A simple hyperbolic dependence of power output on power input will be assumed, saturating at a maximum P sat that is proportional to the amount of, and hence to the energy invested in producing, the required machinery: $$ P_\rm out=1/\left( 1/P_\rm in+1/P_\rm sat\right) $$As

a function of P sat, maximum growth power results when dP G/dP sat = 0, which leads to the condition: $$ \fracP_\rm outP_\rm sat=C_P_\rm out $$In words: the fraction of saturation reached equals the fraction of output power invested in the machinery for chemical storage of the absorbed power. Likewise, if P in were proportional NCT-501 clinical trial to the energy invested in the light-harvesting apparatus and no losses occur, maximum growth power would result when P out/P in = \(C_P_\rm in\): the yield of chemical storage of Selleck TSA HDAC the absorbed power equals the fraction of output power invested in the light-harvesting apparatus. However, adding pigments to a black cell would not help, so this can only be true as long as the attenuation of the light intensity

by the pigments remains negligible. In reality, self-shading will cause diminishing returns and an optimal distribution of the absorbers over the spectrum of the incident light must be sought. The question is what spectral distribution would optimize P G if the organism Rucaparib chemical structure could freely tune the resonance frequency of the electronic transition dipoles that make up its absorption spectrum. In order to express P G in terms of the absorber distribution, we divide the relevant part of the spectrum into n sufficiently small frequency steps with index i. At a light intensity (photon flux density) I sol(ν) the excitation rate becomes: $$ J_\rm L=\sum_i=1^nI_\rm sol,i\left( 1-e^-\sigma_i\right) $$The absorption cross-section σ i is defined here per unit area like I sol, so it is dimensionless and exp(−σ i ) is the transmittance.

The thermal excitation rate at an energy density of black body radiation ρbb(ν) at ambient temperature is: $$ J_\rm D=\sum_i=1^ng_i \cdot B \cdot \rho_\rm bb,i=\sum_i=1^n\sigma_i \cdot I_\rm bb,i $$where B is the Einstein coefficient, which is proportional to dipole strength, and g i the number of dipoles. As indicated, the thermal excitation rate of a dipole Bρ can be written as σI, where I is the light intensity (photon flux density), ρ·c/hν, so that its absorption cross-section σ = B·hν/c, with hν the photon energy and c the speed of light (the weak spectral dependence of the BVD-523 concentration refractive index, and hence of c, in the region of interest will be neglected). The σ i used above, therefore, equals g i ·hν i ·B/c.