single molecule fret experiments were performed using a confocal microscope system ( microtime200 , picoquant ) . the cw mode of a dual mode ( cw / pulsed ) 485 nm diode laser ( ldh - d - c-485 , picoquant ) was used to excite donor dyes through an oil - immersion objective ( planapo , na 1.4 , 100 , olympus ) . donor and acceptor fluorescence was collected by the same objective , split into two channels , and focused through optical filters ( et525/50 m for the donor and e600lp for the acceptor , chroma technology ) onto photon - counting avalanche photodiodes ( spcm - aqr-15 , perkinelmer optoelectronics ) . protein molecules were immobilized on a biotin - embedded , polyethyleneglycol - coated glass coverslip ( bio_01 , microsurfaces inc . ) via a biotin ( surface)-streptavidin - biotin ( protein ) linkage . to reduce dye bleaching and blinking , 1  2 mm of l - ascorbic acid ( a92902 , sigma ) and methyl viologen ( 856117 , sigma ) were added to the 50 mm hepes buffer ( ph 7.6 ) solution for the temperature - dependence experiment . for the experiments in 50% and > 50% glycerol , higher concentrations of 10 mm and 40 mm ascorbic acid / methyl viologen were used , respectively , to increase the bimolecular rates that reduce blinking and bleaching . to collect a large number of trajectories , we used an automated data collection scheme as described in ref . . temperature was controlled by flowing dry nitrogen cooled by liquid nitrogen into a chamber surrounding the microscope objective and sample with a fixed flow rate . the temperature was varied by changing the flow rate and was measured at the confocal spot using the fluorescence lifetime of rhodamine b , excited by a 485 nm laser in the pulsed mode , using the calibration equation : the relative viscosities ( /0 ) of the solutions were obtained from the absolute temperature ( t in k ) and the translational diffusion times (  ) using the relationship t . the translational diffusion times were measured by fluorescence correlation spectroscopy ( fcs ) experiments on an alexa fluor 488- and alexa fluor 594-labeled rigid rod molecule , 20-residue poly - proline , as described previously . to further test the assumption of a two - state model in our data analysis using the maximum likelihood method ( see below ) , we calculated the donor - acceptor cross - correlation function , as cda( ) = -a exp(k ) , as shown in extended data fig . 4 and compared its decay rate with the sum of the rate coefficients determined by the maximum likelihood method ( extended data table 1 ) . to obtain the timescale of the unfolded state dynamics at high viscosity , the donor - acceptor cross - correlation function was calculated ( extended data fig . 6 ) for the photon trajectories in the unfolded segments that were extracted using viterbi algorithm . to determine parameters for the two - state kinetics , we used the gopich - szabo maximum likelihood method and analyzed the photon trajectories without time binning . this method yields the most likely parameters of an assumed model that can reproduce the observed photon trajectories . the likelihood function for the j photon trajectory is . where nj is the number of photons in a trajectory , ci is the color of the i photon ( donor or acceptor ) , and i is a time interval between the i and ( i-1 ) photons ( extended data fig . 2a ) . the photon color matrix f depends on the color of a photon as f(acceptor ) = e and f(donor ) = i  e , where e is a diagonal matrix with elements that are fret efficiencies of the individual states . i is the identity matrix , peq is a vector consisting of the equilibrium population of each state , and 1 is a row vector with elements of 1 . the calculation of likelihood values was performed using the diagonalization of the matrix exponential in equation ( s1 ) as described in ref .. practically , the log likelihood function was calculated and the total log likelihood function of all trajectories was calculated by summing individual log likelihood functions as 
 lnl=jlnlj . for a two - state system , there are four independent parameters - the apparent fret efficiencies and the rate coefficients of the folded and unfolded states ( ef , eu , kf , ku ) . in analyzing the data , instead of varying the two rate coefficients , it is more convenient to vary the sum of the rate coefficients ( k = kf + ku ) and the fractional population of folded molecules ( pf = kf/[kf + ku ] ) . 2b ) , the rate matrix is given by the 4 parameters ( ef , eu , kf , ku ) can be obtained by maximizing the log likelihood ln l. in the measurement of the average transition - path time , ttp , molecules were illuminated at 10  15 times higher laser intensity ( 20  30 kw / cm ) than the intensity used for the determination of two - state kinetics parameters above to collect photons at high photon count rates of 550  710 ms . this high photon count rate permitted clear resolution of transitions by collecting photons in 50 - 100 s bins . the photon trajectory in the short segment near the transition (  1 ms ) was then analyzed using the likelihood function . we adopted a three state model with a virtual intermediate (  step  ) state s in addition to the folded ( f ) and unfolded ( u ) states , which is the simplest discrete representation of how the fret efficiency changes along the transition path . in this model , the fret efficiency of s is midway between the folded and unfolded states ( es = ( ef + eu)/2 ) and the lifetime of s ( s ) corresponds to the average transition - path time , ttp . 2c ) is f and u are used for the folded and unfolded states to distinguish these from those in the two state model with an instantaneous transition . in this analysis ttp = s = 1/2ks , and ttp is the same for folding and unfolding transitions , which is also the case for the real transition path in fig . 1(b ) regardless of the relative stability of the folded and unfolded states as long as qu and qf are the same for the two paths . the likelihood function of this model for the j photon trajectory with a single transition is   where vini and vfin are state vectors at the beginning and the end of the trajectory . for the case of a folding transition in extended data fig . 2a , vini = ( 0 0 1 ) and vfin = ( 1 0 0 ) . in the analysis , the folded and unfolded segments were assigned using the viterbi algorithm , adapted for photon trajectories , and kf and ku were obtained from the low intensity measurements ( extended data table 1 ) . when there are multiple transitions in a trajectory , each pair of segments with a single folding or unfolding transition was analyzed separately . in the calculation of the likelihood function in equation ( s4 ) for these segments , we reduced the rate coefficients by a factor of 1000 to effectively eliminate the possibility of multiple transitions that are not resolvable , i.e. ku (= ku/1000 ) and kf (= kf/1000 ) . this treatment is valid since we use the difference of the log likelihood values ( ln l ) in this paper . the reader is referred to previous publications for an in - depth understanding of the gopich - szabo maximum likelihood method and its experimental application to a two - state protein system and for the transition path analysis . 3d ) using the ratio of the folding time to the transition path time tf / ttp , and the ratio of the curvatures at the barrier top and the bottom of the unfolded state ( */u ) . the value of */u was evaluated for the measured tf / ttp , 200 ( continuous red curve ) , including its error , i.e. 233 ( upper red dashed curve ) , and 167 ( lower red dashed curve ) . the most probable value of */u obtained from the md simulations is 1.3 , the average of 0.94 ( using the fraction of native contacts as the reaction coordinate ( r. b. best . , g. hummer , and w.a . usa ) and 1.6 ( using an optimized reaction coordinate ) , corresponding to gf * = 4.2 . assuming an uncertainty of a factor of 2 for */u , the possible range of gf * becomes 3.2 to 5.1 for */u between 0.65 and 2.6 . using the reported times from the md simulations of tf = 27  8 s , and ttp = 0.9  0.2 s , and */u = 1.6 from the one - dimensional free energy surface constructed with an optimized reaction coordinate , the ratio of equations ( 1 ) and ( 2 ) yields a barrier height of 1.7  0.6 kbt , which is close to the 2.1 kbt ( no error reported ) barrier height for this surface . using the fraction of native contacts as the reaction coordinate for all 3 quantities , the corresponding times are tf= 26.4  8.8 s , ttp = 0.83  0.17 s ( boundaries for the transition path are the two well minima ) , and */u = 0.94  0.18 , the calculated barrier height is 2.5  0.6 kbt , compared to the barrier height of 2.4  0.3 kbt ( r.b . best , unpublished results ) . single molecule fret experiments were performed using a confocal microscope system ( microtime200 , picoquant ) . the cw mode of a dual mode ( cw / pulsed ) 485 nm diode laser ( ldh - d - c-485 , picoquant ) was used to excite donor dyes through an oil - immersion objective ( planapo , na 1.4 , 100 , olympus ) . donor and acceptor fluorescence was collected by the same objective , split into two channels , and focused through optical filters ( et525/50 m for the donor and e600lp for the acceptor , chroma technology ) onto photon - counting avalanche photodiodes ( spcm - aqr-15 , perkinelmer optoelectronics ) . protein molecules were immobilized on a biotin - embedded , polyethyleneglycol - coated glass coverslip ( bio_01 , microsurfaces inc . ) via a biotin ( surface)-streptavidin - biotin ( protein ) linkage . to reduce dye bleaching and blinking , 1  2 mm of l - ascorbic acid ( a92902 , sigma ) and methyl viologen ( 856117 , sigma ) were added to the 50 mm hepes buffer ( ph 7.6 ) solution for the temperature - dependence experiment . for the experiments in 50% and > 50% glycerol , higher concentrations of 10 mm and 40 mm ascorbic acid / methyl viologen were used , respectively , to increase the bimolecular rates that reduce blinking and bleaching . to collect a large number of trajectories temperature was controlled by flowing dry nitrogen cooled by liquid nitrogen into a chamber surrounding the microscope objective and sample with a fixed flow rate . the temperature was varied by changing the flow rate and was measured at the confocal spot using the fluorescence lifetime of rhodamine b , excited by a 485 nm laser in the pulsed mode , using the calibration equation : the relative viscosities ( /0 ) of the solutions were obtained from the absolute temperature ( t in k ) and the translational diffusion times (  ) using the relationship t . the translational diffusion times were measured by fluorescence correlation spectroscopy ( fcs ) experiments on an alexa fluor 488- and alexa fluor 594-labeled rigid rod molecule , 20-residue poly - proline , as described previously . to further test the assumption of a two - state model in our data analysis using the maximum likelihood method ( see below ) , we calculated the donor - acceptor cross - correlation function , as cda( ) = -a exp(k ) , as shown in extended data fig . 4 and compared its decay rate with the sum of the rate coefficients determined by the maximum likelihood method ( extended data table 1 ) . to obtain the timescale of the unfolded state dynamics at high viscosity , the donor - acceptor cross - correlation function was calculated ( extended data fig . 6 ) for the photon trajectories in the unfolded segments that were extracted using viterbi algorithm . to determine parameters for the two - state kinetics , we used the gopich - szabo maximum likelihood method and analyzed the photon trajectories without time binning . this method yields the most likely parameters of an assumed model that can reproduce the observed photon trajectories . the likelihood function for the j photon trajectory is . where nj is the number of photons in a trajectory , ci is the color of the i photon ( donor or acceptor ) , and i is a time interval between the i and ( i-1 ) photons ( extended data fig . the photon color matrix f depends on the color of a photon as f(acceptor ) = e and f(donor ) = i  e , where e is a diagonal matrix with elements that are fret efficiencies of the individual states . i is the identity matrix , peq is a vector consisting of the equilibrium population of each state , and 1 is a row vector with elements of 1 . the calculation of likelihood values was performed using the diagonalization of the matrix exponential in equation ( s1 ) as described in ref .. practically , the log likelihood function was calculated and the total log likelihood function of all trajectories was calculated by summing individual log likelihood functions as 
 lnl=jlnlj . for a two - state system , there are four independent parameters - the apparent fret efficiencies and the rate coefficients of the folded and unfolded states ( ef , eu , kf , ku ) . in analyzing the data , instead of varying the two rate coefficients , it is more convenient to vary the sum of the rate coefficients ( k = kf + ku ) and the fractional population of folded molecules ( pf = kf/[kf + ku ] ) . 2b ) , the rate matrix is given by the 4 parameters ( ef , eu , kf , ku ) can be obtained by maximizing the log likelihood ln l. in the measurement of the average transition - path time , ttp , molecules were illuminated at 10  15 times higher laser intensity ( 20  30 kw / cm ) than the intensity used for the determination of two - state kinetics parameters above to collect photons at high photon count rates of 550  710 ms . this high photon count rate permitted clear resolution of transitions by collecting photons in 50 - 100 s bins . the photon trajectory in the short segment near the transition (  1 ms ) was then analyzed using the likelihood function . we adopted a three state model with a virtual intermediate (  step  ) state s in addition to the folded ( f ) and unfolded ( u ) states , which is the simplest discrete representation of how the fret efficiency changes along the transition path . in this model , the fret efficiency of s is midway between the folded and unfolded states ( es = ( ef + eu)/2 ) and the lifetime of s ( s ) corresponds to the average transition - path time , ttp . 2c ) is f and u are used for the folded and unfolded states to distinguish these from those in the two state model with an instantaneous transition . in this analysis ttp = s = 1/2ks , and ttp is the same for folding and unfolding transitions , which is also the case for the real transition path in fig . 1(b ) regardless of the relative stability of the folded and unfolded states as long as qu and qf are the same for the two paths . the likelihood function of this model for the j photon trajectory with a single transition is   where vini and vfin are state vectors at the beginning and the end of the trajectory . for the case of a folding transition in extended data fig . 2a , vini = ( 0 0 1 ) and vfin = ( 1 0 0 ) . in the analysis , the folded and unfolded segments were assigned using the viterbi algorithm , adapted for photon trajectories , and kf and ku were obtained from the low intensity measurements ( extended data table 1 ) . when there are multiple transitions in a trajectory , each pair of segments with a single folding or unfolding transition was analyzed separately . in the calculation of the likelihood function in equation ( s4 ) for these segments , we reduced the rate coefficients by a factor of 1000 to effectively eliminate the possibility of multiple transitions that are not resolvable , i.e. ku (= ku/1000 ) and kf (= kf/1000 ) . this treatment is valid since we use the difference of the log likelihood values ( ln l ) in this paper . the reader is referred to previous publications for an in - depth understanding of the gopich - szabo maximum likelihood method and its experimental application to a two - state protein system and for the transition path analysis . 3d ) using the ratio of the folding time to the transition path time tf / ttp , and the ratio of the curvatures at the barrier top and the bottom of the unfolded state ( */u ) . the value of */u was evaluated for the measured tf / ttp , 200 ( continuous red curve ) , including its error , i.e. 233 ( upper red dashed curve ) , and 167 ( lower red dashed curve ) . the most probable value of */u obtained from the md simulations is 1.3 , the average of 0.94 ( using the fraction of native contacts as the reaction coordinate ( r. b. best . , g. hummer , and w.a . usa ) and 1.6 ( using an optimized reaction coordinate ) , corresponding to gf * = 4.2 . assuming an uncertainty of a factor of 2 for */u , the possible range of gf * becomes 3.2 to 5.1 for */u between 0.65 and 2.6 . using the reported times from the md simulations of tf = 27  8 s , and ttp = 0.9  0.2 s , and */u = 1.6 from the one - dimensional free energy surface constructed with an optimized reaction coordinate , the ratio of equations ( 1 ) and ( 2 ) yields a barrier height of 1.7  0.6 kbt , which is close to the 2.1 kbt ( no error reported ) barrier height for this surface . using the fraction of native contacts as the reaction coordinate for all 3 quantities , the corresponding times are tf= 26.4  8.8 s , ttp = 0.83  0.17 s ( boundaries for the transition path are the two well minima ) , and */u = 0.94  0.18 , the calculated barrier height is 2.5  0.6 kbt , compared to the barrier height of 2.4  0.3 kbt ( r.b .
kramers developed the theory on how chemical reaction rates are influenced by the viscosity of the medium1,2 . at the viscosity of water , 
 the kinetics of unimolecular reactions are described by diffusion of a brownian particle over a free - energy barrier separating reactants and products . for reactions in solution 
 this famous theory extended eyring 's transition state theory , and is widely applied in physics , chemistry , and biology , including reactions as complex as protein folding3,4 . 
 because the diffusion coefficient of kramers theory is determined by the dynamics in the sparsely - populated region of the barrier top , its properties have not been directly measured for any molecular system . here 
 we show that the kramers diffusion coefficient and free energy barrier can be characterized by measuring the temperature- and viscosity - dependence of the transition path time for protein folding . 
 the transition path is the small fraction of an equilibrium trajectory for a single molecule when the free - energy barrier separating two states is actually crossed ( fig . 
 1a ) . 
 its duration , the transition path time , can now be determined from photon trajectories for single protein molecules undergoing folding / unfolding transitions5 . 
 our finding of a long transition path time with an unusually small solvent viscosity - dependence suggests that internal friction as well as solvent friction determine the kramers diffusion coefficient for -helical proteins , as opposed to a breakdown of his theory that occurs for many small - molecule reactions2 . 
 it is noteworthy that the new and fundamental information concerning kramers theory and the dynamics of barrier crossings obtained here come from experiments on a protein rather than a much simpler chemical or physical system .