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Photosynthesis & Energy Transfer

How nature harvests light with near-perfect quantum efficiency. From exciton dynamics in light-harvesting complexes to quantum coherence in the FMO complex and charge separation at the reaction center.

1. Light-Harvesting Complexes

Photosynthetic organisms capture sunlight using antenna complexes containing hundreds of chlorophyll and carotenoid pigments arranged in precisely organized protein scaffolds. The major light-harvesting complex in plants, LHC-II, is the most abundant membrane protein on Earth, containing 14 chlorophylls and 4 carotenoids per monomer.

Excitonic Coupling

When chromophores are closely spaced (less than ~2 nm), their transition dipoles interact to form delocalized exciton states. The excitonic Hamiltonian for N coupled chromophores is:

$$\hat{H} = \sum_{m=1}^{N} \epsilon_m |m\rangle\langle m| + \sum_{m \neq n} V_{mn} |m\rangle\langle n|$$

where $\epsilon_m$ are site energies and the coupling $V_{mn}$ between chromophores m and n follows the dipole-dipole approximation for moderate distances:

$$V_{mn} = \frac{1}{4\pi\varepsilon_0\varepsilon_r} \left[\frac{\boldsymbol{\mu}_m \cdot \boldsymbol{\mu}_n}{R_{mn}^3} - 3\frac{(\boldsymbol{\mu}_m \cdot \mathbf{R}_{mn})(\boldsymbol{\mu}_n \cdot \mathbf{R}_{mn})}{R_{mn}^5}\right]$$

Derivation: Photon Energy in Photosynthesis

We derive the energy carried by photons at the wavelengths used by the two photosystems and calculate the energy input per electron transferred.

Step 1: Planck-Einstein relation

The energy of a single photon is quantized and proportional to its frequency. Using the wave relation $c = \lambda\nu$:

$$E = h\nu = \frac{hc}{\lambda}$$

where $h = 6.626 \times 10^{-34}$ J$\cdot$s and $c = 3.0 \times 10^8$ m/s.

Step 2: Calculate energy for Photosystem II (P680, $\lambda$ = 680 nm)

$$E_{680} = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^8}{680 \times 10^{-9}} = 2.92 \times 10^{-19} \text{ J} = 1.82 \text{ eV}$$

Per mole: $E = 2.92 \times 10^{-19} \times 6.022 \times 10^{23} = 176$ kJ/mol

Step 3: Calculate energy for Photosystem I (P700, $\lambda$ = 700 nm)

$$E_{700} = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^8}{700 \times 10^{-9}} = 2.84 \times 10^{-19} \text{ J} = 1.77 \text{ eV}$$

Per mole: $E = 171$ kJ/mol

Step 4: Total photon energy input per electron through the Z-scheme

Each electron passing through both photosystems absorbs one photon at each (minimum). The total energy input per electron:

$$E_{\text{total}} = E_{680} + E_{700} = 1.82 + 1.77 = 3.59 \text{ eV} \approx 347 \text{ kJ/mol}$$

Step 5: Energy conversion efficiency

The overall reaction 2H₂O + 2NADP$^+$ + ~3ADP $\to$ O₂ + 2NADPH + ~3ATP requires 4 electrons (and thus 8 photons minimum). The energy stored in products per O₂ is ~480 kJ/mol. Total photon energy input: $8 \times (176 + 171)/2 \approx 1388$ kJ/mol. The maximum thermodynamic efficiency is ~$480/1388 \approx 35\%$.

2. Forster Resonance Energy Transfer (FRET)

At larger distances where weak coupling applies, energy transfer proceeds via the FRET mechanism. The FRET rate depends on the inverse sixth power of the donor-acceptor distance:

$$k_{FRET} = \frac{1}{\tau_D}\left(\frac{R_0}{R}\right)^6$$

The Forster radius $R_0$ (distance at which FRET efficiency equals 50%) depends on spectral overlap, orientation, and medium properties:

$$R_0^6 = \frac{9 Q_D \ln(10) \kappa^2}{128\pi^5 n^4 N_A} \int_0^{\infty} F_D(\lambda)\epsilon_A(\lambda)\lambda^4 \, d\lambda$$

The FRET efficiency as a function of distance is:

$$E = \frac{R_0^6}{R_0^6 + R^6} = \frac{1}{1 + (R/R_0)^6}$$

R < R0

E > 50% — Efficient transfer dominates over donor emission

R = R0

E = 50% — Equal probability of transfer and donor decay

R > 2R0

E < 2% — FRET negligible, donor fluorescence dominates

Derivation: Quantum Yield of Photosynthesis

The quantum yield measures the efficiency of photosynthetic light reactions by comparing the number of O₂ molecules evolved to the number of photons absorbed.

Step 1: Define quantum yield

The quantum yield $\Phi$ is the ratio of the desired photochemical events to the total photons absorbed:

$$\Phi = \frac{\text{moles of O}_2 \text{ evolved}}{\text{moles of photons absorbed}}$$

Step 2: Determine the minimum photon requirement

The water-splitting reaction 2H₂O $\to$ O₂ + 4H$^+$ + 4e$^-$ requires 4 electrons. Each electron requires one photon at PSII and one at PSI. The minimum photon requirement per O₂ is:

$$n_{\text{photons}} = 4 \text{ electrons} \times 2 \text{ photons/electron} = 8 \text{ photons}$$

Step 3: Calculate the maximum quantum yield

The theoretical maximum quantum yield is:

$$\Phi_{\max} = \frac{1 \text{ O}_2}{8 \text{ photons}} = 0.125 \text{ mol O}_2\text{/mol photons}$$

Step 4: Experimental measurements

Emerson and Arnold (1932) measured $\Phi \approx 0.10-0.11$, and modern measurements give $\Phi \approx 0.10-0.125$ under optimal conditions (low light, saturating CO₂). This means the light reactions operate at ~80-100% of the theoretical maximum, reflecting the remarkable near-unity quantum efficiency of energy transfer from antenna to reaction center ($>$95%).

Step 5: Energy efficiency vs. quantum yield

The quantum yield measures photon utilization, not energy efficiency. The energy efficiency must account for the fact that red photons (680-700 nm) carry less energy than blue ones (450 nm), but all are used equally (one photon per excitation). Under white light, many absorbed high-energy photons are thermally relaxed to the P680/P700 level, reducing overall energy efficiency to ~25-30% even while quantum yield remains high.

Derivation: Z-Scheme Energetics — Total Energy Input from Two Photosystems

The Z-scheme describes the non-cyclic electron flow from water to NADP$^+$, powered by two photosystems in series. We derive the total energy balance and the redox potential changes at each step.

Step 1: Define the redox potentials at each stage

The Z-scheme spans from water oxidation to NADP$^+$ reduction. The key redox potentials (at pH 7) are:

$$E^{\circ\prime}(\text{O}_2/\text{H}_2\text{O}) = +0.816 \text{ V}, \quad E^{\circ\prime}(\text{NADP}^+/\text{NADPH}) = -0.320 \text{ V}$$

Step 2: Calculate the total thermodynamic requirement

The minimum energy needed to move one electron from water to NADP$^+$ (uphill in redox potential):

$$\Delta E = E^{\circ\prime}(\text{O}_2/\text{H}_2\text{O}) - E^{\circ\prime}(\text{NADP}^+/\text{NADPH}) = 0.816 - (-0.320) = 1.136 \text{ V}$$

$$\Delta G = nF\Delta E = 1 \times 96485 \times 1.136 = 109.6 \text{ kJ/mol per electron}$$

Step 3: Energy input from PSII

PSII absorbs a 680 nm photon (1.82 eV) and excites an electron from $E \approx +0.82$ V (P680) to $E \approx -0.80$ V (pheophytin). The energy boost per electron:

$$\Delta E_{\text{PSII}} = 0.82 + 0.80 = 1.62 \text{ V} \approx 1.82 \text{ eV (680 nm photon)}$$

Step 4: Energy input from PSI

After electron transport through plastoquinone, cytochrome b₆f, and plastocyanin (dropping from -0.80 V to +0.37 V), PSI absorbs a 700 nm photon (1.77 eV), boosting the electron from +0.37 V to ~-1.30 V (ferredoxin):

$$\Delta E_{\text{PSI}} = 0.37 + 1.30 = 1.67 \text{ V} \approx 1.77 \text{ eV (700 nm photon)}$$

Step 5: Energy budget and proton pumping

Total energy input per electron: 1.82 + 1.77 = 3.59 eV (347 kJ/mol). Energy stored as chemical potential (water to NADPH): 1.136 eV (109.6 kJ/mol). The remaining ~2.45 eV per electron is used to pump protons across the thylakoid membrane (generating pmf for ATP synthesis) and is partially dissipated as heat. For every 4 electrons, approximately 12 H$^+$ are translocated, driving synthesis of ~3 ATP via the chloroplast ATP synthase (14 c-subunits, so H$^+$/ATP $\approx$ 4.67).

3. Quantum Coherence in the FMO Complex

The landmark 2007 experiment by Engel et al. using two-dimensional electronic spectroscopy (2DES) revealed long-lived quantum coherence in the Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria at 77 K. Subsequent studies detected coherence even at physiological temperatures, though with shorter lifetimes (~300 fs at 300 K).

The FMO Hamiltonian

The FMO complex contains 7 bacteriochlorophyll a (BChl a) molecules. Its dynamics are governed by the system-bath Hamiltonian:

$$\hat{H}_{\text{total}} = \hat{H}_{\text{sys}} + \hat{H}_{\text{bath}} + \hat{H}_{\text{sys-bath}}$$

The open quantum system dynamics are described by the Lindblad master equation for the reduced density matrix:

$$\frac{d\hat{\rho}}{dt} = -\frac{i}{\hbar}[\hat{H}_{\text{sys}}, \hat{\rho}] + \sum_k \gamma_k \left(\hat{L}_k \hat{\rho} \hat{L}_k^{\dagger} - \frac{1}{2}\{\hat{L}_k^{\dagger}\hat{L}_k, \hat{\rho}\}\right)$$

The question of whether quantum coherence provides a functional advantage remains debated. Proposed mechanisms include environment-assisted quantum transport (ENAQT), where moderate noise actually enhances transfer efficiency by preventing destructive interference and localization.

Key Experimental Evidence

- 2DES cross-peak oscillations persisting for >660 fs at 77 K (Engel et al., Nature 2007)
- Vibronic coherence coupling electronic and nuclear degrees of freedom
- Near-unity quantum yield of energy transfer (>99%)
- Coherence observed in multiple photosynthetic systems (LHCII, PC645, PE545)

4. Charge Separation & Marcus Electron Transfer

At the reaction center, excitation energy drives charge separation — the conversion of light energy into electrochemical potential. This electron transfer process is described by Marcus theory:

$$k_{ET} = \frac{2\pi}{\hbar}|V_{DA}|^2 \frac{1}{\sqrt{4\pi\lambda k_B T}} \exp\left(-\frac{(\Delta G^0 + \lambda)^2}{4\lambda k_B T}\right)$$

Normal Region

$-\Delta G^0 < \lambda$: Rate increases as driving force increases

Optimal

$-\Delta G^0 = \lambda$: Maximum rate, barrierless transfer

Inverted Region

$-\Delta G^0 > \lambda$: Rate decreases — the Marcus inverted region

In photosynthetic reaction centers, the initial charge separation (P* to P+BA-) occurs in ~3 ps with near-unity quantum yield. The subsequent electron transfers are tuned to lie in the normal region, ensuring rapid forward reactions while charge recombination falls in the inverted region, making it kinetically unfavorable.

Python Simulation: FMO Exciton Dynamics

This simulation models exciton dynamics in the 7-site FMO complex using a simplified Lindblad master equation approach. It diagonalizes the FMO Hamiltonian, computes exciton delocalization, and tracks population transfer from the antenna (site 1) to the reaction center (site 3).

Exciton Dynamics in the FMO Complex

Python

Simplified Lindblad master equation simulation of energy transfer in the 7-bacteriochlorophyll FMO complex

script.py125 lines

import numpy as np

# Exciton Dynamics in the FMO Complex (Simplified Lindblad Model)
# ================================================================
# The Fenna-Matthews-Olson complex contains 7 bacteriochlorophyll
# pigments that transfer excitation energy with near-unit efficiency.

print("=" * 65)
print("EXCITON DYNAMICS IN THE 7-SITE FMO COMPLEX")
print("Simplified Lindblad Master Equation Approach")
print("=" * 65)

# FMO Hamiltonian (cm^-1) - Adolphs & Renger (2006) values
# Site energies on diagonal, couplings off-diagonal
H = np.array([
    [12410, -87.7, 5.5, -5.9, 6.7, -13.7, -9.9],
    [-87.7, 12530, 30.8, 8.2, 0.7, 11.8, 4.3],
    [5.5, 30.8, 12210, -53.5, -2.2, -9.6, 6.0],
    [-5.9, 8.2, -53.5, 12320, -70.7, -17.0, -63.3],
    [6.7, 0.7, -2.2, -70.7, 12480, 81.1, -1.3],
    [-13.7, 11.8, -9.6, -17.0, 81.1, 12630, 39.7],
    [-9.9, 4.3, 6.0, -63.3, -1.3, 39.7, 12440]
])

N = 7  # Number of sites

# Diagonalize the Hamiltonian
eigenvalues, eigenvectors = np.linalg.eigh(H)

print("\n--- Exciton Energies ---")
print(f"{'Exciton':>8} {'Energy (cm^-1)':>16} {'Shift from mean':>16}")
mean_E = np.mean(eigenvalues)
for i in range(N):
    print(f"{i+1:8d} {eigenvalues[i]:16.2f} {eigenvalues[i]-mean_E:16.2f}")

print("\n--- Exciton Delocalization ---")
print("Participation ratio (1=localized, 7=fully delocalized):")
for i in range(N):
    PR = 1.0 / np.sum(eigenvectors[:, i]**4)
    print(f"  Exciton {i+1}: PR = {PR:.3f}")

# Simplified population dynamics using Redfield-like rates
# Dephasing rate and relaxation rate
gamma_deph = 100.0    # cm^-1 (dephasing)
gamma_relax = 10.0    # cm^-1 (relaxation to lower exciton states)

# Time evolution of site populations
# Start with excitation on site 1 (BChl 1, closest to antenna)
dt = 0.001  # picoseconds
n_steps = 5000
t_max = dt * n_steps

# Initial density matrix (site 1 populated)
rho = np.zeros((N, N), dtype=complex)
rho[0, 0] = 1.0

# Convert Hamiltonian to frequency units (cm^-1 to ps^-1)
# 1 cm^-1 = 2*pi*c = 0.0001884 ps^-1 (approximately)
hbar_conv = 5.309  # hbar in cm^-1 * ps
H_freq = H / hbar_conv

print(f"\n--- Time Evolution (dt={dt} ps, total={t_max} ps) ---")
print(f"{'Time (ps)':>10} {'Site 1':>8} {'Site 2':>8} {'Site 3':>8} {'Site 4':>8} {'Site 5':>8} {'Site 6':>8} {'Site 7':>8}")
print("-" * 78)

populations_history = []

for step in range(n_steps + 1):
    t = step * dt
    pops = np.real(np.diag(rho))

if step % 500 == 0:
        print(f"{t:10.3f}", end="")
        for p in pops:
            print(f" {p:8.4f}", end="")
        print()
        populations_history.append((t, pops.copy()))

# Coherent evolution: drho/dt = -i[H, rho]/hbar
    commutator = H_freq @ rho - rho @ H_freq
    drho = -1j * commutator * dt

# Lindblad dissipation (simplified)
    for i in range(N):
        # Dephasing
        for j in range(N):
            if i != j:
                rho[i, j] *= np.exp(-gamma_deph * dt / hbar_conv)

# Downhill energy transfer (relaxation toward site 3 - the trap)
        if i != 2 and pops[i] > 0.001:
            transfer = gamma_relax * dt / hbar_conv * pops[i] * 0.1
            rho[i, i] -= transfer
            rho[2, 2] += transfer

rho += drho

# Ensure trace = 1
    trace = np.real(np.trace(rho))
    if trace > 0:
        rho /= trace

print("\n--- Final Population Distribution ---")
final_pops = np.real(np.diag(rho))
for i in range(N):
    bar = '#' * int(final_pops[i] * 50)
    print(f"  Site {i+1}: {final_pops[i]:.4f} |{bar}")

print(f"\nTotal population: {np.sum(final_pops):.6f}")
print(f"Population at trap (Site 3): {final_pops[2]:.4f}")

# FRET analysis
print("\n--- Forster Resonance Energy Transfer ---")
R0 = 5.4  # nm, typical Forster radius for chlorophyll
distances = [1.2, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 8.0, 10.0]
print(f"R0 (Forster radius) = {R0} nm")
print(f"\n{'R (nm)':>8} {'FRET Efficiency':>16} {'Rate (relative)':>16}")
print("-" * 45)
for R in distances:
    E_fret = 1.0 / (1.0 + (R/R0)**6)
    k_fret = (R0/R)**6
    print(f"{R:8.1f} {E_fret:16.4f} {k_fret:16.4f}")

print("\n[Simulation complete]")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Computation: FRET Efficiency Calculator

This Fortran program calculates FRET efficiency as a function of donor-acceptor distance, computing the Forster radius from physical parameters and analyzing inter-pigment energy transfer in the LHC-II light-harvesting complex.

FRET Efficiency Calculator

Fortran

Forster resonance energy transfer calculations for photosynthetic pigment pairs

fret_calculator.f90104 lines

program fret_calculator
  ! FRET Efficiency Calculator for Photosynthetic Complexes
  ! Calculates Forster resonance energy transfer efficiency
  ! as a function of donor-acceptor distance
  implicit none

integer, parameter :: dp = selected_real_kind(15, 307)
  real(dp) :: R, R0, E_fret, k_fret, tau_D, tau_DA
  real(dp) :: kappa2, n_refr, J_overlap, Q_D
  real(dp) :: pi, c_light, N_A
  real(dp) :: R0_calc
  integer :: i, j

pi = 4.0_dp * atan(1.0_dp)
  c_light = 2.998e10_dp   ! cm/s
  N_A = 6.022e23_dp        ! Avogadro's number

write(*,'(A)') '======================================================'
  write(*,'(A)') ' FRET EFFICIENCY CALCULATOR FOR PHOTOSYNTHETIC SYSTEMS'
  write(*,'(A)') '======================================================'

! Typical parameters for chlorophyll pair
  kappa2 = 2.0_dp / 3.0_dp  ! Orientation factor (random)
  n_refr = 1.4_dp            ! Refractive index (protein medium)
  Q_D = 0.3_dp               ! Donor quantum yield
  J_overlap = 2.5e-13_dp     ! Spectral overlap integral (cm^3/M)
  tau_D = 4.0_dp              ! Donor lifetime (ns)

! Calculate R0 from physical parameters
  ! R0^6 = (9*Q_D*ln(10)*kappa^2*J) / (128*pi^5*n^4*N_A)
  R0_calc = ((9.0_dp * Q_D * log(10.0_dp) * kappa2 * J_overlap) / &
             (128.0_dp * pi**5 * n_refr**4 * N_A))**(1.0_dp/6.0_dp)
  R0_calc = R0_calc * 1.0e7_dp  ! Convert cm to nm

write(*,'(A)') ''
  write(*,'(A)') '--- Physical Parameters ---'
  write(*,'(A,F6.4)')    ' Orientation factor (kappa^2): ', kappa2
  write(*,'(A,F6.3)')    ' Refractive index:             ', n_refr
  write(*,'(A,F6.3)')    ' Donor quantum yield:          ', Q_D
  write(*,'(A,ES12.3)')  ' Spectral overlap J:           ', J_overlap
  write(*,'(A,F6.2,A)')  ' Donor lifetime:               ', tau_D, ' ns'
  write(*,'(A,F8.3,A)')  ' Calculated R0:                ', R0_calc, ' nm'
  write(*,'(A)') ''

! Use a standard R0 for chlorophyll
  R0 = 5.4_dp  ! nm

write(*,'(A,F6.2,A)') ' Using R0 = ', R0, ' nm (chlorophyll standard)'
  write(*,'(A)') ''
  write(*,'(A)') '--- FRET Efficiency vs Distance ---'
  write(*,'(A8,A14,A14,A14,A14)') &
    'R (nm)', 'Efficiency', 'k_FRET/k_D', 'tau_DA (ns)', 'R/R0'
  write(*,'(A)') '--------------------------------------------------------------'

do i = 1, 30
    R = 0.5_dp + real(i-1, dp) * 0.5_dp
    E_fret = 1.0_dp / (1.0_dp + (R/R0)**6)
    k_fret = (R0/R)**6
    tau_DA = tau_D * (1.0_dp - E_fret)
    if (tau_DA < 0.001_dp) tau_DA = 0.001_dp

write(*,'(F8.2,F14.6,F14.4,F14.4,F14.4)') &
      R, E_fret, k_fret, tau_DA, R/R0
  end do

! Multi-chromophore FRET in LHC-II
  write(*,'(A)') ''
  write(*,'(A)') '--- LHC-II Multi-Chromophore Analysis ---'
  write(*,'(A)') 'Nearest-neighbor distances in LHC-II trimer:'
  write(*,'(A)') ''

! Typical inter-pigment distances in LHC-II (nm)
  ! Chl a-Chl a: ~0.9-1.3 nm (strong coupling)
  ! Chl a-Chl b: ~1.0-1.7 nm
  ! Chl b-Chl b: ~1.3-2.0 nm

write(*,'(A,A14,A14)') ' Pair Type          ', 'Dist (nm)', 'FRET Eff'
  write(*,'(A)') '---------------------------------------------'

! Representative pairs
  R = 0.95_dp
  E_fret = 1.0_dp / (1.0_dp + (R/R0)**6)
  write(*,'(A,F14.2,F14.6)') ' Chl a610 - a611    ', R, E_fret

R = 1.05_dp
  E_fret = 1.0_dp / (1.0_dp + (R/R0)**6)
  write(*,'(A,F14.2,F14.6)') ' Chl a611 - a612    ', R, E_fret

R = 1.20_dp
  E_fret = 1.0_dp / (1.0_dp + (R/R0)**6)
  write(*,'(A,F14.2,F14.6)') ' Chl a612 - b608    ', R, E_fret

R = 1.70_dp
  E_fret = 1.0_dp / (1.0_dp + (R/R0)**6)
  write(*,'(A,F14.2,F14.6)') ' Chl b608 - b609    ', R, E_fret

R = 2.10_dp
  E_fret = 1.0_dp / (1.0_dp + (R/R0)**6)
  write(*,'(A,F14.2,F14.6)') ' Chl b609 - a604    ', R, E_fret

write(*,'(A)') ''
  write(*,'(A)') '[Computation complete]'

end program fret_calculator

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Video Lectures

MIT: Photosynthesis (Light Reactions)

MIT lecture covering the light reactions of photosynthesis, photosystems I and II, and the Z-scheme of electron transport.

Quantum Biology: Photosynthesis

Exploration of quantum effects in photosynthetic energy transfer, including the FMO complex experiments and quantum coherence in biological systems.

Key Concepts Summary

Excitonic Coupling

Closely spaced chromophores form delocalized exciton states through transition dipole interactions

FRET Mechanism

Energy transfer efficiency scales as R0^6 / (R0^6 + R^6), with typical R0 of 4-6 nm for chlorophylls

Quantum Coherence

Long-lived coherences in FMO may assist energy transfer through environment-assisted quantum transport

Marcus Theory

Electron transfer rates governed by reorganization energy and driving force, with an inverted region

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