Photosynthesis: from the antenna to the reaction center

ctep_general_public
From the antenna to the reaction center: downhill energy transfer in the photosynthetic apparatus of Chlorobium tepidum.

The photosynthetic apparatus of the green sulfur bacteria (chlorobium tepdidum) spurred a long lasting discussion about quantum coherence in biology, mostly focused on its subunit, the Fenna Matthews Olson complex. An account of the discovery of the FMO Protein is given by Olson in Photosynth. Res 80, 2004.

Recent experiments by Dostál, Pšenčík, and Zigmantas (Nat. Chem 8, 2016) show measured time and frequency resolved 2d-spectra of the whole photosynthetic apparatus. These results allow one to trace the energy flow from the antenna down to the reaction center and relate it to theoretical models.

ctep_animation
Computed two-dimensional spectrum of the antenna and FMO complex of C. tepidum. “A” denotes the location of the antenna peak, 1-7 the FMO complex states. Within tens of picoseconds, the energy is shuffled from the antenna towards the FMO complex.

In our article [Kramer & Rodriguez  Sci. Reports 7, 2017] , open access] we provide a model of the experimental results using the open quantum system dynamics code described previously. In addition we show how the different pathways in 2D spectroscopy (ground state bleaching, stimulated emission, and excited state absorption) affect the spectra and lead to shifts of “blobs” down from the diagonal places. This allows us to infer the effective coupling of the antenna part to the FMO complex and to assess the relative orientations of the different units. The comparison of theory and experimental result is an good test of our current understanding of the physical processes at work.

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The conference focuses on theoretical and experimental new developments in the field of photoactive molecules. If you are working in this field, we encourage you to attend the meeting. Find out more and who already agreed to participate  at photosynthesis2016.wordpress.com.

Please submit your abstract within the next two weeks by filling out the linked template from the conference site.

We (the scientific organizers, Tobias Kramer and Jörg Megow) are grateful for the full support by the Wilhelm and Else Heraeus Foundation, which covers the accommodation for all successful applicants.

Slow or fast transfer: bottleneck states in light-harvesting complexes

lhc2_monomer
Light-harvesting complex II, crystal structure 1RWT from Liu et al (Nature 2004, vol. 428, p. 287), rendered with VMD. The labels denote the designation of the chlorophyll sites (601-614). Chlorophylls 601,605-609 are of chlorophyll b type, the others of type a.

In the previous post I described some of the computational challenges for modeling energy transfer in the light harvesting complex II (LHCII) found in spinach. Here, I discuss the results we have obtained for the dynamics and choreography of excitonic energy transfer through the chlorophyll network. Compared to the Fenna-Matthews-Olson complex, LHCII has twice as many chlorophylls per monomeric unit (labeled 601-614 with chlorophyll a and b types).
Previous studies of exciton dynamics had to stick to simple exponential decay models based on either Redfield or Förster theory for describing the transfer from the Chl b to the Chl a sites. The results are not satisfying and conclusive, since depending on the method chosen the transfer time differs widely (tens of picoseconds vs picoseconds!).

Exciton dynamics in LHCII.
Exciton dynamics in LHCII computed with various methods. HEOM denotes the most accurate method, while Redfield and Förster approximations fail.

To resolve the discrepancies between the various approximate methods requires a more accurate approach. With the accelerated HEOM at hand, we revisited the problem and calculated the transfer rates. We find slower rates than given by the Redfield expressions. A combined Förster-Redfield description is possible in hindsight by using HEOM to identify a suitable cut-off parameter (Mcr=30/cm in this specific case).

Since the energy transfer is driven by the coupling of electronic degrees of freedom to vibrational ones, it of importance to assess how the vibrational mode distribution affects the transfer. In particular it has been proposed that specifically tuned vibrational modes might promote a fast relaxation. We find no strong impact of such modes on the transfer, rather we see (independent of the detailed vibrational structure) several bottleneck states, which act as a transient reservoir for the exciton flux. The details and distribution of the bottleneck states strongly depends on the parameters of the electronic couplings and differs for the two most commonly discussed LHCII models proposed by Novoderezhkin/Marin/van Grondelle and Müh/Madjet/Renger – both are considered in the article Scalable high-performance algorithm for the simulation of exciton-dynamics. Application to the light harvesting complex II in the presence of resonant vibrational modes (collaboration of Christoph Kreisbeck, Tobias Kramer, Alan Aspuru-Guzik).
Again, the correct assignment of the bottleneck states requires to use HEOM and to look beyond the approximate rate equations.

High-performance OpenCL code for modeling energy transfer in spinach

With increasing computational power of massively-parallel computers, a more accurate modeling of the energy-transfer dynamics in larger and more complex photosynthetic systems (=light-harvesting complexes) becomes feasible – provided we choose the right algorithms and tools.

OpenCL cross platform performance for tracking energy-transfer in the light-harvesting complex II found in spinach.
OpenCL cross platform performance for tracking energy-transfer in the light-harvesting complex II found in spinach, see Fig. 1 in the article . Shorter values show higher perfomance. The program code was originally written for massively-parallel GPUs, but performs also well on the AMD opteron setup. The Intel MIC OpenCL variant does not reach the peak performance (a different data-layout seems to be required to benefit from autovectorization).

The diverse character of hardware found in high-performance computers (hpc) seemingly requires to rewrite program code from scratch depending if we are targeting multi-core CPU systems, integrated many-core platforms (Xeon PHI/MIC), or graphics processing units (GPUs).

To avoid the defragmentation of our open quantum-system dynamics workhorse (see the previous GPU-HEOM posts) across the various hpc-platforms, we have transferred the GPU-HEOM CUDA code to the Open Compute Language (OpenCL). The resulting QMaster tool is described in our just published article Scalable high-performance algorithm for the simulation of exciton-dynamics. Application to the light harvesting complex II in the presence of resonant vibrational modes (collaboration of Christoph Kreisbeck, Tobias Kramer, Alan Aspuru-Guzik). This post details the computational challenges and lessons learnt, the application to the light-harvesting complex II found in spinach will be the topic of the next post.

In my experience, it is not uncommon to develop a nice GPU application for instance with CUDA, which later on is scaled up to handle bigger problem sizes. With increasing problem size also the memory demands increase and even the 12 GB provided by the Kepler K40 are finally exhausted. Upon reaching this point, two options are possible: (a) to distribute the memory across different GPU devices or (b) to switch to architectures which provide more device-memory. Option (a) requires substantial changes to existing program code to manage the distributed memory access, while option (b) in combination with OpenCL requires (in the best case) only to adapt the kernel-launch configuration to the different platforms.

The OpenCL device fission extension allows to investigate the scaling of the QMaster code with the number of CPU cores. We observe a linear scaling up to 48 cores.
The OpenCL device fission extension allows us to investigate the scaling of the QMaster code with the number of CPU cores. We observe a linear scaling up to 48 cores.

QMaster implements an extension of the hierarchical equation of motion (HEOM) method originally proposed by Tanimura and Kubo, which involves many (small) matrix-matrix multiplications. For GPU applications, the usage of local memory and the optimal thread-grids for fast matrix-matrix multiplications have been described before and are used in QMaster (and the publicly available GPU-HEOM tool on nanohub.org). While for GPUs the best performance is achieved using shared/local memory and assign one thread to each matrix element, the multi-core CPU OpenCL variant performs better with fewer threads, but getting more work per thread done. Therefore we use for the CPU machines a thread-grid which computes one complete matrix product per thread (this is somewhat similar to following the “naive” approach given in NVIDIA’s OpenCL programming guide, chapter 2.5). This strategy did not work very well for the Xeon PHI/MIC OpenCL case, which requires additional data structure changes, as we learnt from discussions with the distributed algorithms and hpc experts in the group of Prof. Reinefeld at the Zuse-Institute in Berlin.
The good performance and scaling across the 64 CPU AMD opteron workstation positively surprised us and lays the groundwork to investigate the validity of approximations to the energy-transfer equations in the spinach light-harvesting system, the topic for the next post.

Tutorial #1: simulate 2d spectra of light-harvesting complexes with GPU-HEOM @ nanoHub

The computation and prediction of two-dimensional (2d) echo spectra of photosynthetic complexes is a daunting task and requires enormous computational resources – if done without drastic simplifications. However, such computations are absolutely required to test and validate our understanding of energy transfer in photosyntheses. You can find some background material in the recently published lecture notes on Modelling excitonic-energy transfer in light-harvesting complexes (arxiv version) of the Latin American School of Physics Marcos Moshinsky.
The ability to compute 2d spectra of photosynthetic complexes without resorting to strong approximations is to my knowledge an exclusive privilege of the Hierarchical Equations of Motion (HEOM) method due to its superior performance on massively-parallel graphics processing units (GPUs). You can find some background material on the GPU performance in the two conference talks Christoph Kreisbeck and I presented at the GTC 2014 conference (recored talk, slides) and the first nanoHub users meeting.

GPU-HEOM 2d spectra computed at nanohub

GPU-HEOM 2d spectra computed at nanohubComputed 2d spectra for the FMO complex for 0 picosecond delay time (upper panel) and 1 ps (lower panel). The GPU-HEOM computation takes about 40 min on the nanohub.org platform and includes all six Liouville pathways and averages over 4 spatial orientations.
  1. login on nanoHub.org (it’s free!)
  2. switch to the gpuheompop tool
  3. click the Launch Tool button (java required)
  4. for this tutorial we use the example input for “FMO coherence, 1 peak spectral density“.
    You can select this preset from the Example selector.
  5. we stick with the provided Exciton System parameters and only change the temperature to 77 K to compare the results with our published data.
  6. in the Spectral Density tab, leave all parameters at the the suggested values
  7. to compute 2d spectra, switch to the Calculation mode tab
  8. for compute: choose “two-dimensional spectra“. This brings up input-masks for setting the directions of all dipole vectors, we stick with the provided values. However, we select Rotational averaging: “four shot rotational average” and activate all six Liouville pathways by setting ground st[ate] bleach reph[asing , stim[ulated] emission reph[asing], and excited st[ate] abs[orption] to yes, as well as their non-rephasing counterparts (attention! this might require to resize the input-mask by pulling at the lower right corner)
  9. That’s all! Hit the Simulate button and your job will be executed on the carter GPU cluster at Purdue university. The simulation takes about 40 minutes of GPU time, which is orders of magnitude faster than any other published method with the same accuracy. You can close and reopen your session in between.
  10. Voila: your first FMO spectra appears.
  11. Now its time to change parameters. What happens at higher temperature?
  12. If you like the results or use them in your work for comparison, we (and the folks at nanoHub who generously develop and provide the nanoHub platform and GPU computation time) appreciate a citation. To make this step easy, a DOI number and reference information is listed at the bottom of the About tab of the tool-page.

With GPU-HEOM we and now you (!) can not only calculate the 2d echo spectra of the Fenna-Matthews-Olson (FMO) complex, but also reveal the strong link between the continuum part of the vibrational spectral density and the prevalence of long-lasting electronic coherences as written in my previous posts

Oscillations in two-dimensional spectroscopy

Transition from electronic coherence to a vibrational mode.
Transition from electronic coherence to a vibrational mode made visible by Short Time Fourier Transform (see text).

Over the last years, a debate is going on whether the observation of long lasting oscillatory signals in two-dimensional spectra are reflecting vibrational of electronic coherences and how the functioning of the molecule is affected. Christoph Kreisbeck and I have performed a detailed theoretical analysis of oscillations in the Fenna-Matthews-Olson (FMO) complex and in a model three-site system. As explained in a previous post, the prerequisites for long-lasting electronic coherences are two features of the continuous part of the vibronic mode density are: (i) a small slope towards zero frequency, and (ii) a coupling to the excitonic eigenenergy (ΔE) differences for relaxation. Both requirements are met by the mode density of the FMO complex and the computationally demanding calculation of two-dimensional spectra of the FMO complex indeed predicts long-lasting cross-peak oscillations with a period matching h/ΔE at room temperature (see our article Long-Lived Electronic Coherence in Dissipative Exciton-Dynamics of Light-Harvesting Complexes or arXiv version). The persistence of oscillations is stemming from a robust mechanism and does not require adding any additional vibrational modes at energies ΔE (the general background mode density is enough to support the relaxation toward a thermal state). But what happens if in addition to the background vibronic mode density additional vibronic modes are placed within the vicinity of the frequencies related electronic coherences? This fine-tuning model is sometimes discussed in the literature as an alternative mechanism for long-lasting oscillations of vibronic nature. Again, the answer requires to actually compute two-dimensional spectra and to carefully analyze the possible chain of laser-molecule interactions. Due to the special way two-dimensional spectra are measured, the observed signal is a superposition of at least three pathways, which have different sensitivity for distinguishing electronic and vibronic coherences. Being a theoretical physicists now pays off since we have calculated and analyzed the three pathways separately (see our recent publication Disentangling Electronic and Vibronic Coherences in Two-Dimensional Echo Spectra or arXiv version). One of the pathways leads to an enhancement of vibronic signals, while the combination of the remaining two diminishes electronic coherences otherwise clearly visible within each of them. Our conclusion is that estimates of decoherence times from two-dimensional spectroscopy might actually underestimate the persistence of electronic coherences, which are helping the transport through the FMO network. The fine tuning and addition of specific vibrational modes leaves it marks at certain spots of the two-dimensional spectra, but does not destroy the electronic coherence, which is still there as a Short Time Fourier Transform of the signal reveals.

Computational physics & GPU programming: exciton lab for light-harvesting complexes (GPU-HEOM) goes live on nanohub.org

User interface of the GPU-HEOM tool for light-harvesting complexes at nanohub.org.
User interface of the GPU-HEOM tool for light-harvesting complexes at nanohub.org.

Christoph Kreisbeck and I are happy to announce the public availability of the Exciton Dynamics Lab for Light-
Harvesting Complexes (GPU-HEOM) hosted on nanohub.org. You need to register a user account (its free), and then you are ready to use GPU-HEOM for the Frenkel exciton model of light harvesting complexes. In release 1.0 we support

  • calculating population dynamics 
  • tracking coherences between two eigenstates
  • obtaining absorption spectra
  • two-dimensional echo spectra (including excited state absorption)
  • … and all this for general vibronic spectral densities parametrized by shifted Lorentzians.

I will post some more entries here describing how to use the tool for understanding how the spectral density affects the lifetime of electronic coherences (see also this blog entry).
In the supporting document section you find details of the implemented method and the assumptions underlying the tool. We are appreciating your feedback for further improving the tool.
We are grateful for the support of Prof. Gerhard Klimeck, Purdue University, director of the Network for Computational Nanotechnology to bring GPU computing to nanohub (I believe our tool is the first GPU enabled one at nanohub).

If you want to refer to the tool you can cite it as:

Christoph Kreisbeck; Tobias Kramer (2013), “Exciton Dynamics Lab for Light-Harvesting Complexes (GPU-HEOM),” https://nanohub.org/resources/gpuheompop. (DOI:10.4231/D3RB6W248).

and you find further references in the supporting documentation.

I very much encourage my colleagues developing computer programs for theoretical physics and chemistry to make them available on platforms such as nanohub.org. In my view, it greatly facilitates the comparison of different approaches and is the spirit of advancing science by sharing knowledge and providing reproducible data sets.

Good or bad vibrations for the Fenna-Matthews-Olson complex?

Electronic and vibronic coherences in the FMO complex using GPU HEOM by Kreisbeck and Kramer
Time-evolution of the coherence for the FMO complex (eigenstates 1 and 5 ) calculated with GPU-HEOM by Kreisbeck and Kramer, J. Phys. Chem Lett. 3, 2828 (2012).

Due to its known structure and relative simplicity, the Fenna-Matthews-Olson complex of green sulfur bacteria provides an interesting test-case for our understanding of excitonic energy transfer in a light-harvesting complex.

The experimental pump-probe spectra (discussed in my previous post catching and tracking light: following the excitations in the Fenna-Matthews-Olson complex) show long-lasting oscillatory components and this finding has been a puzzle for theoretician and led to a refinement of the well-established models. These models show a reasonable agreement with the data and the rate equations explain the relaxation and transfer of excitonic energy to the reaction center.

However, the rate equations are based on estimates for the relaxation and dephasing rates. As Christoph Kreisbeck and I discuss in our article Long-Lived Electronic Coherence in Dissipative Exciton-Dynamics of Light-Harvesting Complexes (arxiv version), an exact calculation with GPU-HEOM following the best data for the Hamiltonian allows one to determine where the simple approach is insufficient and to identify a key-factor supporting electronic coherence:

Wendling spectral density for FMO complex
Important features in the spectral density of the FMO complex related to the persistence of cross-peak oscillations in 2d spectra.

It’s the vibronic spectral density – redrawn (in a different unit convention, multiplied by ω2)  from the article by M. Wendling from the group of Prof. Rienk van Grondelle. We did undertake a major effort to proceed in our calculations as close to the measured shape of the spectral density as the GPU-HEOM method allows one. By comparison of results for different forms of the spectral density, we identify how the different parts of the spectral density lead to distinct signatures in the oscillatory coherences. This is illustrated in the figure on the rhs. To get long lasting oscillations and finally to relax, three ingredients are important

  1. a small slope towards zero frequency, which suppresses the pure dephasing.
  2. a high plateau in the region where the exciton energy differences are well coupled. This leads to relaxation.
  3. the peaked structures induce a “very-long-lasting” oscillatory component, which is shown in the first figure. In our analysis we find that this is a persistent, but rather small (<0.01) modulation.

2d spectra are smart objects

FMO spectrum calculated with GPU-HEOM
FMO spectrum calculated with GPU-HEOM for a 3 peak approximation of the measured spectral density, including disorder averaging but no excited state absorption.

The calculation of 2d echo spectra requires considerable computational resources. Since theoretically calculated 2d spectra are needed to check how well theory and experiment coincide, I conclude with showing a typical spectrum we obtain (including static disorder, but no excited state absorption for this example). One interesting finding is that 2d spectra are able to differentiate between the different spectral densities. For example for a a single-peak Drude-Lorentz spectral density (sometimes chosen for computational convenience), the wrong peaks oscillate and the life-time of cross-peak oscillations is short (and becomes even shorter with longer vibronic memory). But this is for the experts only, see the supporting information of our article.

Are vibrations good or bad? Probably both… The pragmatic answer is that the FMO complex lives in an interesting parameter regime. The exact calculations within the Frenkel exciton model do confirm  the well-known dissipative energy transfer picture. But on the other hand the specific spectral density of the FMO complex supports long-lived coherences (at least if the light source is a laser beam), which require considerable theoretical and experimental efforts to be described and measured. Whether the seen coherence has any biological relevance is an entirely different topic… maybe the green-sulfur bacteria are just enjoying a glimpse into Schrödinger’s world of probabilistic uncertainty.

Catching and tracking light: following the excitations in the Fenna-Matthews-Olson complex

Peak oscillations in the FMO complex calculated using GPU-HEOM
The animation shows how peaks in the 2d echo-spectra are oscillation and changing for various delay times. For a full explanation, see Modelling of Oscillations in Two-Dimensional Echo-Spectra of the Fenna-Matthews-Olson Complex by B.Hein, C. Kreisbeck, T. Kramer, M. Rodríguez, New J. of Phys., 14, 023018 (2012), open access.

Efficient and fast transport of electric current is a basic requirement for the functioning of nanodevices and biological systems. A neat example is the energy-transport of a light-induced excitation in the Fenna-Matthews-Olson complex of green sulfur bacteria. This process has been elucidated by pump-probe spectroscopy. The resulting spectra contain an enormous amount of information about the couplings of the different pigments and the pathways taken by the excitation. The basic guide to a 2d echo-spectrum is as follows:
You can find peaks of high intensity along the diagonal line which are roughly representing a more common absorption spectrum.  If you delay the pump and probe pulses by several picoseconds, you will find a new set of peaks at a horizontal axis which indicates that energy of the excitation gets redistributed and the system relaxes and transfers part of the energy to vibrational motion. This process is nicely visible in the spectra recorded by Brixner et al.
A lot of excitement and activity on photosynthetic complexes was triggered by experiments of Engel et al showing that besides the relaxation process also periodic oscillations are visible in the oscillations for more than a picosecond.

What is causing the oscillations in the peak amplitudes of 2d echo-spectra in the Fenna-Matthews Olson complex?

A purely classical transport picture should not show such oscillations and the excitation instead hops around the complex without interference. Could the observed oscillations point to a different transport mechanism, possibly related to the quantum-mechanical simultaneous superposition of several transport paths?

The initial answer from the theoretical side was no, since within simplified models the thermalization occurs fast and without oscillations. It turned out that the simple calculations are a bit too simplistic to describe the system accurately and exact solutions are required. But exact solutions (even for simple models) are difficult to obtain. Known exact methods such as DMRG work only reliable at very low temperatures (-273 C), which are not directly applicable to biological systems. Other schemes use the famous path integrals but are too slow to calculate the pump-probe signals.

Our contribution to the field is to provide an exact computation of the 2d echo-spectra at the relevant temperatures and to see the difference to the simpler models in order to quantify how much coherence is preserved. From the method-development the computational challenge is to speed-up the calculations several hundred times in order to get results within days of computational run-time. We did achieve this by developing a method which we call GPU-hierarchical equations of motion (GPU-HEOM). The hierarchical equations of motions are a nice scheme to propagate a density matrix under consideration of non-Markovian effects and strong couplings to the environment. The HEOM scheme was developed by Kubo, Tanimura, and Ishizaki (Prof. Tanimura has posted some material on HEOM here).

However, the original computational method suffers from the same problems as path-integral calculations and is rather slow (though the HEOM method can be made faster and applied to electronic systems by using smart filtering as done by Prof. YiJing Yan). The GPU part in GPU-HEOM stands for Graphics Processing Units. Using our GPU adoption of the hierarchical equations (see details in Kreisbeck et al[JCTC, 7, 2166 (2011)] ) allowed us to cut down computational times dramatically and made it possible to perform a systematic study of the oscillations and the influence of temperature and disorder in our recent article Hein et al [New J. of Phys., 14, 023018 (2012), open access] .