Photosynthesis: Quantum to Calvin

2.1 Light Harvesting & Exciton Physics

The first step of photosynthesis is light absorption by antenna pigment-protein complexes. In higher plants, the major antenna is LHCII (Light Harvesting Complex II), a trimer containing 14 chlorophyll \(a\), 6 chlorophyll \(b\), and 4 carotenoid molecules per monomer. LHCII binds 50–70% of all chlorophyll in the chloroplast.

Chlorophyll \(a\) absorbs strongly at 430 nm (Soret band) and 680 nm (Q\(_y\) band). Chlorophyll \(b\) is blue-shifted (\(\sim\)470 and 650 nm) and funnels energy to Chl \(a\). The absorption cross-section per chlorophyll molecule is \(\sigma \approx 4 \times 10^{-16}\) cm\(^2\)at the Q\(_y\) peak.

Förster Resonance Energy Transfer

Energy migrates between pigments by Förster energy transfer(FRET), a non-radiative dipole–dipole coupling mechanism. To derive the rate, start from Fermi's Golden Rule. The coupling Hamiltonian for two point dipoles separated by \(\mathbf{R}\) is:

where \(\kappa = \cos\theta_{DA} - 3\cos\theta_D\cos\theta_A\) is the orientation factor (\(|\kappa^2| \leq 4\)). By Fermi's Golden Rule, the transfer rate is:

where \(J\) is the spectral overlap integral between donor emission and acceptor absorption. Defining the Förster radius \(R_0\) as the distance at which \(k_{DA} = 1/\tau_D\):

For Chl \(a\)–Chl \(a\) transfer in LHCII, \(R_0 \approx 7\)–\(10\) nm, but inter-pigment distances are only 1–3 nm, giving sub-picosecond transfer rates (\(k_{DA} \sim 10^{12}\) s\(^{-1}\)).

Quantum Coherence (FMO Complex)

Fleming et al. (2007, Nature) observed long-lived quantum beating in the FMO complex of green sulfur bacteria. When pigments are strongly coupled (\(J \gtrsim \lambda_{reorg}\), where \(\lambda_{reorg}\)is the reorganisation energy), excitations become delocalised excitonsrather than localised Frenkel excitations. The system Hamiltonian:

Diagonalising \(H_{sys}\) gives delocalized exciton states \(|\alpha\rangle = \sum_n c_n^\alpha |n\rangle\). The open quantum system dynamics (including bath fluctuations) are described by the Redfield equation for the reduced density matrix\(\rho(t)\) of the electronic degrees of freedom after tracing out the phonon bath:

where \(\mathcal{R}\) is the Redfield relaxation superoperator. The oscillations in cross-peaks of 2D electronic spectroscopy reveal quantum superpositions between exciton states persisting for\(\sim\)660 fs at 77 K. Whether these oscillations enhance energy transfer efficiency in vivo remains debated.

2.2 Photosystem II & the Oxygen-Evolving Complex

PSII is a homodimeric complex (~700 kDa per monomer) embedded in the thylakoid membrane. Its reaction centre chlorophyll, P680, has the highest oxidation potential of any biological molecule:\(E^0(P680^+/P680) \approx +1.26\) V vs. NHE.

S-State Cycle (Kok Cycle)

The Mn\(_4\)CaO\(_5\) water-oxidizing complex (WOC) cycles through five oxidation states S\(_0\)–S\(_4\), accumulating 4 oxidising equivalents before releasing O\(_2\):

Overall: \(2\mathrm{H}_2\mathrm{O} \rightarrow \mathrm{O}_2 + 4\mathrm{H}^+ + 4e^-\). The S1 state is the dark-stable state (Mn oxidation states: Mn\(_2^{III}\)Mn\(_2^{IV}\)). The S4→S0 transition is the rate-limiting O–O bond formation step, releasing O\(_2\)in the \(\sim\)200 \(\mu\)s timescale.

Electron Transport Chain & Redox Potentials

Z-Scheme Energy Diagram

2.3 Chemiosmotic Coupling & ATP Synthesis

Peter Mitchell's chemiosmotic hypothesis (1961) proposes that the free energy stored in the proton electrochemical gradient drives ATP synthesis. The proton-motive force across the thylakoid membrane is:

In chloroplasts, \(\Delta\psi \approx 0\) (membrane potential is small because of counter-ion movements) and the pmf is dominated by \(\Delta\mathrm{pH} \approx 3\) units (stroma pH 8.0, lumen pH 5.0), giving \(\Delta\tilde{\mu}_{H^+} \approx 17.7\) kJ mol\(^{-1}\) per proton.

The chloroplast ATP synthase (CF\(_0\)CF\(_1\)) requires \(n = 4.67\) protons per ATP (14-subunit c-ring, 3 catalytic sites), so:

This matches \(\Delta G_{ATP}^{hydrolysis} \approx 50\)–\(60\) kJ mol\(^{-1}\) in vivo with appropriate efficiency margin.

2.4 Farquhar–von Caemmerer–Berry Model

The FvCB model (1980) predicts leaf-level net CO\(_2\) assimilation \(A\) as:

where \(R_d\) is mitochondrial respiration in the light, \(A_c\) is Rubisco-limited assimilation, and \(A_j\) is RuBP-regeneration-limited assimilation.

Rubisco-Limited Rate \(A_c\)

Rubisco (ribulose-1,5-bisphosphate carboxylase/oxygenase) catalyses carboxylation and oxygenation. The net carboxylation rate by Michaelis–Menten kinetics with competitive inhibition by O\(_2\):

where \(V_{cmax}\) is the maximum carboxylation rate, \(C_i\) is intercellular CO\(_2\),\(K_c\) and \(K_o\) are Michaelis constants for CO\(_2\) and O\(_2\), and \(O\) is intercellular O\(_2\) (21 kPa). The CO\(_2\) compensation point \(\Gamma^*\) (when gross assimilation = photorespiration):

RuBP-Limited Rate \(A_j\)

When electron transport limits RuBP regeneration, the whole-chain electron transport rate \(J\) is:

The factor of 4 reflects that 4 electrons are needed per CO\(_2\) fixed, and the 8 in the denominator accounts for the electron cost of photorespiration. The “co-limitation” at the crossover point\(A_c = A_j\) defines the optimal \(C_i\) at which both processes are equally limiting (predicted by least-cost optimality theory to be ~0.7 of ambient CO\(_2\)).

Photorespiration Cost

When Rubisco oxygenates RuBP, 2-phosphoglycolate is produced. One turn of the photorespiratory C2 cycle recovers only 0.5 CO\(_2\) at the cost of 2 ATP and 2 NADPH. The net carbon cost of one oxygenation event:

2.5 Python: FvCB A-Ci and Light Response Curves