CO2 using Electrochemistry At demonstratioN scale. OCEAN aims to develop an electrochemical process for producing high-value C2 chemicals from carbon dioxide via the following sequence:

1) reduce carbon dioxide (C1,+4) to formate (C1,+2)
2) dimerize formate (C1,+2) to oxalate (C2,+3)
3) protonate oxalate (C2,+3) to oxalic acid (C2,+3)
4) reduce oxalic acid (C2,+3) to glycolic acid (C2,+1)

The oxidation state sequence by which oxalic acid (4→2→3) and glycolic acid (4→2→3→1) are obtained from CO2 looks a bit lossy from an energy efficiency perspective, but this is just my impression. The project is running from October 2017 for four years and is EC-financed to the tune of €5.5 m.

Details of the OCEAN project can be found on these links:
https://www.spire2030.eu/printpdf/projects/our-spire-project/2217
https://www.fabiodisconzi.com/open-h2020/projects/211278/index.html

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P Mander July 2019

mau01

The postmark on this card is Tuesday 25th February 1908 – the date Ronald Ross left Mauritius for England, having spent three months on the island to prepare an official report on measures for the prevention of malaria, while privately thinking about how epidemics can be explained in terms of mathematical principle.

CarnotCycle is a thermodynamics blog but occasionally it ventures into new areas. This post concerns the modeling of disease transmission.

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Calamity

In 1867, a violent epidemic of malaria broke out on the island of Mauritius in the Indian Ocean. In the coastal town of Port Louis 6,224 inhabitants out of a local population of 87,000 perished in just one month. Across the island as a whole there were 43,000 deaths out of a total population of 330,000. It was the worst calamity that Mauritius has ever suffered, and it had a serious impact on the island’s economy which in those days was principally generated by sugar cane plantations.

At the time, Mauritius was ruled by the British. The island had little in the way of natural resources, but perhaps because of its strategic position for Britain’s armed forces, the government was keen to keep the malaria problem under observation. Medical statistics show that following the great epidemic of 1867, deaths from malaria dropped to zero by the end of the century.

In the first years of the 20th century however, a small but significant rise in deaths from malarial fever was observed. And in May 1907 the British Secretary of State for the Colonies requested Ronald Ross, Professor of Tropical Medicine at Liverpool University, to visit Mauritius in order to report on measures for the prevention of malaria there. Ross sailed from England in October 1907 and arrived in Mauritius a month later.

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Genesis

mau02

Just south of Port Louis, on the west coast of Mauritius, lies the township of Albion. Today it is home to a Club Med beach resort, but in 1907 when Ronald Ross visited the island, there were sugar plantations here – Albion Estate and Gros Cailloux estate, employing considerable numbers of Indian laborers. This part of the sea-coast was known for its marshy localities and it was here that the first sporadic cases of malaria were observed in 1865, two years before the great epidemic broke out.

Ronald Ross no doubt toured this area, his mind occupied with the genesis of the outbreak. Just a handful of cases in 1865, then in 1866 there were 207 cases on Albion Estate and 517 cases on Gros Cailloux Estate. From these estates the disease spread north and south, and during 1867 the epidemic broke out with such severity along sixty miles of coastline that those who survived were scarcely able to bury the dead.

How could this rapid increase in cases be explained? Ronald Ross was probably better placed than anyone to furnish an answer. Not only was he the discoverer of the role of the marsh-breeding Anopheline mosquito in spreading malaria (for which he received a Nobel Prize in 1902), he was also a thinker with a mathematical turn of mind.

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Statistics

mau03

Soon after the malaria epidemic broke out on Mauritius, the British government appointed a commission of enquiry, which published a bulky report in 1868. This was followed by numerous other publications, giving Ronald Ross an abundance of statistical data with which to chart the course of the epidemic.

I can picture Ross studying the monthly totals of malaria cases as the epidemic unfolded, and noting how they followed an exponential curve. And I can imagine him seeing the list of figures as the terms of a mathematical sequence, with the question forming in his mind “What is the formula that generates the numbers in this sequence?”.

Although trained in medicine rather than mathematics, Ross nevertheless knew that one route to finding the formula was to construct a first-order difference equation which expresses the next term in a sequence as a function of the previous term. In his 1908 report he adopts this approach, finds a formula, and demonstrates some remarkable results with it. Although at times loosely worded, his pioneering elaboration of what he calls the ‘malaria function’ displays original thinking of a high order.

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Solution

Ronald Ross was a mathematician by nature but not by training, which explains the absence of formal rigor in his mathematical argument. The style of exposition is somewhat saltatory; in fact he never actually states the difference equation, but instead leaps straight to its general solution (the malaria function) without showing the intermediate steps.

Ross begins with the argumentation leading to his famous ‘fsbaimp’ expression (familiarity is assumed; otherwise see Appendix 1), but it is not particularly conducive to understanding his overall scheme since he presents it as an algebraic thing-in-itself rather than a component variable in a first-order difference equation.

To apprehend the architecture of Ross’s thinking, one has to work backwards from the malaria function to obtain the difference equation, which can be expressed in words as

Infections (month n+1) = Infections (month n) – number of recoveries + number of new cases

Now although Ross did not address the matter of dimensions at any point in his argumentation, it was nonetheless a crucial consideration in formulating the above equation. Equality is symmetric, so the dimensions of each RHS term must be the same as the LHS term, which according to Ross’s terminology for infected people is mp. Since Ross is seeking to obtain a difference equation of the form

where α is the growth/decay constant, each of the three RHS terms must be the product of mp and a dimensionless coefficient k:

Clearly k1 is a dimensionless 1 since the total infections in month n is simply m(n)p. The coefficient k2 is the dimensionless recovery constant for the infected population (Ross uses the symbol r), whose value lies in the range 0–1. The real difficulty is with k3 – how to transform fsbai into a dimensionless quantity. Ross achieved this (see Appendix 1) by introducing a one-to-one correspondence constraint which had the effect of changing the units of a from mosquitoes to people, thereby cancelling out the units of b (1/people) and rendering fsbai dimensionless. This could with some justification be regarded as an exercise in artifice, but Ross really had no alternative to employing facilitated convenience if he was to solve this equation.

Putting all these pieces together, the difference equation Ross arrived at, but did not state, was:

where all terms except m (called the malaria rate) are considered constant. In his 1908 report, Ross skipped directly from the above equation, which is of the form

to its solution

which enabled him to compute his malaria function explicitly in terms of the initial value m(0)

or as Ross actually rendered it (by substituting f/p for b; see Appendix 1)

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Ratios

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Ronald Ross and a mosquito trap on Clairfond Marsh, Mauritius

In the above equation, Ross found an explanation not only of the outbreaks of malaria epidemics, but also of why malaria can diminish and even die out – as had happened for example in Europe – despite the continued presence of mosquitoes capable of carrying the disease.

Ross recognized that m(n) would increase or diminish indefinitely at an exponential rate as n increases, according to whether the contents of the parentheses were greater or less than unity, i.e.

Here was the riposte to those who claimed that malaria should persist wherever Anopheline mosquitoes continued to exist, and that anti-malarial strategies which merely reduced mosquito numbers would never eradicate the disease.

Ross could now show that it was the relation of the mosquito-human population ratio in a locality to its threshold value (a/p = r/f2si) that determined growth or decay of the malaria rate m(n), and that mosquito reduction measures, if sufficiently impactful, could indeed result in the disease diminishing and ultimately disappearing. He could even provide a rough estimate of the threshold value of a/p by assigning plausible values to s, i, f and r. In his 1908 report, Ross calculated this value to be 39.6, or about 40 mosquitoes per individual.

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Limitations

Ross’s malaria function was a remarkable result of some brilliantly original thinking, but as with most early forays into uncharted territory it had its limitations. Principal among them was that the equation was valid on the restrictive assumption that infected mosquitoes bit only uninfected human beings.

This clearly lacked credibility in the circumstances of a developed epidemic where a substantial proportion of the local population would be infected. So Ross was forced to preface his equation with the words ‘if … m is small’, which meant that the equation was strictly invalid for charting log phase growth or decay – thereby weakening support for his argument that total eradication of mosquitoes was unnecessary for disease control.

Another significant assumption in Ross’s equation was that the local population p was regarded as constant*, something wildly at variance with the actuality of the Mauritius epidemic of 1867, where a great many deaths occurred in the absence of any significant immigration.

*Although p cancels out from the mp term on both sides of the equation, it remains present in the third coefficient which is a component part of the growth/decay constant.

With limitations like these, it is evident that in his 1908 report Ross had not yet achieved a convincing mathematical argument to support his controversial views on how to control malaria. Ross was well aware of this, and over the next eight years he developed his ideas considerably – both in refining his model and advancing his mathematical approach.

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Extensions

The next phase of Ross’s mathematical thinking was published in a book entitled The Prevention of Malaria (1911) wherein Ross addresses the malaria rate issue using iterated difference equations, from which he computes a limiting value of m. In an addendum to the 2nd edition of this work, under the heading Theory of Happenings, Ross addresses the population variation issue using a systematized set of difference equations, and in the closing pages of the addendum makes the transition from the discrete time period of his difference equations to the infinitesimal time period of a corresponding set of differential equations. This allows him to address variations from the perspective of continuous functions.

Ross could have stopped there, but the instinctive mathematician in him had more to say. This resulted in a lengthy paper published in parts in the Proceedings of the Royal Society of London between July 1915 and October 1916. In this paper, Ross continues from where he left off in 1911, but in a more generalized form. He considers a population of whom a number are affected by something (such as a disease) and the remainder are non-affected; in an element of time dt a proportion of the non-affected become affected and a possibly different proportion of the affected revert to the non-affected group. He then supposes that both groups are subject to possibly different birth rates, death rates, immigration and emigration rates, and asks: What will be the number of affected individuals, the number of new cases, and the number of people living at time t?

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Hilda Hudson (1881-1965) and Ronald Ross (1857-1932)

To answer these questions, Ross attempts to integrate his differential equations; this forms the substance of Part I. For Parts II and III, Ross enlists the assistance of “Miss Hilda P. Hudson, MA, ScD”, a 34-year-old Cambridge mathematician, whom he acknowledges as co-author. In Part II they examine cases where the something that happens to the population (such as a disease) is not constant during the considered period. This propels them into the study of what they call hypometric happenings. In Part III they turn their attention to graphing some of the functions they have obtained, and note the steadily rising curve of a happening that gradually permeates the entire population, the symmetrical bell-shaped curve of an epidemic that dies away entirely, the unsymmetrical bell curve that begins with an epidemic and settles down to a steady endemic level, the periodic curve with its regular rise and fall due to seasonal disturbances, and the irregular curve where outbreaks of differing violence occur at unequal intervals. The conclusion they reach is that “the rise and fall of epidemics as far as we can see at present can be explained by the general laws of happenings, as studied in this paper.”

In summary then, it can be said that having resolved the issues that restricted the applicability of the malaria function, Ross and Hudson found that their generalized model – taking the happening to be a malaria outbreak – endorsed Ross’s original assertions, with the attendant implications for management and prevention.

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But all this lay ahead of Ronald Ross in February 1908 as he completed the groundwork for his first report. We leave him as he packs his bags to depart Mauritius, his mind full of island impressions, malaria statistics and mathematical ideas that he will contemplate at leisure on the month-long journey home.

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Appendix 1

What f.s.b.a.i.m.p means

(terms as defined in the 1908 report; note that Ross later revised some of these definitions)

p = the average population in the locality (units: people)
m = the proportion of p which are already infected with malaria in the start month (dimensionless)
i = the proportion of m which are infectious to mosquitoes (dimensionless)
a = the average number of mosquitoes in the locality (units: mosquitoes)
b = the proportion of a that feed on a single person (units: 1/people)

hence baimp = the average number of mosquitoes infected with malaria in the month

s = the proportion of mosquitoes that survive long enough to bite human beings (dimensionless)
f = the proportion of a which succeed in biting human beings (dimensionless)

hence fsbaimp = the average number of infected mosquitoes which succeed in biting human beings

If the constraint is applied that each of these mosquitoes infects a separate person and only one person, then fsbaimp will denote the average number of persons infected with malaria during the month. Since the constraint imposes a one-to-one correspondence, the units of fsbaimp may equally be taken as ‘infected mosquitoes’ and ‘infected people’.

Note also that, given p, either b or f is technically redundant since p = f/b

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Further reading

Ronald Ross, Report on the Prevention of Malaria in Mauritius (1908)
https://archive.org/details/b21352720

Paul Fine, Ross’s a priori Pathometry – a Perspective (1976)
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1864006/pdf/procrsmed00040-0021.pdf

Smith DL et al., Ross, Macdonald, and a Theory for the Dynamics and Control of Mosquito-Transmitted Pathogens (2012)
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3320609/pdf/ppat.1002588.pdf

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P Mander August 2016

Peter Mander, author and editor of CarnotCycle

“Thank you to everyone who has visited this blog since its inception in August 2012. CarnotCycle’s country statistics show that thermodynamics interests many, many people. They come to this blog from all over the world, and they keep coming.

It’s wonderful to see all this activity, but perhaps not so surprising. After all, thermodynamics is the branch of science that can help us to use energy wisely and to understand and manage the effects of global warming on our atmosphere, our ice caps and our oceans.

Today’s students of physics and physical chemistry, for whom thermodynamics is a core subject, will be at the forefront of future efforts to keep this planet habitable. They must be knowledgeable and competent, and CarnotCycle is one of the many free resources to help them achieve that aim.”

Exploring the mutual solubility of phenol and water at the Faculty of Pharmacy, National University of Malaysia.

The phenol-water system is a well-studied example of what physical chemists call partially miscible liquids. The extent of miscibility is determined by temperature, as can be seen from the graph below. The inverted U-shaped curve can be regarded as made up of two halves, the one to the left being the solubility curve of phenol in water and the other the solubility curve of water in phenol. The curves meet at the temperature (66°C) where the saturated solutions of water in phenol, and phenol in water, have the same composition.

The thermodynamic forces driving the behavior of the phenol-water system are first visible in the upwardly convex mutual solubility curve, showing that the enthalpy of solution (ΔHs) in the saturated solution is positive i.e. that the system absorbs heat and so solubility increases with temperature in accordance with Le Châtelier’s Principle.

More rigorously, one can ascertain whether solubility of the minor component increases or decreases with temperature by computing

where s is the solubility expressed in mole fraction units, f2 is the corresponding activity coefficient and Ts is the temperature at which the solution is saturated.

The line of the umbrella curve charts the variation in composition of saturated solutions – phenol in water and water in phenol – with temperature. The area to the left of the curve represents unsaturated solutions of phenol in water and the area to the right represents unsaturated solutions of water in phenol, while the area above the curve represents solutions of phenol and water that are fully miscible i.e. miscible in all proportions.

But what about the area inside the curve, which is beyond the saturation limits of water in phenol and phenol in water? In this region, the system exhibits its most striking characteristic – it divides into two coexistent phases, the upper phase being a saturated solution of phenol in water and the lower phase a saturated solution of water in phenol. The curious feature of these phases is that for a given temperature their composition is fixed even though the total amounts of phenol and water composing them may vary.

To analyze how this comes about, consider the dotted line on the diagram below, which represents the composition of the phenol-water system at 50°C.

Starting with a system which consists of water only we gradually dissolve phenol in it, maintaining the temperature at 50°C, until we reach the point Y on the curve at which the phenol-in-water solution becomes saturated.

Now imagine adding to the saturated solution a small additional amount of phenol. It cannot dissolve in the solution and therefore creates a separate coexistent phase. Since this newly-formed phenol phase contains no water, the chemical potential of water in the solution provides the driving force for water to pass from the aqueous solution into the phenol phase. This cannot happen on its own however since water passing out of a phenol-saturated solution would cause the solution to become supersaturated. This would constitute change from a stable state to an unstable state which cannot occur spontaneously.

What can be postulated to occur is that the movement of water from the solution into the phenol phase simultaneously lowers the chemical potential of phenol in that coexistent phase, allowing phenol to move with the water in such proportion that the phenol-in-water phase remains saturated – as it must do since the temperature remains constant. In other words, saturated solution passes spontaneously from the aqueous phase into the phenol phase, diminishing the amount of the former and increasing the amount of the latter. Because water is the major component of the phenol-in-water phase, this bulk movement will continuously increase the proportion of water in the coexistent water-in-phenol phase until it reaches the saturation point whose composition is given by point Z on the mutual solubility curve.

In terms of chemical potential in the two-phase system, equilibrium at a given temperature will be reached when:

upper phase = sat. soln. of phenol in water
lower phase = sat. soln. of water in phenol

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Applying the Phase Rule

F = C – P + 2

First derived by the American mathematical physicist J. Willard Gibbs (1839-1903), the phase rule computes the number of system variables F which can be independently varied for a system of C components and P phases in a state of thermodynamic equilibrium.

Applying the rule to the 1 Phase region of the phenol-water system, F = 2 – 1 + 2 = 3 where the system variables are temperature, pressure and composition. So for a chosen temperature and pressure, e.g. atmospheric pressure, the composition of the phase can also be varied.

In the 2 Phase region of the phenol-water system, F = 2 – 2 + 2 = 2. So for a chosen temperature and pressure, e.g. atmospheric pressure, the compositions of the two phases are invariant.

In the diagram below, the compositions of the upper and lower phases remain invariant along the line joining Y and Z, the pressure being atmospheric and the temperature being maintained at 50°C. As we have seen, the upper layer will be a saturated solution of phenol in water where the point Y determines the % weight of phenol (= 11%). Correspondingly, the lower layer will be a saturated solution of water in phenol where the point Z determines the % weight of phenol (= 63%).

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Relationship between X, Y and Z

If a mixture of phenol and water is prepared containing X% by weight of phenol where X is between the points Y and Z as indicated on the above diagram, the mixture will form two phases whose phenol content at equilibrium is Y% by weight in the upper phase and Z% by weight in the lower phase.

Let the mass of the upper phase be M1 and that of the lower phase be M2. The mass of phenol in these two phases is therefore Y% of M1 + Z% of M2. Conservation of mass dictates that this must also equal X% of M1 + M2. Therefore

The relative masses of the upper and lower phases change according to the position of X along the line Y-Z. As X approaches Y the upper phase increases as the lower phase diminishes, becoming one phase of saturated phenol-in-water at point Y. Conversely as X approaches Z the lower phase increases as the upper phase diminishes, becoming one phase of saturated water-in-phenol at point Z.

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Further reading

Logan S.R. Journal of Chemical Education 1998

This paper uses the well-known thermodynamic equation ΔG = ΔH + TΔS as a theoretical basis for determining the circumstances under which spontaneous mixing occurs when two partially miscible liquids are brought together at constant temperature and pressure.

The approach involves the construction of equations for estimating both the enthalpy and entropy of mixing in terms of the mole fraction x of one component, and graphing the change in Gibbs free energy ΔG against x to determine the position of any minimum/minima. The paper goes on to examine the criteria for the existence of two phases on the basis of determining the circumstances under which a system of two phases will have a combined ΔG value that is lower than the corresponding ΔG for a single phase.

The conclusion is reached that the assessment of ΔG on mixing two liquids can provide a qualitative explanation of some of the phenomena observed in relation to the miscibility of two liquids.

The paper is available from the link below (pay to view)
https://pubs.acs.org/doi/abs/10.1021/ed075p339?src=recsys&#

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P Mander November 2018