Two-photon excitation and photolysis by pulsed laser illumination modelled by spatially non-uniform reactions with simultaneous diffusion

Abstract.

Two-photon absorption in the focus of a pulsed laser has the potential for localized photolysis of caged compounds, generating high concentrations of neurotransmitters, hormones and messengers. The concentrations of cage, intermediates and products in the femtolitre focal volume depend on reaction rates and diffusional exchange with the external volume. This problem of reaction with diffusion was analysed with analytical and numerical methods to determine simple relations between parameters useful in the design and interpretation of experiments. The diffraction-limited laser spot is approximated well by a sphere, radius A, in diffusional exchange with either an infinite uniform medium, representing extracellular photolysis, or within a non-permeable sphere, a "cell" of radius B, representing intracellular photolysis. Photolysis is modelled as sequential irreversible reactions, with either the excitation step alone, rate constant k _e, or with a subsequent "dark" reaction, rate constant k _p. For extracellular photolysis, steady-state depletion of a cage averaged in a spherical spot increases hyperbolically with k _e, with half-maximum depletion at k _e=K _0.5=2.5 D/A ², where D is the diffusion coefficient. With measured parameters for spot size A=0.3 µm and diffusion D=800 µm²/s, K _0.5=22,200 s^–1. The optimal exposure for localized photolysis is the characteristic diffusion time τ=A ²/D, 113 µs in this example, and is the time taken to reach 57% of steady state in the diffusion-limited case. In the two-step model, with excitation and "dark" reaction steps, rate constants both exceeding K _0.5 are necessary to generate 50% of maximal product concentration in the illuminated volume. High concentrations of photolysis products depend particularly on a high excitation rate constant % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfKttLearuavTnhis1MBaeXatLxBI9gBam % XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB % Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf % euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9 % q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqaba % WaaqaafaaakeaacqGGOaakcqWGRbWAdaWgaaWcbaGaeeyzaugabeaa % kiabg6da+iabdUealnaaBaaaleaacqaIWaamcqGGUaGlcqaI1aqnae % qaaOGaeiykaKcaaa!44AF! $(k_{\rm e} > K_{0.5} )$ , and localization of the products requires fast dark reactions % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfKttLearuavTnhis1MBaeXatLxBI9gBam % XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB % Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf % euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9 % q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciaacaGaaeqaba % WaaqaafaaakeaacqGGOaakcqWGRbWAdaWgaaWcbaGaeeiCaahabeaa % kiabg6da+iabdUealnaaBaaaleaacqaIWaamcqGGUaGlcqaI1aqnae % qaaOGaeiykaKcaaa!44C5! $(k_{\rm p} > K_{0.5} )$ . If products diffuse faster than the cage, their steady-state concentrations are decreased, and concentration transients may occur. For localized intracellular photolysis, the duration of exposure that generates product concentration at the cell boundary, B, less than 10% of the spot concentration should be shorter than 0.043(B/A)³τ, and is determined by diffusion.