The quantum physics of life in red and green

Life is truly amazing!
Life is beautiful!
...and it involves quantum many-body physics...

There is a beautiful (short) review
Heme: From quantum spin crossover to oxygen manager of life 
Kasper Kepp

The article involves a plethora of topics that I have discussed before on this blog. I have included relevant links.

Kepp starts with the unique (chemically fine-tuned) properties of both iron and porphyrin that enable them to play a central role in two of the most important processes in life: respiration and photosynthesis. He has a beautiful paragraph (perhaps in the style of Roald Hoffmann):

Such ligand-field transitions of iron in porphyrin were familiar to our ancestors as the characteristic red color of blood that largely defines the human psychological and cultural connotations of the color representing courage, war, danger, and suffering. 

Incidentally, pi-pi* transitions within the porphyrin-derived chlorophylls are also responsible for the green color of plants, associated with nature, life and hope, so the reader may perhaps agree that porphyrin has had vast (but alas! rarely appreciated) cultural consequences.

The oxygen molecule is a spin triplet.
Iron(II) porphyrin is in a triplet spin state (S=1). The Fe(II) is a d6 configuration in a D_4h crystal field.
When they bind together the ground state is a spin-singlet.

There are two fundamental quantum chemistry questions that are discussed.

1. What is the electronic structure (many-body wave function) of the ground state for oxygen bound to heme?

2. What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

The first question has a long history. Like almost anything important and profound in quantum chemistry it goes back to Linus Pauling! In 1936 Pauling and Coryell argued that the ground state is

essentially a neutral O=O binding with two of its electrons to iron to produce a formally iron(II) if both the bonding electrons were confined to O2, corre- sponding to the non-bonding limit of neutral parts, but a formally iron(I) if the Fe–O bond were to be considered covalent. 

In 1960, McClure suggested a valence-bond formulation based on triplet–triplet coupling, which is appealing by the low promotion energies required to access these states, rather than the singlet states. In 1964, Weiss suggested, based on analogy to chemical reactions in aqueous solution, that the true ferrous hemeO2 adduct was mainly of the superoxo-iron(III) type caused by ‘‘electron transfer” from iron to O2. 

Goddard and Olafson suggested an ozone model of the adduct in 1975 which emphasized the four-electron three-center bond with maintained triplet state of dioxygen as in the McClure model with less electronic reorganization to explain the reversible binding. 

In 1977, Pauling maintained his original view again, the same year that Huynh, Case, and Karplus did a first attempt to bridge these views by performing early quantum chemical calculations that diplomatically emphasized the importance of both Weiss and Pauling resonance forms. 

However, interpretation depends on model language, orbital localization, and transformation between valence bond and orbital formalisms:  

In terms of molecular orbital theory, the wave function was a multi-configurational state dominated by the Pauling configuration; however, if one uses valence bond theory considerations, it can be interpreted as having large Weiss character. Thus, the multi-configurational state produced from CASPT2 is interpreted differently by different models. This partly explains why the trenches were so deeply dug during the exchange between Pauling, Goddard, McClure, and Weiss; all were right, and all were wrong. 

This is just another example of unnecessary conflicts about valence bond vs. molecular orbital (VB vs. MO). 

In terms of valence structures, the ground state was summarized by Shaik and Chen as having contributions from both Weiss, Pauling, and McClure forms, the first .. dominating. 

Ironically DFT ends up providing a useful language after all! 

The charge assignments to O2 are very dependent on calculation scheme, and both the orbitals, valence structures, and atomic charges that defined the Weiss-Pauling debate are non-observable. In contrast, the electron density is observable as are the geometries and spectroscopic data...

Molecular orbitals are not physical observables but calculational constructs. MO's don't exist.

In different words, one can take a many-body wave-function and make a linear unitary transformation of the molecular orbitals. The Slater determinants do not change. [The value of a determinant is invariant to a change of basis.]

Now. Question 2.
What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

Kepp talks about spin-orbit coupling and the fact that it is small for oxygen, motivating a discussion of a "broad crossing mechanism".  However, I am not sure this is relevant. I don't see the binding as necessarily spin forbidden. As the oxygen approaches the heme the two triplet states can mix to form a total spin singlet.
This is analogous to bringing two hydrogen atoms (each of which is spin 1/2) together to form a hydrogen molecule (which is spin zero). A multi-configurational wavefunction has no problem with this. But DFT-based approximations, which use a single determinant cannot describe this smooth crossover.

Other things of particular interest to me that are discussed in the review include the central role of back bonding and the success of the TPSSh functional in DFT calculations for organometallics.

Unfortunately, the review does not mention recent work by Weber et al, applying DMFT to the problem of oxygen binding to haemoglobin.

Talk on "crackpot" theories

At UQ there is a great student physics club, PAIN. Today they are having a session on "crackpot" theories in science. Rather than picking on sincere but misguided amateurs I thought I would have a go at "mainstream" scientists who should know better. Here are my slides on quantum biology.

A more detailed and serious talk is a colloquium that I gave six years ago. I regret that the skepticism I expressed then seems to have been justified.

Postscript.
I really enjoyed this session with the students. Several gave interesting and stimulating talks, covering topics such as flat earth, last thursdayism, and The Final Theory of gravity [objects don't fall to the earth but rather the earth rises up to them...]. There were good discussions about falsifiability, Occam's razor, Newton's flaming laser sword, ...
There was an interesting mixture of history, philosophy, humour, and real physics.

I always find to encouraging to encounter students who are so excited about physics that they want to do something like this on a friday night.

Photosynthesis is incoherent

Beginning in 2007 luxury journals published some experimental papers making claims that quantum coherence was essential to photosynthesis. This was followed by a lot of theoretical papers claiming support. I was skeptical about these claims and in the first few years of this blog wrote several posts highlighting problems with the experiments, theory, interpretation, and hype.

Here is a recent paper that repeats one of the first experiments.

Nature does not rely on long-lived electronic quantum coherence for photosynthetic energy transfer Hong-Guang Duan, Valentyn I. Prokhorenko, Richard Cogdell, Khuram Ashraf, Amy L. Stevens, Michael Thorwart, R. J. Dwayne Miller

During the first steps of photosynthesis, the energy of impinging solar photons is transformed into electronic excitation energy of the light-harvesting biomolecular complexes. The subsequent energy transfer to the reaction center is understood in terms of exciton quasiparticles which move on a grid of biomolecular sites on typical time scales less than 100 femtoseconds (fs). Since the early days of quantum mechanics, this energy transfer is described as an incoherent Forster hopping with classical site occupation probabilities, but with quantum mechanically determined rate constants. This orthodox picture has been challenged by ultrafast optical spectroscopy experiments with the Fenna-Matthews-Olson protein in which interference oscillatory signals up to 1.5 picoseconds were reported and interpreted as direct evidence of exceptionally long-lived electronic quantum coherence. Here, we show that the optical 2D photon echo spectra of this complex at ambient temperature in aqueous solution do not provide evidence of any long-lived electronic quantum coherence, but confirm the orthodox view of rapidly decaying electronic quantum coherence on a time scale of 60 fs. Our results give no hint that electronic quantum coherence plays any biofunctional role in real photoactive biomolecular complexes. Since this natural energy transfer complex is rather small and has a structurally well defined protein with the distances between bacteriochlorophylls being comparable to other light-harvesting complexes, we anticipate that this finding is general and directly applies to even larger photoactive biomolecular complexes.

I do not find the 60 fsec timescale surprising. In 2008, Joel Gilmore and I published a review of experiment and theory on a wide range of biomolecules (in a warm wet environment) that suggested that tens of femtoseconds is the relevant time scale for decoherence.

I found the following section of the paper (page 7) interesting and troubling.

The results shown in Figs. 3 (a) and (b) prove that any electronic coherence vanishes within a dephasing time window of 60 fs. It is important to emphasize that the dephasing time determined like this is consistent with the dephasing time of τhom = 60 fs independently derived from the experiment (see above). It is important to realize that this cross-check constitutes the simplest and most direct test for the electronic dephasing time in 2D spectra. In fact, the only unique observable in 2D pho- ton echo spectroscopy is the homogeneous lineshape. The use of rephasing processes in echo spectroscopies removes the inhomogeneous broadening and this can be directly inferred by the projection of the spectrum on the antidiagonal that shows the correlation between the excitation and probe fields. This check of self-consistency has not been made earlier and is in complete contradiction to the assertion made in earlier works. Moreover, our direct observation of the homogeneous line width is in agreement with independent FMO data of Ref. 12. This study finds an ∼ 100 cm−1 homogeneous line width estimated from the low-temperature data taken at 77 K, which corresponds to an electronic coherence time of ∼ 110 fs, in line with our result given the difference in temperature. In fact, if any long lived electronic coherences were operating on the 1 ps timescale as claimed previously (1), the antidiagonal line width would have to be on the order of 10 cm−1, and would appear as an extremely sharp ridge in the 2D inhomogeneously broadened spectrum (see Supplementary Materials). The lack of this feature conspicuously points to the misassignment of the long lived features to long lived electronic coherences where as now established in the present work is due to weak vibrational coherences. The frequencies of these oscillations, their lifetimes, and amplitudes all match those expected for molecular modes (41, 42) and not long-lived electronic coherences.

The challenge of intermediate coupling

The point here is a basic one. But, it is important to keep in mind.

One might tend to think that in quantum many-body theory the hardest problems are strong coupling ones. Let g denote some dimensionless coupling constant where g=0 corresponds to non-interacting particles. Obviously for large g perturbation theory is most unreliable and progress will be difficult. However, in some problems one can treat 1/g as a perturbative parameter and make progress. But this does require the infinite coupling limit be tractable.

Here are a few examples where strong coupling is actually tractable [but certainly non-trivial]

• The Hubbard model at half filling. For U much larger than t, the ground state is a Mott insulator. There is a charge gap and the low-lying excitations are spin excitations that are described by an antiferromagnetic Heisenberg model. Except for the case of frustration, i.e. on a non-bipartite lattice, the system is well understood.
• BEC-BCS crossover in ultracold fermionic atoms, near the unitarity limit.
• The Kondo problem at low temperatures. The system is a Fermi liquid, corresponding to the strong-coupling fixed point of the Kondo model.
• The fractional quantum Hall effect.

But, many of the problems of greatest interest are in an intermediate coupling region.

• Cuprate superconductors. For a long time it was considered that they are in the large U/t limit [i.e. strongly correlated] and that the Mottness was essential. However, Andy Millis and collaborators argue otherwise, as described here. It is interesting that one gets d-wave superconductivity both from a weak-coupling RG approach and a strong coupling RVB theory.
• Quantum chemistry. Weak coupling corresponds to molecular orbital theory. Strong coupling corresponds to valence bond theory. Real molecules are somewhere in the middle. This is the origin of the great debate about the relative merits of these approaches.
• Superconducting organic charge transfer salts. Many can be described by a Hubbard model on the anisotropic triangular lattice at half filling. Superconductivity occurs in proximity to the Mott transition which occurs for U ~ 8t. Ring exchange terms in the Heisenberg model may be important for understanding spin liquid phases.
• Graphene. It has U ~ bandwidth and long range Coulomb interactions. Perturb it and you could end up with an insulator.
• Exciton transport in photosynthetic systems. The kinetic energy, thermal energy, solvent reorganisation energy, and relaxation frequency [cut-off frequency of the bath] are all comparable.
• Water. This is my intuition but I find it hard to justify. It is not clear to me what the "coupling constants" are.

Aside from "brute force" numerical methods one is forced to attack these problems from either the weak-coupling or strong-coupling sides, hoping that one is capturing the essential physics. Sometimes one can come up with clever approximations that capture both the weak and strong coupling limits, and one hopes interpolates between the two. An example is Iterative Perturbation Theory used as an "impurity solver" in Dynamical Mean-Field Theory (DMFT). But, the big question arises whether there are intermediate coupling fixed points / phase transitions.

Intermediate coupling is both a blessing and a curse. It is a blessing because there is lots of interesting physics and chemistry associated with it. It is a curse because it is so hard to make reliable progress.

I welcome suggestions of other examples.

Desperately seeking quantum coherence

There is a paper in Science

Engineering Coherence Among Excited States in Synthetic Heterodimer Systems

Dugan Hayes, Graham B. Griffin, Gregory S. Engel

Six years ago Engel was first author of a Nature paper claiming that photosynthetic systems used quantum computing to maximise efficiency. The claims of this paper are more modest. The abstract begins:

The design principles that support persistent electronic coherence in biological light-harvesting systems are obscured by the complexity of such systems. Some electronic coherences in these systems survive for hundreds of femtoseconds at physiological temperatures, suggesting that coherent dynamics MAY play a role in photosynthetic energy transfer. Coherent effects MAY increase energy transfer efficiency relative to strictly incoherent transfer mechanisms. 

The key data is in the Figure below. It shows a Fourier transform of the "cross peaks in the two-dimensional spectra". The three boxes correspond to the different heterodimers. 

The vertical dashed lines are all present in monomers and are identified as vibronic features of those monomers.

[n.b. In the past some people mistakenly identified such features with coherences between different chromophores in photosynthetic complexes.]

The solid coloured vertical lines are identified as a quantum beating coherence between the two monomors. [But note smaller versions of these peaks are also present in two of the heterodimers]. The corresponding frequencies correspond to the difference frequency epsilon between the two monomers. This is the main result of the paper, the presence of electronic coherences between two dimers.

The decoherence time of the coherences is estimated to be of the order of tens of femtoseconds.
Although, not stated in the paper, this is just what one expects for typical chromophores in polar solvents. This timescale is much shorter than the timescale [1-100 psec] for many of the exciton transport processes in photosynthetic systems.

In a heterodimer the quantum beat frequency is given by

where Delta* is the coupling energy between the two chromophores and epsilon is the energy difference between the excitons in the individual chromophores. Since the authors observe a beat frequency of close to epsilon this means that Delta* is very small, at most tens of cm-1 which is less than thermal energy scales at room temperature.

It should also be pointed out that in the regime where epsilon is much larger than Delta* that the spectral weight of the quantum coherences becomes very small and they are easily decohered.  This can  be seen in Figure 2 of this PRL by Costi and Kieffer.

Details do matter in photosynthesis

In Telluride I heard David Coker give a fascinating talk about his recent work on quantum decoherence in photosynthetic proteins, reported in a recent paper with Jeremy Moix, Jianlan Wu, Pengfei Huo, and Jianshu Cao.

A 2007 Nature paper claimed to report evidence for quantum coherence of electronic excitations for a few hundred femtoseconds at liquid nitrogen temperatures in the FMO complex. This led to a flurry of theoretical activity [and various silly (at least to me) and grandiose claims about "quantum biology" and "green quantum computers"].
[Aside: it is often overlooked that the claimed coherence is just between 2 chromophores not all seven in the complex].

However, these experiments were not done on the complete photosynthetic complex, but a subset with 7 chromophores. Recent structural studies have shown that there is actually an 8th chromophore. Surely, such small details don't matter...
But, they do. Coker's group has shown that this extra chromophore changes the energy transfer pathway through the complex. Moreover, coherent oscillations between the two chromophores are no longer present. The energy transfer is incoherent in the native complex. It functions fine without quantum coherence.

Is photosynthesis highly efficient?

One should be careful about comparing apples and oranges!

There is a helpful and interesting article in Science Comparing Photosynthetic and Photovoltaic Efficiencies and Recognizing the Potential for Improvement
written by an Aussie Rules football team (18 co-authors!).

It points out that quantifying the efficiency of photosynthesis is not completely straightforward. It is sometimes claimed that it has evolved to have an optimum efficiency and that it has a quantum efficiency of 100% because every photon that is absorbed produces a desired chemical product. The authors state:

For comparison with PV electrolysis over an annual cycle, the energy efficiency of photosynthesis is a more useful parameter and is defined as the energy content (heat of combustion of glucose to CO₂ and liquid H₂O at STP) of the biomass that can be harvested annually divided by the annual solar irradiance over the same area. Using this definition, solar energy conversion efficiencies for crop plants in both temperate and tropical zones typically do not exceed 1%, a value that falls far below the benchmark for PV-driven electrolysis.

The authors note that the main evolutionary pressure on photosynthetic organisms is that they survive not that they have the optimum thermodynamic efficiency!

If one wants to compare photosynthesis to photovoltaics one should not consider the efficiency of the latter to produce electrical energy but rather chemical energy. The authors suggest an appropriate measure is the efficiency of photovoltaics to produce hydrogen gas from the water splitting reaction.

Coupled electron-proton transfer II

I had a nice visit this morning at University of Washington with Jim Mayer who has worked extensively on coupled electron-proton transfer [see this post for an earlier discussion]. Here are a few things I learnt.

CPET is involved in one of the most important processes in biology, whereby we get all of our oxygen! This is the Kok S-state mechanism of Photosystem II: the amino acid tyrosine-Z is oxidised to yield a neutral tyrosyl radical. Specifically, the electron is transferred 14 Angstroms (i.e. a long way!) to a photoexcited chlorophyll radical and the proton is transferred across a hydrogen bond to a nearby histidine residue (e.g. see this 2003 PNAS paper for evidence).

It is important to note that the proton and the electron are spatially separated and "attached" to different atoms. Nevertheless, their motion is concerted, i.e. the transfer is not sequential.

A major question concerns whether this process is adiabatic or non-adiabatic. Uncertainty about the answer is highlighted in a recent issue of Chemical Reviews. One article is by Hammes-Schiffer and Stuchebrukov which advocates a non-adiabatic approach.
A different article by Siegbahn and Blomberg considers DFT based calculations and implicitly assumes an adiabatic approach.

We agreed there is a need for some simple models to describe this fascinating phenomena.

Aside: a recent Science paper from Mayer's group shows that similar chemistry occurs in transition metal oxides which are important in energy research. Titanium dioxide has been the subject of 58,000 papers!

Do photosynthetic proteins protect quantum coherence?

A 2007 paper in Science Coherence Dynamics in Photosynthesis: Protein Protection of Excitonic Coherence by Lee, Cheng, and Fleming has attracted considerable interest, particularly from people enthusiastic about "quantum biology."
However, some recent papers based on molecular dynamics simulations cast doubt on the main claims of that paper.

The conclusion of the paper, Quest for Spatially Correlated Fluctuations in the FMO Light-Harvesting Complex by Carsten Olbrich, Johan Strumpfer, Klaus Schulten, and Ulrich Kleinekathofer

The comparison between present results and the reported experimental findings is difficult. It seems to be clear, though, that site correlations do not play a role at physiological conditions and that the biological function of the FMO complex is not affected by spatial site energy correlations. A similar conclusion has already been drawn for the light-harvesting II complex of Rhodospirillum molischianum in a similar study.

What do you conclude from this graph?

The graph below (taken from a paper Environment assisted quantum transport by Patrick Rebentrost, Masoud Mohseni, Ivan Kassal, Seth Lloyd and Alán Aspuru-Guzik) shows the calculated efficiency of transport (blue curve) of an exciton through at network in the presence of a dissipative and dephasing environment. The horizontal scale is the magnitude of the dephasing rate. n.b. it is a logarithmic scale and spans 10 orders of magnitude. The vertical line is the estimated dephasing rate at room temperature. 

To me the curve shows that the efficiency is quite insensitive to the environment, i.e. it is greater than 80 per cent for dephasing rates varying by eight orders of magnitude! In terms of biological functionality I would say that the environment matters little to the efficiency, except when the system couples very strongly to the environment.

However, other people interpret this graph in a very different way. In a 2010 paper in PNAS,  Long-lived quantum coherence in photosynthetic complexes at physiological temperature, the authors (led by Greg Engel from University of Chicago) state

theoretical studies incorporating both incoherent and coherent transfer as well as thermal dephasing predict that environmentally assisted quantum transfer efficiency peaks near physiological temperature

we ... observe quantum coherence lasting beyond 300 fs, showing that evolution has had the opportunity to exploit the theorized environmentally assisted quantum transport (EnAQT) mechanism for biological function. 

What do you think?

Colloquium on quantum biology

Here are my slides for the UQ Physics Colloquium. I have tried to build it around 5 key ideas:

1. Truly quantum dynamics requires phase coherence.

2. We can quantify quantum decoherence of excited states of optically active biomolecules. (Decoherence arises due to dielectric relaxation of the surrounding protein and water.)

3. Quantum dynamics is determined by competition between system timescale [which creates superposition state] and time scales of the environment.

4. There are significant scientific reasons to be skeptical about these claims of “quantum biology”.

5. The real scientific challenges for understanding are defining and solving realistic effective Hamiltonians for specific functional processes.

The key reference is a 2008 Review article I wrote with Joel Gilmore.

Overall I feel there is too much material, some of it is too technical, ...
I welcome feedback.

Going beyond the data?

In a 2007 Nature paper Engel et al. reported the data below showing the amplitude of an optical signal versus time.

The lower curve is the Fourier transform [using a new numerical method they developed explicitly for this paper] of the upper data.
They interpreted this data as evidence for quantum coherence between the excited states of different chromophores in a photosynthetic protein, since an oscillatory signal is a signature of quantum interference (Rabi oscillations). Engel et al. went on to claim that this coherence enabled the biomolecule to function in a highly efficient manner because:

…the system is essentially performing a single quantum computation, sensing many states simultaneously and selecting the correct answer, as indicated by the efficiency of the energy transfer. In the presence of quantum coherence transfer, such an operation is analogous to Grover's algorithm, …such a scheme can provide efficiency beyond that of a classical search algorithm.

This Nature paper greatly excited many in the quantum information community and has led to a host of theoretical papers about "quantum biology". The paper has already been cited hundreds of times.

But is this enthusiasm and hyperactivity justified? Given the noisy data and the absence of any concrete measurements of entanglement (e.g., violation of Bell inequalities) are the conclusions and claims really justified? I do not think so, as I have expressed several times in previous blog posts.

There is an interesting new preprint A critical view on transport and entanglement in models of photosynthesis by Tiersch, Popescu, and Briegel which has the abstract:

Quantum effects in biological light-harvesting molecules, such as quantum coherence of excitonic states and entanglement have recently gained much attention. We observe a certain discrepancy between the original experimental work and several theoretical treatments of coherent excitation transport in light-harvesting molecules. Contrary to what is generally stated, we argue that entanglement in such molecules is generally not equivalent to the presence of coherence but mostly introduced by initial assumptions underlying the models, and that entanglement, as opposite to coherence, seems to play no role in the transport efficiency.

Seeing the light from a single biomolecule

Single molecule spectroscopy has opened up a whole new vista on the photophysics and photochemistry of proteins whose functionality depends on optically active chromophores. A Nature Chemistry paper by Goldsmith and Moerner reports their study of the allophycocyanin (APC) protein in solution. It consists of a trimer, each of which contains two phycocyanonobilin (PCB) chromophores (shown in red below). There is a strong excitonic (Forster) coupling within each pair.

I found this particularly interesting because this is a symmetric methine dye, similar to those studied in this recent J. Chem. Phys. paper by Seth Olsen and I.

They observe significant fluctuations in the radiative lifetimes of the chromophore, arguing that this is due to changes in the local environment due to conformational change of the protein. They suggest there are four relevant electronic states.

The results can be compared to A Microscopic Model for the Fluctuations of Local Field and Spontaneous Emission of Single Molecules in Disordered Media

I thank Seth Olsen for bringing the paper to my attention.

Modelling electron transfer in photosynthesis

This follows up on a previous post about measurements of the rate at which electron transfer occurs in a photosynthetic protein. I noted several deviations of the experimental results from what is predicted by Marcus-Hush electron transfer theory. This is not necessarily surprising because one is not in quite in the right parameter regime.

In principle [at least to me] this should be described by a spin boson model which has the Hamiltonian

and the spectral density contains all the relevant information about the protein dynamics,

So the question I have is: if one has the correct parameters and spectral density can one actually describe all the experiments? Below is the spectral density found by Parson and Warshel in molecular dynamics simulations.

Non-Markovian quantum dynamics in photosynthesis

Understanding how photosynthetic systems convert photons into separated charge is of fundamental scientific interest and relevant to the desire to develop efficient photovoltaic cells. Systematic studies could also provide a laboratory to test theories of quantum dynamics in complex environments, for reasons I will try and justify below. When I was at U. Washington earlier this year Bill Parson brought to my attention two very nice papers from the group of Neal Woodbury at Arizona State.

Protein Dynamics Control the Kinetics of Initial Electron Transfer in Photosynthesis

Unusual Temperature Dependence of Photosynthetic Electron Transfer due to Protein Dynamics

The Figures below are taken from the latter paper. I think the basic processes involved here are 

P + H_A + photon ->  P* + H_A -> P+ + H_A-

where a photon is absorbed by the P, producing the excited state P* which then decays non-radiatively by transfer of an electron to a neighbouring molecule H_A. The graphs below show the electron population on P as a function of time, for a range of different temperatures and with different mutants of the protein.  

The different mutants correspond to substituting amino acids which are located near the special pair. These change the relative free energy of the P+ state.
The simplest possible theory which might describe these experiments is Hush-Marcus theory. However, it predicts

* the decay should be exponential with a single decay constant
* the rate should decrease with decreasing temperature
* the activation energy for the rate should be smallest when the energy difference between the two charge states epsilon equals the environmental re-organisation energy.

The above two papers contained several important results which are inconsistent with the simplest version of Hush-Marcus theory:

• The population does not exhibit simple exponential decay, but rather there are several different times scales, suggesting that the dynamics is non-Markovian, and may be correlated with the dynamics of the protein environment.
• The temperature dependence: the rate increases with decreasing temperature.
• A quantitative description of the data could be given in terms of a modification of Hush-Marcus to include the slow dynamics of the protein environment. This allows extraction of epsilon for the different mutants.

Later I will present my perspective on this....

Disentangling extraordinary claims about photosynthesis

Shaul Mukamel has a helpful paper, Signatures of quasiparticle entanglement in multidimensional nonlinear optical spectroscopy of aggregates

It addresses some important issues that have been discussed (ranted about?) on this blog before. Here is a good paragraph:

The fact that exciton states may be delocalized among chromophores is obviously interesting and implies quantum communication between them, provided it survives decoherence effects due to the surrounding medium. One could then justifiably argue that the chromophores are entangled and that the dynamics is profoundly quantum in nature. However, as shown here, these quantum effects can be easily eliminated by basing the description on the delocalized exciton modes.

This paper is a concrete realisation of a point Shaul made at a Workshop on Quantum Efficiency in the Black Forest last year.

I thank Seth Olsen for bringing the paper to my attention.

How does protein structure optimise function?

It actually has something to do with strong electronic correlations and the Jahn-Teller effect. Furthermore, it illustrates chemistry is very local.

There is a nice review article, Molecular modelling for transition metal complexes: Dealing with d-electron effects by Deeth, Anastasi, Diedrich, and Randell. It discusses specific examples of the Jahn-Teller effect in d9 Cu(II) complexes. [Aside: this the same complex as in cuprate superconductors].

Type I blue copper proteins are involved in electron transfer in both plant and bacterial photo-synthesis. A key question concerns how the structure around the Cu active site might be optimised in order to facilitate electron transfer. It was proposed that the local environment was arranged to minimise the reorganisation energy in the Marcus-Hush electron transfer theory. However, using quantum chemistry calculations, Ryde, Olssen, Pierloot, and Roos, showed that this was not the case in 1996. In fact the local geometry was the same in vacuo as in the protein. Furthermore, the structure was dominated by the strong Cu-thiolate bond.

Hence, Deeth et al., state

We begin with the presumption that although protein molecules may be large and complex, they form simple M–L coordinate bonds – i.e. a Cu–N(imidazole) bond can be handled in exactly the same way irrespective of whether it corresponds to an isolated imidazole or to a histidine.

This allows them to develop force fields for molecular mechanics.
Chemistry is local. This is why localised approaches such as valence bond theory are often preferable to molecular orbital or DFT-based approaches which tend to delocalise electrons.

Quantum effects and noise in biomolecules

New Journal of Physics has a Focus issue on Quantum Effects and Noise in Biomolecules. This includes a paper by myself and my former students Jacques Bothma and Joel Gilmore, The role of quantum effects in proton transfer reactions in enzymes: quantum tunneling in a noisy environment?

Artificial photosynthesis

Accounts of Chemical Research just published a special issue on Artificial Photosynthesis and Solar Fuels.

Articles I would particularly like to read include:

Structures and Energetics for O2 Formation in Photosystem II Per E. M. Siegbahn Theory of Proton-Coupled Electron Transfer in Energy Conversion Processes Sharon Hammes-Schiffer

Dye Sensitization of Single Crystal Semiconductor Electrodes Mark T. Spitler and B. A. Parkinson

Quantum coherence in photosynthesis?

In the latest PNAS Peter Wolynes has a nice Commentary, Some quantum weirdness in physiology on a paper by Ishizaki and Fleming, Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature.

Wolynes rightly cautions about invoking quantum effects to explain biological functions but seems convinced by work from Flemings group.

Here, I just explain what I believe is the essential physics in the theoretical paper since it can lost in all the detail. The key time (energy) scales in the calculation are:

• relaxation time of the environment (50 fsec)
• period of coherent oscillations (150 fsec)
• reorganisation energy of the environment , E_R (35 cm-1 ~ 1 psec)
• thermal energy, kB T (200 cm-1 ~ 150 fsec)

A relatively simple calculation using an independent boson model (see for example this paper) shows that there is initially a Gaussian decay of quantum coherence on the time scale, tau_g


Using the estimate E_R ~ 35 cm-1 gives a time scale of about 300 fsec for the decoherence time comparable to that in the detailed simulations of Ishizaki and Fleming. Decreasing the temperature by a factor of 4 will result in a two-fold increase in the decoherence time.

It should also be stressed that this reorganisation energy is an order of magnitude smaller than that estimated for other chromophores in proteins (see Table 2 in this paper). This will only be possible if the chromophore is well isolated from water in the environment.