the problem of protein folding kinetics is formidable from a purely statistical mechanics point of view : the unfolded protein , in other words the entire ensemble of micro - states that significantly deviate from the native state , transits via a myriad of distinct pathways to the folded ( native ) state , and trying to predict folding times from basic principles is obviously hopeless . yet, robust features have emerged both from experiments and theoretical concepts  .a key fact is that any experiment that probes protein folding or unfolding projects protein micro - states onto a low - dimensional ( typically one - dimensional ) observable .for example , circular dichroism in the far ultraviolet and infrared adsorption spectroscopy basically measure the average helicity , while fluorescence is sensitive to side chain contacts or local solvent structure around tryptophan residues  .kinetic information at ambient conditions and on short time scales relevant for fast folding events can be obtained by time - resolved spectroscopy after flash photoheating  or by fret and ttet correlation studies that couple to the distance between a donor and acceptor linked to two positions along the peptide chain  . more recently, single - molecule spectroscopic techniques have allowed the observation of time - dependent folding / unfolding of individual proteins , thus going beyond ensemble averaging  .likewise , single molecule studies where forces are applied at two points along the peptide backbone probe the distance between those two anchoring points  .all these experimental observables in fact constitute distinct _ reaction coordinates _ ( rcs ) .exponential distributions of folding times found for many ( but not all ) proteins using different techniques suggest two - state - folding as a quite general paradigm of folding kinetics : here the folded and unfolded states are separated by a free energy barrier along the respective rc  .even proteins folding via many intermediate states can produce a single exponential folding time if there exists a rate - limiting transition .therefore , as long as the reaction coordinate of choice distinguishes the two states connected by the rate - limiting step , using different kinds of measurement / reaction coordinate would likely generate similar single - exponential kinetics even in such a case .similar conclusions can been drawn from the direct observation of population distributions , where a free energy barrier means that folding intermediates are rarely observed  .the recent observation that different experimental techniques yield different kinetics  or distribution functions  when applied to the same protein casts doubt on the clear division between two - state ( exhibiting a free - energy barrier ) and down - hill folders ( without such a barrier ) . in this paperwe argue that such inconsistencies can arise when implicitly referring to different rcs , and show a way of how to reconcile conflicting results .in theoretical studies , various rcs have become popular to characterize the folding transition , either because they approximately correspond to an experimentally accessible observable or because they are simple to calculate .the radius of gyration , the fraction of native contacts between residues , or the mean distance from the native state are typical examples  .more complex topological order parameters such as the contact order have been suggested for describing universal features of protein folding kinetics  . in the theoretical framework that naturally emerges , the protein diffuses along the rc , governed by a stochastic equation and subject to deterministic forces encripted in the free energy landscape , as well as stochastic forces due to the random environment  .early on , it was realized that the diffusion constant in this coarse - grained picture is an effective quantity that takes into account the connectivity between states ( i.e. the number of possible connecting paths ) , the energetic ruggedness of such paths  , as well as orthogonal degrees of freedom  . as folding progresses ,internal friction starts to play a more dominating role  , while solvent friction becomes less important as more and more peptide groups lose solvent contact  .recently , the simplification of a constant diffusivity was abandoned and a diffusivity profile was extracted from simulations of peptides : these works either considered proteins without solvent ( and thus exclude variations of the solvent friction )  or considered exclusively short - time dynamics and thus are not applicable to global folding kinetics  . the trifold coupling between the choice of a specific rc and the free energy and diffusivity profiles in the presence of explicit solvent has remained elusive .  in this paperwe perform an in - depth analysis of long md trajectories of an -helix forming oligo - peptide including explicit water .such model peptides form the subject of detailed experimental studies and constitute some of the simplest peptides that exhibit non - trivial folding kinetics  .they are thus interesting in their own right and at the same time due to their minute size allow for realistic modelling over times much longer than their folding times , including solvent degrees of freedom  .as a prerequisite for our analysis , we introduce a simple way of extracting diffusivity profiles from time series data for an arbitrary rc , that can be conveniently applied to experimental spectroscopic data  , or force spectroscopic data for rna  , or proteins  as well .we demonstrate that different rcs for one and the same protein trajectory are associated with substantially different free energy profiles , some showing a barrier separating the folded and unfolded helix state , some showing no barrier at all ( which is not surprising and has been found in different contexts before  ) .this resembles the experimental findings in connection with the dispute on down - hill versus two - state folding  , but is resolved by accounting for the spatially inhomogeneous diffusivity : the diffusivity profiles are full of structure and show considerable variation among different rcs .no simple connection between the free energy and diffusivity profiles seems to exist .yet , the folding kinetics predicted using a stochastic approach based on the free energy landscape is largely independent of the rc if and only if the diffusivity profile is taken into account .thus , the variance between free energy profiles along different rcs gives rise to kinetic universality if the coupling to diffusivity is included ( where we distinguish between reaction coordinates that contain knowledge of the native state and those that are purely geometric in nature ) .this specifically means that the presence of a free energy barrier ( i.e. absence of intermediate states ) is in principle compatible with both exponential and non - exponential kinetics , and that different experimental probes are bound to measure different free energy profiles .the same conclusions also apply to more refined or optimized rcs  .full understanding of protein folding kinetics thus requires measuring both average distributions and kinetic trajectories .similar conclusions were very recently drawn from a bayesian analysis of folding trajectories of simple coarse - grained model peptides based on implicit - solvent simulations  . since -helices are a prominent folding motif , the features we find are most likely relevant for more complex proteins as well .* simulations - * standard all - atom md simulations provide 1.1 trajectories of an alanine ( a)-based peptide with sequence ace - aeaaakeaaaka - nme in explicit water  , which is a shortened version of similar sequences with charged glu ( e ) and lys ( k ) residues at positions and that experimentally are known to spontaneously form -helices  .the mechanism for -helix formation involves , in addition to the stabilizing influence of e - k salt bridges , hydration effects  .the md simulations utilize the parallel module sander.mpi in the amber 9.0 package with the ff03 force - field and the tip3p water model at a pressure of 1 bar and a temperature fixed by a berendsen barostat and langevin thermostat , respectively  .the periodically repeated cubic simulation box has an edge length  including 1500 water molecules .electrostatic interactions are calculated by particle mesh ewald summation and real - space electrostatic and van der waals interactions are cut off at 9 . as a check on the convergence of the standard md simulation ,replica - exchange md ( remd ) simulations are performed with the amber10 simulation package  . here the same force - field and system parameters as in the other standard md simulations are employed , apart from switching to a constant volume ensemble .32 replicas are considered in a temperature range between 265 and 520 k , with each replica simulated for 22.5 ns , amounting to a total sampling time of 720 ns .temperature exchanges between neighboring replicas are attempted every 250 integration steps , leading to an exchange rate of 10 - 30% .+  .list of reaction coordinates ( rcs ) used in the paper . [ cols="^,^",options="header " , ]   [ rcdefinition ]  * reaction coordinates - * trajectory analysis is performed using the _ ptraj _ tool in the amber package . the helicity ( i.e. , the -helical fraction ) is identified using the dssp method by kabsch and sander  .in addition , we focus on five different rcs to follow the folding kinetics :  \(i ) , defined as the root - mean - square distance from a fully helical reference structure , averaged over all atoms of the peptide . the reference structure was chosen randomly from configurations which display 100% helicity , with little variation depending on the specific choice .\(ii ) the mean native hydrogen bond ( hb ) length , , averaged over all residues including the acetyl ( ace ) and amine ( nme ) end caps , where is the distance between hb forming atoms , and , in the peptide backbone .\(iii ) the mean inverse hb length , , where  is the native hb length in the folded state , defined by the most probable length of each hb .\iv ) the radius of gyration , ^{1/2}$ ] , a measure for the average peptide size and accessible in scattering .\v ) , the distance between the centres of mass of the end caps .trajectories are recorded with a resolution of 20 ps , giving a total of 54171 data points . to compare different rcs with each other ,we exclude for each rc the 11 smallest and 11 largest values , and define rescaled rcs such that the minimal and maximal values of the remaining 54149 data points , denoted as and , are projected on the rc values and , respectively .+  * diffusion constant - * we assume that the stochastic time evolution of a given rc is described by the one - dimensional fokker - planck ( fp ) equation  where is the probability of having a configuration with rc value at time , is the ( in general -dependent ) diffusivity , and is the free energy where is the time - averaged probability distribution .a few methods to extract from time - series data based on bayesian analysis of transition rates  or short - time fluctuations have been described  .our method extracts directly from folding times .define as the mean first passage ( mfp ) time to go from a state to some final state without recrossing , corresponding to an adsorbing boundary condition at . for the case onefinds  and for one has where at and reflective ( zero - flux ) boundary conditions hold . by differentiation with respect to , we obtain the diffusivity for  and for as an even simpler procedure employs the _ round - trip time _ ,\ ] ] the magnitude of which is the time needed to start at , reach for the first time , start from again and reach back to for the first time .one finds where is the partition function .the diffusivity profile based on the round - trip time reads intuitively , the slope of the round - trip time function is inversely proportional to : for a given , a larger slope implies a slower return to the starting point , or equivalently a smaller local diffusivity .the fp approach assumes an underlying markovian process , meaning that and thus are independent of . we exploit ( and check ) this by defining a mean round - trip time function that results from an average of round - trip times over their final states . since on the fp level curves for different differ only by an additive constant , we should be able to collapse all such curves onto .the assumption of markovian behavior breaks down at short times and for unsuitable reaction coordinates ( i.e. rcs that do not single out the transition state , as will be explained in detail later on ) and is clearly indicated by deviations of the round - trip time functions for varying , , from the mean .insight into this can be gained with a simpler definition of the diffusivity based on the variance in rc space  where denotes one specific realization of a path that starts at at time .as we will demonstrate , sensitively depends on the lag time . to get accurate results , should be small enough that the region explored by the rc in this time interval has an approximately constant free energy ; however if is below a threshold time scale , the resulting may be dominated by non - markovian properties. we will mostly use the round - trip method for determining , but compare to the other methods as well .  in our analysis of the simulation time series data we discretize rcs in typically intervals andnormalize probability distributions according to . +* fit of round - trip times - * to extract from the simulation data requires estimating the derivative .we start by fitting a smooth function to the numerical results , exploiting the fact that should be a monotonically increasing function of .thus the fitting function can be expressed in the form : where is an arbitrary function .we expand out in a basis of cubic b - splines defined over the range to , and use the coefficients of the expansion as fitting parameters .the size of the basis is fixed at splines .the full expression for is fit to the simulation estimate for using a standard least squares technique , with one modification : the quantity to be minimized is the sum of squared residuals plus another term which penalizes roughness in the fitted function .this additional term has the form , with smoothing parameter .larger values of lead to progressively smoother fits to the data .the entire fitting procedure is implemented through the functional data analysis package in the r programming language  .for all the results shown below we set , since we found that varying in the range 10 - 200 had minimal effect on the resulting diffusion profiles .the range is unsuitable because we fit to jagged features in the simulation curve which are the result of statistical noise . for the range ,we over - smooth the curve , losing most of the local slope information and resulting in poor fits to the round - trip function . + * reparameterization - * as is well - known  , the fp eq .( [ eq:1 ] ) is invariant under an arbitrary rc rescaling according to if the functions , , are simultaneously rescaled as , , and . here, is assumed positive .thus an arbitrary diffusivity profile can be obtained , while the kinetics on the fp level and the partition function stay invariant , as long as the folding free energy is adjusted accordingly . for the particular choice of a constant diffusivity , , we get and thus .fig . 1 shows the complete times series data for the simulated oligopeptide . in all five rcs and in the helicity data frequent switching between the folded state ( large helicity and small values ) and the unfolded stateis observed , meaning that the simulation is converged and allows drawing conclusions on the folding and unfolding kinetics ( further evidence is provided by the excellent comparison between straight md and replica - exchange simulations , as shown in fig .the fine resolution data ( fig . 1 , right panel ) in terms of the rms - deviation from the fully helical state , rc , suggest that an intermediate state and two barriers are present .as the snapshots indicate , in the fully helical state ( ) roughly three -helical turns are stabilized by salt bridges between the glu - 2 and lys - 6 and the glu - 7 and lys - 11 residues , respectively . in the intermediate state ( )only one of the two salt bridges stabilizes two turns , while in the unfolded state ( ) no bridge is present .note that the characteristic transition time for unfolding of one helical turn , i.e. for the transition from to in ( d ) , is roughly 200 ps and thus about 100 times shorter than the corresponding unfolding time in fig .while a high degree of correlation between different rcs can be inferred from fig . 1 , there is no one - to - one mapping , e.g. , in fig .1(c ) shows pronounced fluctuations in intervals where stays virtually constant .this is already evident from the average distribution function shown in fig . 2 as a function of all different rcs .while the distribution in the leftmost panel as a function of shows three broad peaks ( corresponding roughly to none , one and two intact salt bridges ) , clearly separated peaks are absent when is shown as a function of , , or .the reason is simple : states that are separated when , e.g. , described by , are mixed when they are projected onto different rcs .this is demonstrated by the coloured regions in fig . 2 that for correspond to pure states , i.e. narrow intervals of values .while for and the colored regions are smeared out but the ordering along the rc is preserved , for and the ordering is lost .this points to a fundamental difference between the rcs , that embody knowledge of the native state , and the rcs , which are purely geometric .  in fig .3 we focus on rc .the free energy profile in a ) reveals the intermediate state and two barriers at and .3(b ) shows the roundtrip times for various final states as a function of , directly extracted from the simulation time series  .the data sets are shifted vertically ( which according to eq .( [ eq:5 ] ) is irrelevant for extracting ) to illustrate the predicted collapse onto a single mean round - trip time function .the smooth fit is shown as a blue curve .the collapse of for different is a strong check on the consistency of the fp approach .the red curve denotes the round - trip time from the bayesian approach  , obtained for optimized time interval and smoothing parameters and  .3(c ) shows the diffusivity extracted from via eq .( [ eq:5 ] ) ( blue curve ) .most notably , varies considerably along : it is reduced by an order of magnitude around the intermediate state at and seems correlated with .the profile from the bayesian approach ( red curve ) reproduces the coarse features of our round - trip approach with slight difference that will be discussed below .we stress that we have fitted the two parameters in the bayesian approach , namely the time interval and the smoothing parameter , by a comparison with the simulation mean - first passage times ( see supplement for further details  ) .the diffusivity profiles resulting from the bayesian approach sensitively depend on these parameters , and without such a comparison it is not easy to see what are sensible parameter values .this highlights an advantage of our method based on the round - trip time , since the only parameter is a smoothing factor that operates directly on the round - trip time , a physical observable , and sensible parameter values are straightforwardly estimated .the variance method eq .( [ eq:6 ] ) for lag time  fs ( upper green curve ) overestimates by two orders of magnitude , yet for  ps ( lower green curve ) approaches the results of the other two methods quite nicely .thus for  ps , is dominated by non - markovian events that are unrelated to the long - time folding / unfolding dynamics ; interestingly , this threshold time is similar to the transition time for helix unwrapping inferred from fig .1(d ) . in figs .3(d ) and ( e ) , we show mfp times for ( folding ) and ( unfolding ) calculated from eq .( [ eq:2 ] ) and the various profiles shown in ( c ) . directly extracted from simulation data ( circles ) in fig .3(d ) is most accurately reproduced by the bayesian fitting approach ( red curve ) , as expected since the probability distribution and thus the frequency of transitions is maximal in the range ( see fig .the rt approach ( blue curve ) considers an equal balance of folding and unfolding events and consequently describes unfolding mfp times in fig .3(e ) better .noteworthy , the rt approach is simple to implement , directly works on the property one wishes to describe ( namely folding / unfolding times ) and has apart from the functional form of the fitted round - trip time no freely adjustable parameter .the combined deviations between simulation data and fokker - planck predictions in figs .3(d , e ) are due to a combination of non - markovian processes at short times and insufficient trajectory sampling .  in fig .4(a ) we compare the diffusivities based on the round - trip time approach ( blue curve ) and the bayesian approach ( red curve ) , already presented in fig .3(c ) , with results obtained from the mfp times via eq .( [ eq:3 ] ) , shown as a green curve . for the fit we used a final state and considered folding events from to .it is seen that the three curves roughly coincide , which testifies to the robustness of methods for deriving diffusivities from folding times . in fig .4(b ) we compare diffusivities from the variance method , eq .( [ eq:6 ] ) , to the round - trip time method eq .( [ eq:5 ] ) ( blue curve ) . herewe present results for for a wider range of lag times of  fs , 20 ps , 200 ps , 2 ns and 10 ns ( green curves , from top to bottom ) .it is seen that for lag times between  ps and  ns , agrees with the round - trip time approach . as already discussed , for smaller lag times is too large . for larger lag times loses structure and becomes too small , which has to do with the fact that at those times the peptides explores a considerable subsection of the free energy space and the effect of the energetic barriers encountered are spuriously accounted for by a reduction of the diffusivity .the situation is similar to the bayesian approach : there is no a - priori way of knowing what the suitable parameter value for the lag time is , unless one compares to a physical observable , which might be the folding or round - trip time . in that case , however , a direct fitting of based on folding times as suggested by us seems more direct and transparent .  a free energy barrier , as exhibited by in fig .3(a ) , was argued to arise from a subtle compensation of energy and entropy effects , which both increase upon unfolding  .this scenario , developed in the context of lattice models , is basically confirmed by our explicit water simulations . in fig .5(a ) , we show free energy profiles at different temperatures from replica - exchange simulations .indeed , the entropic contribution , estimated from the free energy difference between and 320k , shows considerable numerical error but rises across the unfolding transition . in fig .5(b ) we show the number of backbone - bound water molecules that have a distance to a backbone oxygen smaller than 0.35 nm .apart from the loss of one bound water molecule at ( paralleled by a helicity increase ) , steadily rises from about in the folded state to in the unfolded state .so we conclude that the entropy increase upon unfolding results from a competition of water binding and conformational effects . the overall good comparison between the free energy profile from a standard md simulation run ( for a length of 1.1 ) and results from a replica exchange md simulation ( trajectory length 22.5 ns and equilibrated with 32 replicas at different temperatures ) at t=300k in fig .6 gives good evidence that the times series considered in our kinetic analysis is long enough .the appearance of a free energy barrier , as seen in in fig .3(a ) , is often interpreted as equivalent to exponential kinetics , which is not necessarily true as we will now discuss .in fact , even the presence of a free energy barrier depends on the specific rc employed and thus is a much less robust feature than often assumed : in fig .7 we show the free energy and diffusivity profiles of all five rcs .we separate rcs that embody knowledge of the native state and the unbiased rcs . in the columns original we use the bare rcs as defined in the methods section , in the columns transformed we use rescaled rcs such that the diffusivities are constant , .two features strike the eye :  \i ) most diffusivity profiles are full of structure and vary substantially along the reaction path ; it immediately transpires that a description of the folding kinetics without consideration of the diffusivity profile can fail .\ii ) the profiles and vary considerably among different rcs . in fact , while shows pronounced barriers and an intermediate state , the profiles and are free of barriers : we conclude that the presence of barriers depends on the rc chosen .do the kinetics within an effective fokker - planck description also vary among rcs , possibly showing exponential for some and non - exponential behavior for other rcs ? while the free energy profiles as a function of the original rcs show large variations , the profiles after the transformation are quite similar ( this is most striking for the radius of gyration , , and the end - to - end radius , ) , and thus the kinetics as characterized by the mfp times in the bottom row are very similar .this at first surprising result can be easily rationalized : the round - trip method is designed to optimally reproduce the complete set of round - trip times and thus the slowest conformational transitions in the system .the different diffusivities and free energy profiles together uniquely determine the folding times . assuming that different rcs yield a comparable separation of states into the unfolded and folded basins, it follows that the folding times must be very similar .this in fact holds for the rcs on the one hand and for the rcs on the other hand .since after the rescaling the entire kinetic information is contained in the free energy profile , those profiles must be quite similar .it follows that the presence of a free energy barrier does not necessarily imply exponential kinetics ; for that statement to be true the free energy barrier must persist after a rc transformation that makes the diffusivity profile flat .although there are still differences among the free energy profiles for after the transformation , they are small enough that the kinetics are not particularly distinguished .   .starting from the rc exhibiting a flat diffusivity ( shown in blue ) , we arbitrarily rescale according to eq .( [ fantasy ] ) such as to increase the barrier ( green ) , decrease the barrier ( red ) and to reolocate the stable minimum ( turquoise ) ., width=272 ]  to highlight the implications of these results , we now turn the argumentation around . consider a general rc transformation - 1),\ ] ] that is assumed to be a monotonic function which implies that .this rescaling corresponds to a local stretching / compression of the rc around and via the reparametrization properties of the fokker - planck equation also modifies the diffusivity and the free energy profiles . in fig .8 we show three different rescaled and profiles , all generated via eq .( [ fantasy ] ) from the rc for which is flat ( shown in blue ) .depending on the parameters we generate free energy profiles that either exhibit a more pronounced barrier ( green curve ) , a reduced barrier ( red curve ) , or a free energy profile where the position of the minimum is moved from the folded to the unfolded state ( turquoise curve ) .we mention that by construction , the kinetics as characterized by the round trip or mfp time are invariant under this rescaling .what this figure demonstrates is that under a combined rescaling of and one can generate a bewildering variety of free energy curves which share the identical kinetics , meaning that the free energy profile without the diffusivity is not sufficient to even qualitatively predict protein folding kinetics .much of the discussion in the preceding sections and the usage of one - dimensional rcs presumes that the reaction coordinates are `` good '' in the sense that i ) the ensemble of transition states is assigned to a narrow region of rc values and ii ) that the probability of finding a transition state in that region is maximal  . to make that notion more concrete , one introduces the splitting probabilities and for each value of the rc , where is the probability to reach , starting from rc value , region a before region b  . in the context of transition states , the regionsa and b would denote regions corresponding to the folded and unfolded domains flanking the transition region .the splitting probabilities are normalized as since eventually any state will diffuse out towards the boundaries . for a trajectory that passes through state there are four choices , it can be trajectory starting in a and returning to a , starting in b and returning b , starting in a and ending up in b or starting in b and ending in a. the respective probabilities are normalized as for non - ballistic stochastic motion , the transition path probability , i.e. the probability that the trajectory connect regions a and b , can be maximally 1/2 .a maximum close to 1/2 characterizes a good reaction coordinate , a significantly smaller number points to a bad reaction coordinate . in fig .9 we show a detailed reaction coordinate analysis for rc with a resolution of 25 bins in the range and using the full time resolution of 20 ps . in ( a )we show again the complete time series and in ( b ) the corresponding probability distribution .region a for is the folded region , region b for is a region where one helical turn is unfolded . in ( c )we show the splitting probabilities and ( orange and blue lines ) .the behavior is as expected , with the probabilities switching from zero to unity between the boundaries of the regions a and b , and a rather large slope in the region around .the maximum of the transition path probability means that is quite close to a perfect reaction coordinate and that the fokker - planck analysis performed in this paper is appropriate for long times on the order of folding and unfolding events .note that is close to a minimum in the equilibrium distribution , see fig.9b , at which position the free energy thus exhibits a maximum .this is coincidental , since as we have shown in fig.8 , one can easily change the free energy profile by a reaction - coordinate rescaling , which however leaves the splitting probabilities and the transition path probabilities invariant .in the naive approach towards protein kinetics , folding times are deduced from the free energy profile alone . as has been argued before , such an approach is unreliable since for the simplest non - trivial folder , namely a single short -helix in explicit solvent , the diffusivity profile varies substantially along the folding path .our variation comes out somewhat stronger than from similar simulations with implicit solvent , suggesting that explicit solvent further increases the importance of diffusivity inhomogeneities  .in fact , to match experimental folding times of simple alpha - helix forming oligo - peptides within solvent - implicit simulations , an overall correction factor to the time scales is typically applied . a detailed microsopic justification for this is lacking ; on the contrary , it has been shown that in many cases explicit solvent strongly influences the free energy landscape and introduces novel kinetic mechanisms that are completely absent in solvent - implicit simulations .when extending the analysis to five different popular reaction coordinates , we find free energy and diffusivity profiles to vary substantially among different rc representations . yet ,the kinetics that follows from a fokker - planck description is largely independent of the rc chosen , if and only if is properly accounted for .a similar conclusion was reached recently based on coarse - grained , solvent - implicit simulations  .this means that a quasi - universal ( i.e. rc independent ) description of protein folding kinetics necessarily involves . for this quasi - universality to hold we have to distinguish between reactioncoordinates that are based on the distance to the native state ( such as ) and those that are purely geometric in nature ( such as ) . by considering generalized rcs and using the reparametrization invariance of the fokker - planck equation, we can design arbitrary profiles with no barrier at all , an enhanced barrier , or an interchange of the naive stable and unstable states .this means that the concept of a free energy profile is to some degree arbitrary , which might be relevant with regards to recent discussions in the experimental literature  .the kinetics , embodied in the folding time , and dependent on and , is less arbitrary .our simulations are for a single -helix fragment , one of the shortest oligopeptides which shows non - trivial folding .there is no reason to believe that for larger proteins the situation will simplify ; we therefore argue that the diffusivity profile will be full of features and thus important in those more complicated situations as well .our conclusions also apply to optimized or otherwise carefully selected rcs  , since the reparametrization can be done for any rc and thus arbitrarily create , annihilate and shift barriers in the folding landscape ( incidentally , rc turns out to be a quite good reaction coordinate according to the definition of ref . , as shown in fig . 9 ) .our method of extracting the diffusivity profile via the mean - 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trip times and the diffusivity profiles of all five reaction coordinates . in fig .3 we show the mean - first passage times for folding and unfolding events for all reaction coordinates , as extracted from the fitted diffusivity profiles and the fokker - planck description . the final states were chosen such that of the probability distribution is contained in the range ( folding ) , or ( unfolding ) .the noise and non - monotonicity in the curves extracted from the simulation data are due to the statistical effects of insufficient trajectory sampling ( particularly at the edges of the free energy landscape ) and time discretization .we briefly review the optimization method introduced in ref . , and used previously to extract diffusivity profiles for protein folding dynamics in implicit solvent  .   when discretized in reaction coordinate space , the fp equation takes the form of a master equation  where the probability of being in bin is denoted by , the bin width is , the bin index ranges from to , and the transition rate from bin to bin is .the rates fulfill detailed balance , i.e. , where the equilibrium probability of each bin is denoted by ; the loss in bin is caused by transitions to neighboring bins , i.e. . the rates in the master equation [ eq : mastereqfp ] are related to the free energy and the diffusivity profile in the fp equation via :  with being the diffusivity between the bins . for bins the system is consequently characterized by independent parameters : rates for transitions from the neighboring bin on the right hand side and equilibrium probabilities .   in a system described by eq .[ eq : mastereqfp ] the conditional probability of landing in bin in time given a start in bin is : where is the matrix with entries . in our case is tridiagonal and the transition probabilities are easily obtained numerically by diagonalization of the symmetrized matrix defined by the entries  . for a process described by eq .[ eq : mastereqfp ] , the likelihood of observing a certain sequence with transitions at equidistant time intervals is : where is the total number of transitions from to observed along the trajectory and the time intervals .bayesian inference ( bi ) can be used to determine the underlying free energy and diffusivity profile from a stochastic trajectory .bayes theorem states that for a given trajectory ( ) the probability of certain parameters to be correct is : where is the likelihood of eq .[ eq : likelihood ] ; in our case the prior just depends on the diffusivity profile , penalizing large deviations of the diffusivity at adjacent grid points .a standard simulated annealing scheme is used to optimize the probability in eq .[ eq : bayes ] by iterative variation of the parameters of the system .the quantity to be minimized is the `` energy '' defined by : at each step the parameters and are slightly perturbed giving rise to a new configuration with energy , which is always accepted for and accepted with probability for ; the `` temperature '' of the system is subsequently lowered until the optimized and are reached .several independent simulated annealing runs are performed ; variations in the results obtained in different runs allow drawing conclusions on the quality of the estimate and the suitability of the process for a fp type description .( a ) , diffusivity ( b ) and mfp times ( panels ( c ) and ( d ) ) for fixed value of and different times intervals used in the optimization procedure . the free - energy obtained from the equilibrium analysis of the trajectoryis shown as a solid black curve in ( a ) , the target states and are denoted as vertical dashed black lines , and the values of extracted directly from the simulation data as black circles in ( c ) and ( d ) . ][ [ dependence - of - dq - on - the - time - interval - delta - t - and - the - smoothing - parameter - gamma ] ] dependence of on the time interval and the smoothing parameter ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~  the bayesian optimization method is applied to the dynamics of the reaction coordinate . in fig .[ fig : lagtime ] we compare results obtained for different time intervals ; in fig .[ fig : gamma ] results for different values of the smoothing parameter weighting the prior in eq .[ eq : bayes ] are shown .we show results for bins along the rc , and show average values of and from independent optimization runs .though being a fit quantity , the free energy profile does not significantly differ from obtained from the equilibrium analysis of the trajectory ( black lines in the upper panel of the figures ) .we note that the diffusivity profile is strongly sensitive on the time interval : while almost identical profiles like in the variance method analysis are obtained for , the diffusivity subsequently decreases for larger .the parameter can compensate insufficient sampling by externally requiring a smoothness of the diffusivity ; however , strong external constraints corresponding to low -values tend to erase any structure in .  reasonable choices of the parameters and are not evident a priori to ensure that the long - time dynamics are correctly reproduced by the optimization result , we compute the position dependent mfp times for a folded state ( ) and an unfolded one ( ) for each of the optimized diffusivity profiles and compare these curves to the one directly extracted from the simulation data .this comparison shows that in our case and are sensible values .

Abstract: the concept of a protein diffusing in its free energy folding landscape has been fruitful for both theory and experiment . yet the choice of the reaction coordinate ( rc ) introduces an undesirable degree of arbitrariness into the problem . we analyze extensive simulation data of an -helix in explicit water solvent as it stochastically folds and unfolds . the free energy profiles for different rcs exhibit significant variation , some having an activation barrier , others not . we show that this variation has little effect on the predicted folding kinetics if the diffusivity profiles are properly taken into account . this kinetic quasi - universality is rationalized by an rc rescaling , which , due to the reparameterization invariance of the fokker - planck equation , allows the combination of free energy and diffusivity effects into a single function , the rescaled free energy profile . this rescaled free energy indeed shows less variation among different rcs than the bare free energy and diffusivity profiles separately do , if we properly distinguish between rcs that contain knowledge of the native state and those that are purely geometric in nature . our method for extracting diffusivity profiles is easily applied to experimental single molecule time series data and might help to reconcile conflicts that arise when comparing results from different experimental probes for the same protein .


quantum computers have been much studied in recent years due mainly to the possibility of solving some important problems more efficiently on quantum computers than is possible on classical machines .however quantum computers and quantum robots are also interesting from other viewpoints .for example , a study of these systems may help to determine what properties a quantum system must have to conclude that it has significant characteristics of intelligence .if quantum mechanics is universally applicable , then many intelligent quantum systems exist ( e.g. the readers of this paper ) .the fact that these systems are macroscopic , which may be necessary , does not contradict the fact that they are also quantum systems .another aspect originates in the fact that basis states of multiqubit systems in quantum computers are product states of the form where is a function from to .and is the state of the jth qubit . herethe state is supposed to be a binary representation of a nonnegative integer . even though this representation is assumed implicitly in the literature , it is not trivial , especially regarding conditions that a composite physical system must satisfy in order that it admits states representing numbers .  the notion that states represent the nonnegative integers can be extended to representations of more general types of numbers , such as all integers and rational numbers .the representations can also be extended by considering the states to represent symbols in some language .the language can be formal as is the case for axiom systems studied in mathematical logic , or informal as is the case in english . in this casethe symbol basis states would include orthogonal states for each of the 26 letters in the alphabet and states for some punctuation symbols .word states would consists of products of the symbol basis states excluding the spacer symbol state .these would be used to separate the different words .these representations are of interest for several reasons . as part of an attempt to characterize intelligent quantum systems ,one wants to understand what physical properties a quantum system must have so that it can be said to be creating text that has meaning to the system generating the text .another is related to the need to develop a coherent theory of mathematics and physics together that is maximally internally self consistent .in such a theory one would expect mathematical logical concepts to be closely integrated with quantum mechanics or some generalization such as quantum field theory .since mathematical logic deals with systems of axioms as words in a language and their interpretation , such an integration would require the representation of words by quantum states .the potential importance of this has been recognized by other work .included are recent work on theories of everything , an attempt to use feynman diagrams to represent expressions in propositional logic , and other relevant work .  in order to see howmathematical logical concepts such as truth , validity , consistency , and completeness might be used in quantum mechanics , an example will be studied that is based on a simplification of a simple machine described by smullyan .the description of the simplified machine , given in the next section , is followed by a description of a quantum machine that is similar to a quantum turing machine .additional details are given in .the simplified version of smullyan s machine m , used here prints , one symbol at a time , a nonterminating string of any one of the five symbols .words are defined as any finite strings of symbols that exclude the which denotes a spacer symbol .based on this the machine prints a steadily growing string of words separated by finite spacer strings .some of the words , which are separated from other words by spacer strings , are assigned a meaning .these words , referred to as sentences , are and where is any word that is not a sentence .the strings and where and are examples of these two types of sentences with separating shown .the intended meaning of is that is printable and that of is that is not printable .the restriction that is not a sentence is not present in smullyan s original example .it is made here to keep things simple and avoid inference chains generated by words of the form , etc .. removal of the restriction is discussed in .the term `` printable '' refers to the dynamical description of m. if the dynamical description of m correctly predicts the behavior of , then any word that is printable will be printed sooner or later . if the word is not printable , then it will never be printed .  based on the assigned meaning of these words , is defined to be _ true _ if is printable .it is false if is not printable . is _ true _ if is not printable and false if is printable .these definitions relate truth and falseness of the sentences to the dynamics of .however nothing is said so far about whether these statements are true or false .this is accounted for by defining the dynamics of m to be _valid _ if any printable sentence is true .it follows that false sentences are not printable .it is also the case that the dynamics is valid if no sentences are printable .this possibility is avoided by requiring the dynamics to be _complete_. that is , it must be such that for all that are not sentences , either or is printable .it is _ consistent _ if at most one of or is printable .let m be a multistate quantum system or a head moving along a one dimensional lattice of quantum systems at sites .the basis states of the head m have the form where is an internal head state label and the lattice position of m. the hilbert space of states associated with the system at lattice site is spanned by a basis of five states or as . the state latexmath:[ ] denote the product state for the symbol states in the lattice interval .there is an obvious map of }\rangle ] which has at all sites outside  ] where for each in  ] where for each in  ] is over all length symbol string states ( including the spacer ) of lattice systems at sites .the sums are over all states and over all five symbol states for the site . the states }\rangle ] are the constant states at all lattice positions and at lattice positions respectively .the separation of the state of the first systems from the in the sums is based on the fact that in future time steps m is at positions and no longer interacts with the lattice systems at sites .  to obtain eq .[ feyps ] one first notes that }\rangle ] , which includes spacer symbol states, one sees that each state }\rangle ] , then . for each value of }\rangle ] is a product of symbol states , the must satisfy .each of the states has at least one symbol state , so for .  based on this each state }\rangle ] is a word path state with ( empty and nonempty ) words with as a sum over word path states in a form similar to the sum over symbol string states shown in eq .[ feyps ] .  to achieve thislet here is the projection operator for a lattice system in state being at a lattice position two sites behind that of the head m if and at position if the head is at sites or . is the projection operator for the lattice system , at the same position relative to that of , in a symbol state different from .note that the two projection operators are orthogonal and .also \neq 0. ] are projection operators on the word in path and on the spacer string over the lattice interval  ] are projection operators for finding at site and the word starting at site and ending at site .the reason for the before and after is to exclude cases where is part of a longer word .the requirement that the terminal in be at least two sites behind m means that the probability that appears at a fixed location is independent of for sufficiently large .more exactly is independent of for all .the meanings chosen for the sentences and are based on the path description of eq .[ wdpthsum ] or [ pthexp ] .a sentence is a word that has a meaning .the domain of meaning for is the set of paths containing .similarly the meaning domain for is the set of paths containing . is _ true _ on its domain if all paths containing also contain .it is false if some paths containing do not contain . is _ true _ on its domain if no path containing contains .it is false if some path containing contains .note that and are distinct words in a path , so they are separated by at least one spacer string .also the order of appearance of and in the path is immaterial .these definitions can be expressed as limits of matrix elements . is true if and is true if here is the projection operator for not occurring in any path two or more sites behind the head .note that inside these matrix elements . and are false if these equations hold with replaced by .these equations show that if a measurement at time of finds then is true [ false ] if the asymptotic conditional probability is unity [ that will be found in a subsequent measurement of .similarly if is found in a measurement of , then is true [ false ] if the asymptotic conditional probability that will not be found is unity [ .based on the choice of meaning used here , says nothing about the occurrence or nonoccurrence of in paths not containing the sentence .the same holds for . for a measurement at time that does not find ,the truth or falseness of has no meaning for any following measurement of , relative to the time measurement .however a later measurement may find .this shows that the meaning domain is nondecreasing with increasing .one concludes from this that the domain of meaninglessness of is not empty if and only if a similar statement holds for .the definitions of truth and falseness given above relate these concepts to the dynamics .but nothing so far requires the sentences to be true .this is the case even if is a correct theoretical description of the dynamics of .this is taken care of by defining the dynamics to be _ valid _ if for all paths and all sentences , if contains then is true in .an equivalent statement is that is valid if all printable sentences are true on their domain of meaning . in terms of limits of matrix elementsone has that is valid if for all sentences for which eq . [ xpr ] holds , so do eqs .[ pxtr ] for and [ notpxtr ] for . is _ consistent _ if for all that are not sentences , no path contains both and . in terms of limits of matrix elements, is consistent if   the definition of validity has some interesting aspects .it is satisfying to note that , as is the case for the classical , one can prove that if is valid , then it is consistent .the converse does not necessarily hold , though .the requirement that is valid is a restriction on as it limits what can and can not appear in paths containing the sentences .this requirement corresponds to conditions that must be satisfied by the amplitudes associated with the paths in the path sum , eq .[ wdpthsum ] or [ pthexp ] .  one property of the definition is that it says nothing about how many , if any , of the words and are printable by .for instance is valid if no sentence is printable .one would like to avoid this possibility .also it is desirable for to maximize the printing of words that give information about the dynamics by telling what can and can not be printed .  to this endone defines to be _ complete _ if all sentences are printable .if is complete then , for the interpretation considered here , the amount of information provided by about what it can and can not print is maximal .whether or not this condition can be satisfied , or should be relaxed in the presence of conditions that exclude printing of some sentences , may depend on the interpretation given to the words chosen to be sentences .more generally one defines to be _ maximally complete _ if all sentences that are not excluded by these conditions , if any , are printable .it should be noted that , for the path interpretation used here , completeness is quite different for the quantum than for the classical single path . in particular completeness does not restrict the printability of sentences and their negation .both and can appear provided they are on different paths .this ensures that consistency is satisfied .this is impossible for a classical with only one path as at most one of or can appear .the relationship among these concepts is shown in the figure which is a schematic tree representation of very few of the paths in the sum over word paths of eqs .[ wdpthsum ] or [ pthexp ] . for illustrative purposes ,only a very few of the relevant words are shown in some of the path segments .a full tree representation would be very complex with 5 branches at each time step node .this is based on the sum over symbol paths of eq .[ feyps ] .the paths all have the same height as they each contain symbols .words next to the paths denote that the paths contain these words .the two paths with circled words show examples of invalidity .the path containing and is inconsistent and the path containing and shows that is false .the validity status of the path containing and no is still open as may appear later on .it should be emphasized that the word states chosen to have meaning and the meaning assigned to these states were chosen or imposed arbitrarily from the outside .they were done to illustrate properties of some mathematical logical concepts in quantum mechanics .in particular the requirement that the dynamics of be valid imposes a restriction on the dynamics that depends on the meaning assigned to the states and to the truth definitions used .the quantum system is itself completely silent on which expressions , if any , have meaning and how they are to be interpreted . is also silent on what basis is to be used to to assign meaning to the states in the basis .  in this connectionit is worth investigating the dependence of validity of a dynamics on a change of basis .intuitively one would expect that validity of a dynamics would not be preserved under a change of basis . reading a word state in different basiswould give a different outcome with a finite probability that depends on the relationship between the two basis .also , as is well known , the state is changed by the reading so that it can not be reread to determine the outcome in the original basis .  as a simple example of this ,suppose is generating output as described earlier in section [ qmm ] .assume that the output lattice is a lattice of spin systems with the spin projection eigenstates of each system along some axis corresponding to the 5 symbol states .initially each system is in a spin projection eigenstate corresponding to the symbol state . then meaningful output word states }\rangle ] in the rotated basis steps after finding }\rangle ] for all if }\rangle = | \sim p(x)_{[a , b]}\rangle ] ( original basis ) . in this casethe other terms in the sums give the nonzero contributions for the amplitude .more generally , let be any unitary operator on the five dimensional hilbert space spanned by the symbol basis and let be independent of the lattice site . define the symbol projection operators for the observer in terms of the original symbol projection operators by here and are projection operators for the symbol at site in the observer reading basis and in the original basis .the corresponding projection operators for words in the observer basis and the original basis are obtained as tensor products of these operators .define a new dynamics by where . here is the projection operator for at site and is the operator restricted to symbols at site of the lattice .it is clear that is unitary on the subspace of all states with at positions .since this is the space of states attained by iteration of on any initial state with at sites one can consider to be unitary .define the projection operator by replacing }} ] in eq .[ qproj ] where }} ] .replacement of by and each operator by in eqs .[ pxtr ] and [ notpxtr ] shows that if is valid then so is but for a different initial state latexmath:[\lim_{n\rightarrow\infty}\langle\psi(n )  if latexmath:[$\lim_{n\rightarrow\infty}\langle\psi(n )  is true if is false and conversely , and there is no restriction on the meaning domain of these sentences .however it is still the case that if these words are not printable then does not provide information about its own dynamics in that it says nothing about what can or can not be printed .as was noted already the choice of which states have meaning and what these states mean was imposed externally . was completely silent about the meaning of its output .as such this work is a prelude to examining some much deeper and potentially more interesting problems . consider to be a complex quantum system , such as a quantum robot , , moving in and interacting with a complex environment of quantum systems .as moves about it generates output or signals .the state of the cumulative output at time can be represented by a density operator .the time dependence of allows for the increase of the length or complexity of the output with increasing .this increase with is the case for the dynamics of m described by eq .[ wdpthsum ] where the length of the word path states generated by m increases with the time step number .the density operator description is used to account for the possibility that states of the output systems are entangled with states of other quantum systems in m or in the environment .a basic question is `` what properties must have so that we as external observers conclude that has meaning ? '' even more important is the question `` what properties must have so that we would conclude that it has meaning to m , the system that generated it ? '' and `` would the two meanings be the same ? ''if we interpreted to be a theoretical and experimental description of m s environment , and the interpretation was valid , one might expect , and perhaps may even require that have the same meaning and interpretation for m as for us as external observers .  in essence this problem is faced all the time by each human being in interactions with other humans .all writing and speaking and use of other means of communication can be described in terms of some system m generating output that in essence creates systems described by a time dependent state .the state is time dependent because new output is being generated either continuously or sporadically by m. each of us must be able to assign meaning to the states of the output of others .this meaning is , for the most part , the same as the meaning assigned by the system generating the output .a potentially important aspect of _ existing _ systems m that generate output with meaning and of the output systems also is that they are all large quantum systems for which the relevant degrees of freedom are macroscopic or essentially classical .it is suspected that that this may be a necessary condition .the main reason is that if the output consists of quantum systems in some time dependent quantum state that is not quickly stabilized by decoherent interactions with the environment , then reading the output to determine if it has meaning or not , requires knowledge of what basis to use for the reading . as was shown in section [ dvb ] ,reading the output state in another basis will give the wrong result with a finite probability that depends on the relation between the two bases .also the state will be changed so that one can not read the state another time to get the original answer .in this paper an example of a machine generating output , analyzed by smullyan to illustrate various mathematical logical concepts was described quantum mechanically .symbol strings states of the form and where was any symbol string state without that did not have this form were assigned a path meaning .for these words , referred to as sentences , was defined to be true if all paths containing also contained ; was true if no path containing also contained .the dynamical evolution of described by iteration of a unitary step operator was represented by a feynman sum over word paths .based on this was defined to be valid if each sentence was true on any path containing it . was defined to be complete if each sentence appeared on some path , and was defined to be consistent if no path contained both and .based on these definitions it was seen that the mathematical logical concepts of truth , validity , completeness , and consistency , have different properties than in the classical case .for instance the domain of meaning of a sentence was limited to the paths containing it .sentences had no truth value for paths not containing them .also , contrary to the classical case for which there is just one path , it is possible for to be valid and consistent and for both a sentence and its negation to appear on some paths .however no sentence and its negation can have any path in common .it was also seen that a slight extension of the model to include self referential sentences gives an incompleteness result similar to that of the first gdel incompleteness theorem .a main purpose of this paper was to present and emphasize the main results of without the extensive intervening mathematics .the figure was presented to illustrate more clearly the above definitions and their relationships .new material includes showing that the properties defined above depend on the basis used to define symbol states .for instance it was seen that validity of a dynamics was not preserved under a unitary change of the symbol basis .however there is a transformed dynamics that does preserve validity provided the initial state is changed suitably . as was seen in the discussionthis result is relevant to the question regarding how one determines if output generated by a quantum system moving in a quantum environment has meaning , if any , to .if the output system states are not stabilized by interaction with the environment , one must know what basis to use to examine the output to answer this question .another new result was obtained by examining the relation between the potential meaning of word string states in general and the algorithmic complexity of the systems generating the word string states .two word string states were described that had essentially the same algorithmic complexity . for one string statethe component word states had meaning . for the other they had no meaning .the contextual basis of the two states was the same in that they were both theorem enumerations based on two different axiom sets , one consistent and the other inconsistent .this shows that the relationship between the potential meaning of a word string state and the algorithmic complexity of the dynamics generating the string must be quite complex , if any relationship even exists .in conclusion it is noted that the work done in this paper is a small initial part of a larger attempt to combine mathematical logical concepts with quantum mechanics .this is one approach to the questions presented in the discussion section , and towards the goal of construction of a coherent theory of mathematics and physics together .this work is supported by the u.s .department of energy , nuclear physics division , under contract w-31 - 109-eng-38 .p. shor , in _ proceedings , 35th annual symposium on the foundations of computer science _ , s. goldwasser ( ed ) , ieee computer society press , los alamitos , ca , 1994 , pp 124 - 134 ; siam j. computing , * 26 * , 1484 - 1510 ( 1997 ) .

Abstract: the representation of numbers by product states in quantum mechanics can be extended to the representation of words and word sequences in languages by product states . this can be used to study quantum systems that generate text that has meaning . a simple example of such a system , based on an example described by smullyan , is studied here . based on a path interpretation for some word states , definitions of truth , validity , consistency and completeness are given and their properties studied . it is also shown by means of examples that the relation between the potential meaning , if any , of word states and the quantum algorithmic complexity of the process generating the word states must be quite complex or nonexistent .


the modern study of the `` habitability '' of circumstellar environments started almost half a century ago ( huang 1959 ) .the concept of a circumstellar habitable zone ( chz ) is relatively well defined , being tightly related to the requirement of the presence of liquid water as a necessary condition for life - as - we - know - it ; the corresponding temperature range is a function of the luminosity of the star and of the distance of the planet from it .an important amount of recent work , drawing on various disciplines ( planetary dynamics , atmospheric physics , geology , biology etc . ) refined considerably our understanding of various factors that may affect the chz ; despite that progress , the subject should still be considered to be in its infancy ( see.g .chyba and hand 2005 or gaidos and selsis 2005 , and references therein ) .habitability on a larger scale was considered a few years ago , by gonzalez et al .( 2001 ) who introduced the concept of galactic habitable zone ( ghz ) .the underlying idea is that various physical processes , which may favour the development or the destruction of complex life , may depend strongly on the temporal and spatial position in the milky way .for instance , the risk of a supernova explosion sufficiently close to represent a threat for life is , in general , larger in the inner galaxy than in the outer one , and has been larger in the past than at present .another example is offered by the metallicity ( the amount of elements heavier than hydrogen and helium ) of the interstellar medium , which varies across the milky way disk , and which may be important for the existence of earth - like planets .indeed , the host stars of the extrasolar giant planets detected so far are , on average , more metal - rich than stars with no planets in our cosmic neighborhood ( e.g. fischer and valenti 2005 ) . several other factors , potentially important for the ghz , are discussed in gonzalez ( 2005 ) .the concept of ghz is much less well defined than the one of chz , since none of the presumably relevant factors can be quantified in a satisfactory way .indeed , the role of the metallicity in the formation and survival of earth - like planets is not really understood at present , while the `` lethality '' of supernovae and other cosmic explosions is hard to assess .the first study attempting to quantitatively account for such effects is made by lineweaver et al .( 2004 , herefater l04 ) , with a detailed model for the chemical evolution of the milky way disk .they find that the probability of having an environment favourable to complex life is larger in a `` ring '' ( a few kpc wide ) surrounding the milky way center and spreading outwards in the course of the galaxy s evolution .we repeat that exercice here with a model that reproduces satisfactorily the major observables of the milky way disk ( sec .2 ) . in sec . 3we discuss the role of metallicity and we quantify it in a different ( and , presumably , more realistic ) way than l04 , in the light of recent simulations of planetary formation . in sec .4 we discuss the risk of sn explosions and conclude that it can hardly be quantified at present , in view of our ignorance of how robuste life really is .for comparison purposes , though , we adopt the same risk factor as the one defined in l04 . in sec .5 we present our results , showing that the ghz may , in fact , extend to the whole galactic disk today .we conclude that , at the present stage of our knowledge , the ghz may extend to the entire mw disk .the evolution of the solar neighborhood is rather well constrained , due to the large body of observational data available .in particular , the metallicity distribution of long - lived stars suggests a slow formation of the local disk , through a prolonged period of gaseous infall ( e.g. goswami and prantzos 2000 ; see however , haywood 2006 for a different view ) . for the rest of the disk , available data concern mainly its current status and not its past history ; in those conditions , it is impossible to derive a unique evolutionary path .still , the radial properties of the milky way disk ( profiles of stars , gas , star formation rate , metallicity , colours etc . ) , constrain significantly its evolution and point to a scenario of `` inside - out '' formation ( e.g. prantzos and aubert 1995 ) .such scenarii fit `` naturally '' in the currently favoured paradigm of galaxy formation in a cold dark matter universe ( e.g. naab and ostriker 2006 ) .some results of a model developed along those lines ( boissier and prantzos 1999 , hou , prantzos and boissier 2000 ) are presented in fig .1 . note that the evolution of the galactic bulge ( innermost kpc ) is not studied here , since it is much less well constrained than the rest of the disk ; the conclusions , however , depend very little on that point .the model assumes a star formation rate proportional to ( where is the galactocentric distance ) , inspired from the theory of star formation induced by spiral waves in disks .it satisfies all the available constraints for the solar neighborhood and the rest of the disk ; some of those constraints appear in fig .1 . among the successes of the model, one should mention the prediction that the abundance gradient should flatten with time , due to the `` inside - out '' formation scheme ( bottom right panel in fig .1 ) , a prediction which is quantitatively supported by recent measurements of the metallicity profile in objects of various ages ( maciel et al ..  it should be stressed , however , that the output of the model is only as good as the adopted observational constraints .for instance , there is still some uncertainty concerning the level of the oxygen abundance gradient : the `` standard '' value of dlog(o / h)/dr = 0.07 dex / kpc is challenged by the recent evaluations of daflon and cunha ( 2004 ) , who find only half that value .if the latter turns out to be true , some of the model parameters ( e.g. the adopted radial dependence of the infall rate and/or the star formation rate ) should have to be revised .  in any case, it appears rather well established now that the star formation rate and the metallicity are more important in the inner disk than in the outer one , and that this trend has been even stronger in the past ( see left panel in fig .some authors have suggested that those two parameters play an important role in the `` habitability '' of a given region of the galactic disk ( gonzalez et al .2001 , l04 ) ; we discuss briefly that role in the next two sections .since the first detection of an extrasolar planet around peg 51 ( mayor and queloz 1995 ) , more than 170 stars in the solar neighborhood have been found to host planets ( e.g. butler et al .the masses of those planets range from 0.03 to 18 jupiter masses and their distances to the host star lie in the range of 0.03 - 6 au .these ranges of mass and distance , however , are at present due to selection effects , resulting from the current limitations of detection techniques .continuous improvement of those techniques may well reveal the presence of smaller , earth - like , planets at small distances from the host stars ( as well as massive ones further away ) .in fact , the mass distribution of the detected planets is ( butler et al .2006 ) and suggests that earth - like planets should be quite common ( unless a yet unknown physical effect truncates the distribution from the low mass end ) .  the unexpected existence of `` hot jupiters '' is usually interpreted in terms of planetary migration ( see e.g. papaloizou and terquem 2006 for a review ) : those gaseous giants can , in principle , be formed at a distance of several au from their star ( where gas is available for accretion onto an already rapidly formed rocky core ) ; their subsequent interaction with the proto - planetary disk leads , in general , to loss of angular momentum and migration of the planets inwards . on their way ,those planets destroy the disk and any smaller planets that may have been formed there .thus , the presence of hot jupiters around some stars implies that the probability of life ( at least as we know it ) in the corresponding stellar system is rather small .star as a function of their mass ( from simulations of mordasini et al .results are displayed for three different values of the metallicity of the star ( and of the corresponding proto - planetary disk ) , as shown on the figure .it is seen that , high metallicities favour larger fractions of massive planets ( in rough agreement with observations ) , while at low metallicities the presence of earth - like planets is enhanced rather than suppressed .( courtesy y. alibert ) . , scaledwidth=80.0% ]  a key feature of the stars hosting planets is their high metallicity ( compared to stars wih no planets ) .that feature was first noticed by gonzalez ( 1997 ) , who intrepreted it in terms of accretion of ( metal enriched ) planetesimals onto the star .however , subsequent studies found no correlation of the stellar metallicity with the depth of the convective zone of the star , invalidating that idea ( see e.g. fischer and valenti 2005 , hereafter fv05 , and references therein ) .thus , it appears that the high metallicity is intrinsic to the star , and it presumably plays an important role to the giant planet formation .fv05 quantified the effect , finding that the fraction of fgk stars with hot jupiters increases sharply with metallicity , as = 0.03 , where designates fe abundance .note that this function has lower values and is much less steep than the one suggested by lineweaver ( 2001 , hereafter l01 ) , which reaches a value of = 0.5 at a metallicity of 2 z ; the latter function is used in the calculations of l04 .the impact of metallicity on the formation of earth - like planets is unknown at present .metals ( in the form of dust ) are obviously necessary for the formation of planetesimals , but the required amount depends on the assumed scenario ( and its initial conditions , e.g. size of protoplanetary disk ) . inspired by the early results on extrasolar planet host stars , several authors argued for an important role of metallicity .thus , in l01 it is suggested that the probability of forming earth - like planets should simply have a linear dependence on metallicity . on the other hand , zinnecker ( 2003 )argues for a threshold on metallicity , of the order of 1/2 z , for the formation of earth - sized planets .however , the situation may be more complicated than the one emerging from simple analytical arguments .recent ( and yet unpublished ) numerical simulations by the bern group find that at low metallicities , the decreasing probability of forming giant planets leaves quite a lot of metals to form a significant number of earth - sized planets ( see fig .those simulations cover a factor of in metallicity ( from .6 z to 2 z ) and they roughly reproduce the observations concerning ( see above ) ; their predictions for have to be confirmed by further simulations and , ultimately , by observations .they suggest , however , that the formation of earth - like planets may be quite common , even at low metallicity environments like the outer galaxy and the early inner galaxy .  assuming that and are known , the probability of having stars with earth - like planets ( but not hot jupiters , which destroy them ) is simply : .however , it is clear from the previous paragraphs that neither of the terms of the right member of that equation can be accurately evaluated at present to form in the habitable zone '' . ] .just for illustration purposes , we adopt in this work a set of metallicity - dependent probabilities different from the one adopted in l04 ( who assumed linearly dependent on , and strongly increasing with ) : we assume that =0.4 ( so that the metallicity integrated probability is the same as in l04 ) for .1 and from fv05 .the two sets appear in fig .3 , which also displays ( bottom panel ) : with our set , extends to non - zero values at metallicities much lower and higher than in in the work of l04 . from that factor aloneit is anticipated that any ghz ( found through a chemical evolution model ) will be much larger with our set of data than with the one of l04 .the potential threat for complex life on earth that represent nearby sn explosions was first studied by ruderman ( 1974 ) .he pointed out that energetic radiation from such events , in the form of hard x - rays , gamma - rays or cosmic rays , may ( partially or totally ) destroy the earth s atmospheric ozone , leaving land life exposed to lethal does of uv fluxes from the sun .the paper went completely unnoticed , with just one citation for about 20 years . in the last few years , however , a large number of studies were devoted to that topic ( for reasons that are not quite clear to the author of this paper ) .two factors are , perhaps , at the origin of that interest : the availability of complex models of earth s atmospheric structure and chemistry ; and the discovery of extrasolar giant planets .the latter suggests that earth - like planets may also be abundant in the galaxy ; however , complex life on them may be a rare phenomenon , because of various cosmic threats like sn explosions .this is the basic idea underlying the concepts of ghz ( gonzalez et al . 2001 ) and of the rare earth ( brownlee and ward 2002 ) : complex life in the universe may be rare , not for intrinsic reasons ( i.e. improbability of development of life on a planet ) , but for extrinsic ones , related to the hostile cosmic environment .despite the simplicity of the idea , however , the studies devoted to the topic revealed that it is very difficult to quantify the sn threat for life ( or any other cosmic threat ) . studies of that kind evaluate the energetic particle flux / irradiation impinging on earth , which could induce a significant number of gene mutations ( based on our understanding of such mutations in various organisms on earth ) . knowing the intrinsic emissivity of energetic particles from a typical sn ( which depends essentially on the configuration of the progenitor star and the energetics of the explosion ) , one may then calculate a lower limit for the distance to the sn , for such fluxes / irradiations to occur .the distribution of the rates of various sn types in the milky way ( presumably known to within a factor of a few ) is then used to evaluate the frequency / probability of such events in our vicinity and elsewhere in the galaxy .numerous studies in the last few years explored several aspects of the scenario , with the use of models of various degrees of complexity for the atmosphere of the earth ( or mars ) and the transfer of high energy radiation through it ( see e.g. ejzak et al . 2006 and references therein ) .thus , smith et al .( 2004 ) find that a substantial fraction ( % ) of the high energy radiation ( x- and - rays ) impinging on a thick atmosphere ( column density g , like the earth s ) may reach the ground and induce biologically important mutations .ejzak et al .( 2006 ) find that the ozone depletion depends mainly on the total irradiation ( for durations in the 10 - 10 s range ) and only slightly on the received flux ; this result allows one to deal with both short ( -ray bursts and long ( sn explosions ) events .they also , find that the overall result depends significantly on the shape of the spectrum ( with harder spectra being more harmful ) .  even if one assumes that such calculations are realistic (which is far from being demonstrated ) , it is hard to draw any quantitative conclusions about the probability of _ definitive sterilisation of a habitable planet _ ( which is the important quantity for the calculation of a ghz ) . even if 100% lethality is assumed for all land animals after a nearby sn explosion , marine life will certainly survive to a large extent ( since uv is absorbed from a couple of meters of water ) . in the case of the earth, it took just a few hundred million years for marine life to spread on the land and evolve to dinosaurs and , ultimately , to humans ; this is less than 4% of the lifetime of a g - type star . even if land life on a planet is destroyed from a nearby sn explosion , it may well reappear again after a few 10 yrs or so .life displays unexpected robustness and a cosmic catastroph might even accelerate evolution towards life forms that are presently unknown40 myr after the last `` snowball earth '' ( which presumably occured in the neoproterozoic era , 750 myr ago ) . ] . only an extremely high frequency of such catastrophic events ( say , more than one every few 10 yr )could , perhaps , ensure permanent disappearence of complex life from the surface of a planet .gehrels et al .( 2003 ) find that significant biological effects due to ozone depletion , i.e. doubling of the uv flux on earth s surface , may arise for sn explosions closer than pc .taking into account the estimated sn frequency in the milky way ( a couple per century ) , the frequency of the sun crossing spiral arms ( gyr ) , and the vertical to the galactic plane density profile of the supernova progenitor stars ( scaleheight pc , comparable to the current vertical displacement of the sun pc ) , gehrels et al .( 2003 ) find that a sn should explode closer than 8 pc to the sun with a frequency 1.5 gyr .repeating their calculation with a realistic radial density profile for sn resulting from massive stars ( see profile of sfr in fig .1 ) , we find a number closer to gyr .note , however , the sensitivity of that number to the assumed critical distance ( ) : reducing that distance from 8 pc to 6 pc , would correspondingly reduce the frequency to 0.4 gyr . in any case , it appears that no such catastrophic event occured close to the earth in the past gyr or so .  in an attempt to circumvent the various unknowns , l04 quantify the risk for life represented by sn explosions by using the time integrated rate of sn within 4 gyr ( see fig .4 ) . adopting such a variable avoids considering the spiral arm passage and allows for a rapid implementation of the sn risk factor in a a galactic chemical evolution model ; however , the assignment of probabilities is rather arbitrary .for instance , it is assumed that for tir=1 local units , a rather strange assumption in view of the fact that life on earth has well survived the local sn rate .also , it is assume that for tir=4 local units , implying that if the local sn time integrated rate were just 4 times larger than what it has actually been , then our planet would have been permanently unable to host complex life .but such an increase in the sn rate would increase the frequency of `` lethal sn '' ( closer than 8 pc , according to gehrels et al .2003 ) , from 1 gyr to 4 gyr , leaving still 250 myr between lethal events , which is probably more than enough for a `` renaissance '' of land life .assuming for substantially larger tir values ( say , for tir=10 local units ) seems a safer bet , but it still constitutes a very imperfect evaluation of the sn threat .for comparison purposes with l04 , the `` sn risk factor '' of fig . 4is adopted also here .using the chemical evolution model of sec .2 , the probabilities for forming earth - like planets that survive the presence of hot jupiters of sec . 3 and the risk factor from sn explosions from sec . 4 , we have calculated the distribution of stars potentially hosting earth - like planets with complex life in the milky way .  the distribution of various probabilities as a function of galactocentric radius appears in fig . 5 , for five different epochs of the galaxy s evolution , namely 1 , 2 , 4 , 8 and 13 gyr . due to the inside - out formation scheme of the milky way ,all metallicity - dependent probabilities peak early on in the inner disk and progressively increase outwards .the time - integrated sn rate is always higher in the inner than in the outer galaxy .however , at late times its absolute value in the inner disk is smaller than at early times ; as a result , the corrseponding probability for surviving sn explosions , which is null in the inner disk at early times , becomes quite substantial at late times .the bottom line is depicted in the bottom right panel of fig .it shows that the overall probability of earths surviving hot jupiters and sn has indeed a ring - like shape , quite narrow early - on ( as it should , since there is basically no star formation in the outer disk at that time ) and progressively `` migrating '' outwards and becoming more and more extended .today , that `` ring '' peaks at kpc , but it is quite large , since even the inner disk ( the molecular ring ) the sn risk factor is not much larger ( just a factor of a few ) than in the solar neighborhood .obviously , the ghz extends to the quasi - totality of the galactic disk today .( left panel ) shows the same result as fig . 5 ( bottom right ) in a different way ( space - time diagram ) , comparable with fig . 3 of l04 .  in the right panel of fig6 , the radial probability of the left panel is multiplied with the corresponding number of stars created up to time . since there are more stars in the inner disk , this latter probability peaks in the inner galaxy ( in other terms : the left panel displays the probability to have complex life around _ one star _ at a given position , while the right panel displays the probability of having complex life _ per unit volume _ in a given position ) .thus , despite the high risk from sn early on in the inner disk , that place becomes later relatively `` hospitable '' .because of the large density of stars in the inner disk , it is more interesting to seek complex life there than in the outer disk ; the solar neighbohood ( at 8 kpc from the center ) is not privileged in that respect .the results obtained in this section depend heavily on the assumptions made in sec . 2 and 3 , which are far from being well founded at present .it is clear that the contours of a ghz in the milky way can not be , even approximately , defined , either in space or in time ; it may well be that most of our galaxy is ( and has been ) suitable for life .thus , the concept of a ghz may have little or no significance at all .it should be considered , at best , as a broad framework , allowing us to formulate our thoughts / educated guesses / knowledge about a very complex phenomenon such as life ( origin , development and survival ) in the milky way .zinnecker , h. ( 2003 ) , in : norris , r. and stootman , f. ( ed . ) `` bioastronomy 2002 : life among the stars '' , proceedings of iau symposium no 213 , ( san francisco : astronomical society of the pacific ) , p.45

Abstract: the concept of galactic habitable zone ( ghz ) was introduced a few years ago as an extension of the much older concept of circumstellar habitable zone . however , the physical processes underlying the former concept are hard to identify and even harder to quantify . that difficulty does not allow us , at present , to draw any significant conclusions about the extent of the ghz : it may well be that the entire milky way disk is suitable for complex life .   example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore