Excitation dynamics in Photosystem I trapped in TiO2 mesopores.

Excitation decay in closed Photosystem I (PSI) isolated from cyanobacterium Synechocystis sp. PCC 6803 and dissolved in a buffer solution occurs predominantly with a ~ 24-ps lifetime, as measured both by time-resolved fluorescence and transient absorption. The same PSI particles deposited in mesoporous matrix made of TiO2 nanoparticles exhibit significantly accelerated excitation decay dominated by a ~ 6-ps component. Target analysis indicates that this acceleration is caused by ~ 50% increase of the rate constant of bulk Chls excitation quenching. As an effect of this increase, as much as ~ 70% of bulk Chls excitation is quenched before the establishment of equilibrium with the red Chls. Accelerated quenching may be caused by increased excitation trapping by the reaction center and/or quenching properties of the TiO2 surface directly interacting with PSI Chls. Also properties of the PSI red Chls are affected by the deposition in the TiO2 matrix: they become deeper traps due to an increase of their number and their oscillator strength is significantly reduced. These effects should be taken into account when constructing solar cells' photoelectrodes composed of PSI and artificial matrices.

Ultrafast energy transfer and excited state coupling in an artificial photosynthetic antenna.

We have studied the energy transfer dynamics in an artificial light-harvesting dyad composed of a phthalocyanine (Pc) covalently linked to a carotenoid (Car). The combination of high temporal resolution transient absorption spectroscopy with global and target analysis allowed us to quantify the efficiency of the energy transfer from the S2 excited state of the Car to the Pc at 37%, close to values observed in some natural light-harvesting complexes. In addition, following selective excitation of the Pc, we have identified the spectral signatures of the S1 excited state of the Car which appear within the ≈30 fs time resolution of our measurement. This strongly indicates excited state coupling between the S1 state of Car and the Qx state of Pc, with important implications for the regulation of photosynthetic activity.

Charge separation is virtually irreversible in photosystem II core complexes with oxidized primary quinone acceptor.

X-ray structures of the Photosystem II (PSII) core revealed relatively large interpigment distances between the CP43 and CP47 antenna complexes and the reaction center (RC) with respect to the interpigment distances in a single unit. This finding questions the possibility of fast energy equilibration among the antenna and the RC, which has been the basic explanation for the measured PSII fluorescence kinetics for more than two decades. In this study, we present time-resolved fluorescence measurements obtained with a streak-camera setup on PSII core complexes from Thermosynechococcus elongatus at room temperature (RT) and at 77 K. Kinetic modeling of the RT data obtained with oxidized quinone acceptor Q(A), reveals that the kinetics are best described by fast primary charge separation at a time scale of 1.5 ps and slow energy transfer from the antenna into the RC, which results in an energy equilibration time between the antenna and the RC of about 44 ps. This model is consistent with structure-based computations. Primary radical pair formation was found to be a virtually irreversible process. Energy equilibration within the CP43 and CP47 complexes is shown to occur at a time scale of 8 ps. Kinetic modeling of the 77 K data reveals similar energy transfer time scales in the antenna units and among the antenna and the RC as at RT, respectively, 7 and 37 ps. We conclude that the energy transfer from the CP43/CP47 antenna to the RC is the dominant factor in the total charge separation kinetics in intact PSII cores.

Charge separation and energy transfer in the photosystem II core complex studied by femtosecond midinfrared spectroscopy.

The core of photosystem II (PSII) of green plants contains the reaction center (RC) proteins D1D2-cytb559 and two core antennas CP43 and CP47. We have used time-resolved visible pump/midinfrared probe spectroscopy in the region between 1600 and 1800 cm(-1) to study the energy transfer and charge separation events within PSII cores. The absorption difference spectra in the region of the keto and ester chlorophyll modes show spectral evolution with time constants of 3 ps, 27 ps, 200 ps, and 2 ns. Comparison of infrared (IR) difference spectra obtained for the isolated antennas CP43 and CP47 and the D1D2-RC with those measured for the PSII core allowed us to identify the features specific for each of the PSII core components. From the presence of the CP43 and CP47 specific features in the spectra up to time delays of 20-30 ps, we conclude that the main part of the energy transfer from the antennas to the RC occurs on this timescale. Direct excitation of the pigments in the RC evolution associated difference spectra to radical pair formation of PD1+PheoD1- on the same timescale as multi-excitation annihilation and excited state equilibration within the antennas CP43 and CP47, which occur within approximately 1-3 ps. The formation of the earlier radical pair ChlD1+PheoD1-, as identified in isolated D1D2 complexes with time-resolved mid-IR spectroscopy is not observed in the current data, probably because of its relatively low concentration. Relaxation of the state PD1+PheoD1-, caused by a drop in free energy, occurs in 200 ps in closed cores. We conclude that the kinetic model proposed earlier for the energy and electron transfer dynamics within the D1D2-RC, plus two slowly energy-transferring antennas C43 and CP47 explain the complex excited state and charge separation dynamics in the PSII core very well. We further show that the time-resolved IR-difference spectrum of PD1+PheoD1- as observed in PSII cores is virtually identical to that observed in the isolated D1D2-RC complex of PSII, demonstrating that the local structure of the primary reactants has remained intact in the isolated D1D2 complex.

Pigment spectra and intermolecular interaction potentials in glasses and proteins.

A model is proposed for chromophore optical spectra in solids over a wide range of temperatures and pressures. Inhomogeneous band shapes and their pressure dependence, as well as baric shift coefficients of spectral lines, selected by the frequency, were derived using Lennard-Jones potentials of the ground and excited states. Quadratic electron-phonon coupling constants, describing the thermal shift and broadening of zero-phonon lines, were also calculated. Experimentally, thermal shift and broadening of spectral holes were studied between 5 and 40 K for a synthetic pigment, chlorin, embedded in polymer hosts. The baric effects on holes were determined by applying hydrostatic He gas pressure up to 200 bar, at 6 K. Absorption spectra of pheophytin a, chlorophyll a, and beta-carotene in polymers and plant photosystem II CP47 complex were measured between 5 (or 77) and 300 K, and subject to Voigtian deconvolution. A narrowing of inhomogeneous bandwidth with increasing temperature, predicted on the basis of hole behavior, was observed as the shrinking of Gaussian spectral component. The Lorentzian broadening was ascribed to optical dephasing up to 300 K in transitions with weak to moderate linear electron-phonon coupling strength. The thermal broadening is purely Gaussian in multiphonon transitions (S(2) band of beta-carotene, Soret bands of tetrapyrrolic pigments), and the Lorentz process appears to be suppressed, indicating a lack of exponential dephasing. Density, polarity, polarizability, compressibility, and other local parameters of the pigment binding sites in biologically relevant systems can be deduced from spectroscopic data, provided that sufficient background information is available.

Excitation energy transfer in native and unstacked thylakoid membranes studied by low temperature and ultrafast fluorescence spectroscopy.

In this work, the transfer of excitation energy was studied in native and cation-depletion induced, unstacked thylakoid membranes of spinach by steady-state and time-resolved fluorescence spectroscopy. Fluorescence emission spectra at 5 K show an increase in photosystem I (PSI) emission upon unstacking, which suggests an increase of its antenna size. Fluorescence excitation measurements at 77 K indicate that the increase of PSI emission upon unstacking is caused both by a direct spillover from the photosystem II (PSII) core antenna and by a functional association of light-harvesting complex II (LHCII) to PSI, which is most likely caused by the formation of LHCII-LHCI-PSI supercomplexes. Time-resolved fluorescence measurements, both at room temperature and at 77 K, reveal differences in the fluorescence decay kinetics of stacked and unstacked membranes. Energy transfer between LHCII and PSI is observed to take place within 25 ps at room temperature and within 38 ps at 77 K, consistent with the formation of LHCII-LHCI-PSI supercomplexes. At the 150-160 ps timescale, both energy transfer from LHCII to PSI as well as spillover from the core antenna of PSII to PSI is shown to occur at 77 K. At room temperature the spillover and energy transfer to PSI is less clear at the 150 ps timescale, because these processes compete with charge separation in the PSII reaction center, which also takes place at a timescale of about 150 ps.

Ultrafast infrared spectroscopy reveals a key step for successful entry into the photocycle for photoactive yellow protein.

Photoactive proteins such as PYP (photoactive yellow protein) are generally accepted as model systems for studying protein signal state formation. PYP is a blue-light sensor from the bacterium Halorhodospira halophila. The formation of PYP's signaling state is initiated by trans-cis isomerization of the p-coumaric acid chromophore upon the absorption of light. The quantum yield of signaling state formation is approximately 0.3. Using femtosecond visible pump/mid-IR probe spectroscopy, we investigated the structure of the very short-lived ground state intermediate (GSI) that results from an unsuccessful attempt to enter the photocycle. This intermediate and the first stable GSI on pathway into the photocycle, I0, both have a mid-IR difference spectrum that is characteristic of a cis isomer, but only the I0 intermediate has a chromophore with a broken hydrogen bond with the backbone N atom of Cys-69. We suggest, therefore, that breaking this hydrogen bond is decisive for a successful entry into the photocycle. The chromophore also engages in a hydrogen-bonding network by means of its phenolate group with residues Tyr-42 and Glu-46. We have investigated the role of this hydrogen bond by exchanging the H bond-donating residue Glu-46 with the weaker H bond-donating glutamine (i.e., Gln-46). We have observed that this mutant exhibits virtually identical kinetics and product yields as WT PYP, even though during the I0-to-I1 transition, on the 800-ps time scale, the hydrogen bond of the chromophore with Gln-46 is broken, whereas this hydrogen bond remains intact with Glu-46.

How light-induced charge transfer accelerates the receptor-state recovery of photoactive yellow protein from its signaling state.

Stark (electroabsorption) spectra of the M100A mutant of photoactive yellow protein reveal that the neutral, cis cofactor of the pB intermediate undergoes a strikingly large change in the static dipole moment (|Deltamu| = 19 Debye) on photon absorption. The formation of this charge-separated species, in the excited state, precedes the cis --> trans isomerization of the pB cofactor and the regeneration of pG. The large |Deltamu|, reminiscent of that produced on the excitation of pG, we propose, induces twisting of the cis cofactor as a result of translocation of negative charge, from the hydroxyl oxygen, O1, toward the C7-C8 double bond. The biological significance of this photoinduced charge transfer reaction underlies the significantly faster regeneration of pG from pB in vitro, on the absorption of blue light.

Excitation energy transfer pathways in Lhca4.

EET in reconstituted Lhca4, a peripheral light-harvesting complex from Photosystem I of Arabidopsis thaliana, containing 10 chlorophylls and 2 carotenoids, was studied at room temperature by femtosecond transient absorption spectroscopy. Two spectral forms of Lut were observed in the sites L1 and L2, characterized by significantly different interactions with nearby chlorophyll a molecules. A favorable interpretation of these differences is that the efficiency of EET to Chls is about two times lower from the "blue" Lut in the site L1 than from the "red" Lut in the site L2 due to fast IC in the former case. A major part of the energy absorbed by the "red" Lut, approximately 60%-70%, is transferred to Chls on a sub-100-fs timescale from the state S(2) but, in addition, minor EET from the hot S(1) state within 400-500 fs is also observed. EET from the S(1) state to chlorophylls occurs also within 2-3 ps and is ascribed to Vio and/or "blue" Lut. EET from Chl b to Chl a is biphasic and characterized by time constants of approximately 300 fs and 3.0 ps. These rates are ascribed to EET from Chl b spectral forms absorbing at approximately 644 nm and approximately 650 nm, respectively. About 25% of the excited Chls a decays very fast-within approximately 15 ps. This decay is proposed to be related to the presence of the interacting Chls A5 and B5 located next to the carotenoid in the site L2 and may imply some photoprotective role for Lhca4 in the photosystem I super-complex.

Circular dichroism of carotenoids in bacterial light-harvesting complexes: experiments and modeling.

In this work we investigate the origin and characteristics of the circular dichroism (CD) spectrum of rhodopin glucoside and lycopene in the light-harvesting 2 complex of Rhodopseudomonas acidophila and Rhodospirillum molischianum, respectively. We successfully model their absorption and CD spectra based on the high-resolution structures. We assume that these spectra originate from seven interacting transition dipole moments: the first corresponds to the 0-0 transition of the carotenoid, whereas the remaining six represent higher vibronic components of the S2 state. From the absorption spectra we get an estimate of the Franck-Condon factors of these transitions. Furthermore, we investigate the broadening mechanisms that lead to the final shape of the spectra and get an insight into the interaction energy between carotenoids. Finally, we examine the consequences of rotations of the carotenoid transition dipole moment and of deformations in the light-harvesting 2 complex rings. Comparison of the modeled carotenoid spectra with modeled spectra of the bacteriochlorophyll QY region leads to a refinement of the modeling procedure and an improvement of all calculated results. We therefore propose that the combined carotenoid and bacteriochlorophyll CD can be used as an accurate reflection of the overall structure of the light-harvesting complexes.

Energy and electron transfer in photosystem II reaction centers with modified pheophytin composition.

Energy and electron transfer in Photosystem II reaction centers in which the photochemically inactive pheophytin had been replaced by 13(1)-deoxo-13(1)-hydroxy pheophytin were studied by femtosecond transient absorption-difference spectroscopy at 77 K and compared to the dynamics in untreated reaction center preparations. Spectral changes induced by 683-nm excitation were recorded both in the Q(Y) and in the Q(X) absorption regions. The data could be described by a biphasic charge separation. In untreated reaction centers the major component had a time constant of 3.1 ps and the minor component 33 ps. After exchange, time constants of 0.8 and 22 ps were observed. The acceleration of the fast phase is attributed in part to the redistribution of electronic transitions of the six central chlorin pigments induced by replacement of the inactive pheophytin. In the modified reaction centers, excitation of the lowest energy Q(Y) transition produces an excited state that appears to be localized mainly on the accessory chlorophyll in the active branch (B(A) in bacterial terms) and partially on the active pheophytin H(A). This state equilibrates in 0.8 ps with the radical pair. B(A) is proposed to act as the primary electron donor also in untreated reaction centers. The 22-ps (pheophytin-exchanged) or 33-ps (untreated) component may be due to equilibration with the secondary radical pair. Its acceleration by H(B) exchange is attributed to a faster reverse electron transfer from B(A) to. After exchange both and are nearly isoenergetic with the excited state.

Stark spectroscopy on photoactive yellow protein, E46Q, and a nonisomerizing derivative, probes photo-induced charge motion.

The change in the electrostatic properties on excitation of the cofactor of wild-type photoactive yellow protein (WT-PYP) have been directly determined using Stark-effect spectroscopy. We find that, instantaneously on photon absorption, there is a large change in the permanent dipole moment, /Delta(-->)mu/, (26 Debye) and in the polarizability, (-)Deltaalpha, (1000 A(3)). We expect such a large degree of charge motion to have a significant impact on the photocycle that is associated with the important blue-light negative phototactic response of Halorhodospira halophila. Furthermore, changing E46 to Q in WT-PYP does not significantly alter its electrostatic properties, whereas, altering the chromophore to prevent it from undergoing trans-cis isomerization results in a significant diminution of /Delta(-->)mu/ and (-)Deltaalpha. We propose that the enormous charge motion that occurs on excitation of 4-hydroxycinnamyl thioester, the chromophore in WT-PYP, plays a crucial role in initiating the photocycle by translocation of the negative charge, localized on the phenolate oxygen in the ground state, across the chromophore. We hypothesize that this charge motion would consequently increase the flexibility of the thioester tail thereby decreasing the activation barrier for the rotation of this moiety in the excited state.

Dynamics of excitation energy transfer in the LH1 and LH2 light-harvesting complexes of photosynthetic bacteria.

Photosynthetic light harvesting is a unique life process that occurs with amazing efficiency. Since the discovery of the structure of the bacterial peripheral light-harvesting complex (LH2), this process has been studied using a variety of advanced laser spectroscopic methods. We are now in a position to discuss the physical origins of excitation energy transfer and trapping in the LH2 and LH1 antennae of photosynthetic purple bacteria. We demonstrate that the time evolution of the state created by the light is determined by the combined action of excitonic pigment-pitment interactions, energetic disorder, and coupling to nuclear motion in a pigment-protein complex. A quantitative fit of experimental data using Redfield theory allowed us to determine the pathways and time scales of exciton and vibrational relaxation and analyze separately different contributions to the measured transient absorption dynamics. Furthermore, these dynamics were observed to be strongly dependent on the excitation wavelength. A numerical fit of this dependence turns out to be extremely critical to a variation of the structure and disorder parameters and, therefore, can be used as a test for different antenna models (disordered ring, elliptical deformations, correlated disorder, etc.). The calculated equilibration dynamics in the exciton basis allow a visualization of the exciton motion using a density matrix picture in real space.

Oligomerization of light-harvesting I antenna peptides of Rhodospirillum rubrum.

We investigated the oligomerization of the core light-harvesting complex (LH1) of Rhodospirillum rubrum from the separated alpha beta BChl(2) subunits (B820) and the oligomerization of the B820 subunit from its monomeric peptides. The full LH1 complex was reversibly associated from B820 subunits by either varying the temperature in the range 277-300 K or by varying the detergent concentration in the buffer from 0.36 to 0.52% n-octyl-beta-D-glucopyranoside. Temperature-induced transition measurements showed hysteresis: raising the temperature induced dissociation of B873 directly into B820 subunits whereas upon recooling an intermediate spectral form was observed with an absorption maximum located around 850 nm. This intermediate form was also observed in detergent-induced transitions. It is speculated that the B850 form is a small aggregate of B820, for instance a dimer. Additionally, during a temperature-mediated transition at low detergent concentration, a set of spectral forms with maxima slightly blue-shifted from 873 nm were observed, possibly due to opened rings with one or only a few alpha beta BChl(2) units missing. The temperature-induced transition of LH1 is discussed in terms of a simple assembly model. It is concluded that a moderately cooperative assembly explains the formation of small aggregates of B820 as well as of incomplete rings. Furthermore, the B820 subunits were reversibly dissociated into the monomeric B777 form by increasing either the temperature or the detergent concentration. Estimations of the enthalpy and entropy changes for the dimeric association reaction of B777 into B820 yielded an enthalpy change of -216 kJ mol(-1) and an entropy change of -0.59 kJ mol(-1)K(-1), at a detergent concentration of 0.8% n-octyl-beta-D-glucopyranoside.

Probing the structure and dynamics of a DNA hairpin by ultrafast quenching and fluorescence depolarization.

DNA hairpins have been investigated in which individual adenines were replaced by their fluorescent analog 2-aminopurine (2AP). The temperature dependence of the time evolution of polarized emission spectra was monitored with picosecond time resolution. Four isotropic decay components for each oligonucleotide indicated the coexistence of at least four conformations. The fluorescence for three of these was significantly quenched, which is explained by hole transfer from 2AP to guanine(s). An approximately 8-ps component is ascribed to direct hole transfer, the approximately 50-ps and approximately 500-ps components are ascribed to structural reorganization, preceding hole transfer. At room temperature, a fraction remains unquenched on a 10-ns timescale, in contrast to higher temperatures, where the flexibility increases. Besides quenching due to base stacking, a second quenching process was needed to describe the data. Evidence for both intrastrand and interstrand hole transfer was found. The extracted probability for stacking between neighboring bases in double-stranded regions was estimated to be approximately 75% at room temperature and approximately 25% at 80 degrees C, demonstrating structural disorder of the DNA. Fluorescence depolarization revealed both local dynamics of the DNA and overall dynamics of the entire oligonucleotide. Upon raising the temperature, the C-N terminus of the hairpin appears to melt first; the rest of the hairpin denatures above the average melting temperature.

Time-resolved fluorescence emission measurements of photosystem I particles of various cyanobacteria: a unified compartmental model.

Photosystem I (PS-I) contains a small fraction of chlorophylls (Chls) that absorb at wavelengths longer than the primary electron donor P700. The total number of these long wavelength Chls and their spectral distribution are strongly species dependent. In this contribution we present room temperature time-resolved fluorescence data of five PS-I core complexes that contain different amounts of these long wavelength Chls, i.e., monomeric and trimeric photosystem I particles of the cyanobacteria Synechocystis sp. PCC 6803, Synechococcus elongatus, and Spirulina platensis, which were obtained using a synchroscan streak camera. Global analysis of the data reveals considerable differences between the equilibration components (3.4-15 ps) and trapping components (23-50 ps) of the various PS-I complexes. We show that a relatively simple compartmental model can be used to reproduce all of the observed kinetics and demonstrate that the large kinetic differences are purely the result of differences in the long wavelength Chl content. This procedure not only offers rate constants of energy transfer between and of trapping from the compartments, but also well-defined room temperature emission spectra of the individual Chl pools. A pool of red shifted Chls absorbing around 702 nm and emitting around 712 nm was found to be a common feature of all studied PS-I particles. These red shifted Chls were found to be located neither very close to P700 nor very remote from P700. In Synechococcus trimeric and Spirulina monomeric PS-I cores, a second pool of red Chls was present which absorbs around 708 nm, and emits around 721 nm. In Spirulina trimeric PS-I cores an even more red shifted second pool of red Chls was found, absorbing around 715 nm and emitting at 730 nm.

Fluorescence lifetime heterogeneity in aggregates of LHCII revealed by time-resolved microscopy.

Two-photon excitation, time-resolved fluorescence microscopy was used to investigate the fluorescence quenching mechanisms in aggregates of light-harvesting chlorophyll a/b pigment protein complexes of photosystem II from green plants (LHCII). Time-gated microscopy images show the presence of large heterogeneity in fluorescence lifetimes not only for different LHCII aggregates, but also within a single aggregate. Thus, the fluorescence decay traces obtained from macroscopic measurements reflect an average over a large distribution of local fluorescence kinetics. This opens the possibility to resolve spatially different structural/functional units in chloroplasts and other heterogeneous photosynthetic systems in vivo, and gives the opportunity to investigate individually the excited states dynamics of each unit. We show that the lifetime distribution is sensitive to the concentration of quenchers contained in the system. Triplets, which are generated at high pulse repetition rates of excitation (>1 MHz), preferentially quench domains with initially shorter fluorescence lifetimes. This proves our previous prediction from singlet-singlet annihilation investigations (Barzda, V., V. Gulbinas, R. Kananavicius, V. Cervinskas, H. van Amerongen, R. van Grondelle, and L. Valkunas. 2001. Biophys. J. 80:2409-2421) that shorter fluorescence lifetimes originate from larger domains in LHCII aggregates. We found that singlet-singlet annihilation has a strong effect in time-resolved fluorescence microscopy of connective systems and has to be taken into consideration. Despite that, clear differences in fluorescence decays can be detected that can also qualitatively be understood.

Energy transfer in the peridinin chlorophyll-a protein of Amphidinium carterae studied by polarized transient absorption and target analysis.

The peridinin chlorophyll-a protein (PCP) of dinoflagellates differs from the well-studied light-harvesting complexes of purple bacteria and green plants in its large (4:1) carotenoid to chlorophyll ratio and the unusual properties of its primary pigment, the carotenoid peridinin. We utilized ultrafast polarized transient absorption spectroscopy to examine the flow of energy in PCP after initial excitation into the strongly allowed peridinin S2 state. Global and target analysis of the isotropic and anisotropic decays reveals that significant excitation (25-50%) is transferred to chlorophyll-a directly from the peridinin S2 state. Because of overlapping positive and negative features, this pathway was unseen in earlier single-wavelength experiments. In addition, the anisotropy remains constant and high in the peridinin population, indicating that energy transfer from peridinin to peridinin represents a minor or negligible pathway. The carotenoids are also coupled directly to chlorophyll-a via a low-lying singlet state S1 or the recently identified SCT. We model this energy transfer time scale as 2.3 +/- 0.2 ps, driven by a coupling of approximately 47 cm(-1). This coupling strength allows us to estimate that the peridinin S1/SCT donor state transition moment is approximately 3 D.

Singlet-singlet annihilation kinetics in aggregates and trimers of LHCII.

Singlet-singlet annihilation experiments have been performed on trimeric and aggregated light-harvesting complex II (LHCII) using picosecond spectroscopy to study spatial equilibration times in LHCII preparations, complementing the large amount of data on spectral equilibration available in literature. The annihilation kinetics for trimers can well be described by a statistical approach, and an annihilation rate of (24 ps)(-1) is obtained. In contrast, the annihilation kinetics for aggregates can well be described by a kinetic approach over many hundreds of picoseconds, and it is shown that there is no clear distinction between inter- and intratrimer transfer of excitation energy. With this approach, an annihilation rate of (16 ps)(-1) is obtained after normalization of the annihilation rate per trimer. It is shown that the spatial equilibration in trimeric LHCII between chlorophyll a molecules occurs on a time scale that is an order of magnitude longer than in Photosystem I-core, after correcting for the different number of chlorophyll a molecules in both systems. The slow transfer in LHCII is possibly an important factor in determining excitation trapping in Photosystem II, because it contributes significantly to the overall trapping time.