Physiological conversion of the maternal spiral arteries is key to a successful human pregnancy. It involves loss of smooth muscle and the elastic lamina from the vessel wall as far as the inner third of the myometrium, and is associated with a 5–10-fold dilation at the vessel mouth. Failure of conversion accompanies common complications of pregnancy, such as early-onset preeclampsia and fetal growth restriction. Here, we model the effects of terminal dilation on inflow of blood into the placental intervillous space at term, using dimensions in the literature derived from three-dimensional reconstructions. We observe that dilation slows the rate of flow from 2 to 3 m/s in the non-dilated part of an artery of 0.4–0.5 mm diameter to approximately 10 cm/s at the 2.5 mm diameter mouth, depending on the exact radius and viscosity. This rate predicts a transit time through the intervillous space of approximately 25 s, which matches observed times closely. The model shows that in the absence of conversion blood will enter the intervillous space as a turbulent jet at rates of 1–2 m/s. We speculate that the high momentum will damage villous architecture, rupturing anchoring villi and creating echogenic cystic lesions as evidenced by ultrasound. The retention of smooth muscle will also increase the risk of spontaneous vasoconstriction and ischaemia–reperfusion injury, generating oxidative stress. Dilation has a surprisingly modest impact on total blood flow, and so we suggest the placental pathology associated with deficient conversion is dominated by rheological consequences rather than chronic hypoxia. First, the blood supply to the uterus is enhanced and a maternal circulation to the placenta is established, effectively diverting blood away from the lower limbs. Second, several haemodynamic adjustments in the mother's circulation occur; her blood volume and cardiac output increase quickly by over one third, yet her blood pressure falls A range of common complications of pregnancy, encompassing recurrent first and second trimester losses, fetal growth restriction (FGR), early-onset preeclampsia, spontaneous preterm labour and preterm premature rupture of the membranes, are associated with varying degrees of failure to anatomically transform the spiral arteries of the placental bed Most of these pregnancy complications are unique to the human, and may relate to the fact that our invasive form of implantation, and subsequent haemochorial placentation, poses special haemodynamic challenges. These fall into two principal categories. Firstly, the spiral arteries of the human uterus perform contrasting roles during the menstrual cycle and in pregnancy. During the cycle they supply the endometrial stroma and the glands in preparation for implantation by the blastocyst, yet must retain the ability to contract as the upper layers of decidualized endometrium are shed during menstruation, in order to limit blood loss a. FollowSeveral recent publications have comprehensively reviewed the cellular mechanisms by which the spiral arteries undergo physiological conversion during early pregnancy 22.1The uterus is supplied by the left and right uterine arteries, which ascend along the lateral aspect within the broad ligament , and terIn the non-pregnant state the walls of the spiral and radial arteries contain large quantities of smooth muscle equipped with a rich autonomic innervation. Hence, they are highly responsive to both exogenous and endogenous adrenergic stimuli 2.2During early pregnancy endovascular extravillous trophoblast cells migrate down the lumens of the spiral arteries, while interstitial trophoblast cells migrate through the endometrial stroma and penetrate the vessel walls from their outside. In normal pregnancy, the interstitial trophoblast cells invade as deep as the inner third of the myometrium, where they progressively transform into immotile giant cells . Both enThe final outcome was elegantly demonstrated by Harris and Ramsey, who performed three-dimensional reconstructions of a number of spiral arteries from hysterectomy specimens at various stages of pregnancy Equally, it should be remembered that trophoblast invasion reaches neither the radial arteries, nor the arcuate and uterine arteries. Nonetheless, all these undergo profound dilation during pregnancy, particularly beneath the implantation site in vitro and in vivo data confirms that administration of hCG causes vasodilation, mediated through increased production of eicosanoids This non-trophoblast induced dilation is most likely a combined response to endocrine stimulation and nitric oxide-mediated flow-dilation signals, as uterine blood flow increases from approximately 45 ml/min in the follicular phase to approximately 750 ml/min at term The loss of smooth muscle from the walls of the spiral arteries will indeed convert them into flaccid conduits. The normal extent of trophoblast invasion as far as the inner third of the myometrium means that the highly vasoreactive myometrial segment of the spiral arteries is included in the conversion process. Indeed, it is intriguing that trophoblast invasion stops at this level, suggesting the function it performs has been completed. The myometrial segment of a spiral artery is responsive to steroid hormones, and is substantially remodelled before and during pregnancy It is notable that trophoblast invasion does not penetrate as far in the rhesus monkey, being restricted to the endometrium Equivalent studies have not been performed in the human, and so the extent to which intermittent perfusion of the intervillous space occurs in our species is uncertain. Radiographic images of contrast medium entering the intervillous space reveal the characteristic ‘doughnut rings’ in apparently random areas of the placenta, suggesting that it may be the case. We have also observed ultrasound evidence of brief periods of discontinuous flow Recently, a combination of sonography, vascular casting and oxygen measurements has shown that extensive shunting occurs within the myometrium under the placental bed 2.3From the mouth of a spiral artery the maternal blood is delivered into the villus-free central cavity of a lobule, as demonstrated by vascular casting 3Given the substantial changes in the more proximal uterine vasculature during pregnancy, the unique conversion of the distal portions of the spiral arteries must have additional physiological advantages other than to add in a small way to the overall reduction in uteroplacental and systemic vascular resistance. In an attempt to answer this question, we have modelled flow in the spiral arteries at term, and will consider the impact of conversion on four aspects of flow into the intervillous space that are critical for placental exchange, namely the rate, pressure, constancy and volume of flow. We will then consider the impact on uterine arterial vascular resistance.Q through each artery is governed by the local Poiseuille flow relationr(x) is the local radius of the artery, with x the downstream distance, p the pressure and μ the viscosity of the blood. Here, we have assumed that the change of the radius of the artery over a distance comparable to the radius is small compared to the radius so that, to leading order, the flow is parallel to the downstream direction x.The spiral arteries are modelled as being 10 mm in length, with a pressure drop of 80 mmHg along their length. We have taken the total flow supplied to the uterus to be 750 ml/min If the radius is constant along the length of the artery then the pressure gradient is constant, and in this case This simple calculation suggests that in the absence of any dilation at the end of the artery a very high speed jet, of 1–2 m/s depending on the viscosity and the exact radius, would enter the intervillous space, and this would have considerable momentum. Such a flow rate would imply a Reynolds number of the flow of about 20–80, and so the viscous stress will dominate the flow resistance.a < x < b, where x = b is the terminal end of the artery, with the radius increasing according to r = r0 + c(x − a) in this terminal zone, from a fixed radius r0 upstream to a final radius r0 + c(b − a), then for a given pressure drop across the artery, the speed of the flow entering the intervillous space decreases substantially as the increase in radius becomes larger, as shown in the blue line in QHowever, if there is a region of dilation in the artery near the opening into the intervillous space, then the flow speed will decrease, and the overall flow for a given pressure gradient will increase. For example, in the simplified case that there is a linearly divergent section of the artery in the region ν where ν is the kinematic viscosity of the blood, also decreases as it enters the intervillous space as the amount that the distal end of the artery diverges increases. In In turn, the Reynolds number of the jet, given by Re = ur/It is notable that the volume of flow remains similar in both these cases, with the non-diverging artery of radius 0.25 mm supplying a flow of 0.27 ml/s, and that with the dilated end providing 0.37 ml/s in the present calculations. If we assume a flow of 0.2–0.4 ml/s and a total uterine blood flow of 750 ml/min, then we estimate that 30–60 spiral arteries are required to deliver that flow. This is towards the lower end of the range given by Boyd and Hamilton Finally, it is of interest to note how the pressure varies along the spiral artery. It is seen that the pressure losses occur primarily upstream of the dilation zone, and in this distal segment the pressure is comparable to that of the intervillous space b. This e4Although we recognise the values derived from these calculations are approximations due to assumptions inherent in the model, nonetheless they allow us to appreciate the physiological consequences dependent on the terminal dilation. These will now be considered in association with the other changes in arterial structure associated with conversion.4.1The potential impact of the jets of maternal blood emerging from the arteries within the intervillous space depends on the distance they travel prior to meeting other tissues. Jets emerging from arteries with a diameter of 0.4 mm at a speed in the order of 1.0 m/s will tend to mix and entrain blood within the intervillous space, but may require distances of a few millimetres to centimetres to decelerate to speeds in the order of 10 cm/s. In contrast, the slower flow from the dilated arteries will already emerge at these lower speeds, and will continue to slow as it moves out. Therefore, the potential damage of the momentum flux in the inflowing blood will be considerably smaller if the distal end of the artery dilates . Reynold2/s. Therefore, the normal rate of flow through the lobule provides sufficient time for oxygen exchange, and this may set the scale for the lobules.It is also of interest to estimate the time of travel through the villous meshwork comprising the wall of a lobule before the blood drains into the openings of the uterine veins. Typically, lobules have a diameter of 2–3 cm, and so assuming they take the shape of a hemisphere, and the incoming flow passes uniformly through the lobule, then the travel time with a flow of order 0.2–0.4 ml/s is about 10–20 s. This fits closely with the value of 25 s for the clearance of radiopaque dye from the intervillous space at term 4.2in vivo until around 14 weeks of gestation in vivo evidence of a continuous systolic umbilical circulation from 7 weeks of gestation For effective diffusional exchange a thin barrier between the maternal and fetal circulations needs to be maintained. In the human, this is facilitated by the formation of vasculosyncytial membranes where dilated fetal capillaries obtrude from the surface of the villus, stretching the trophoblast covering and causing local thinning in vivo comes from analysis of the effects of maternal posture on umbilical arterial waveforms. In the supine position the pregnant uterus compresses the inferior vena cava, leading to a build-up in pressure and a reduction in intervillous blood flow Experiments in which the maternal and fetal circulations of term placentas were pressurised to different levels revealed that the critical determinant of fetal capillary size, and hence mean membrane thickness, is the pressure differential between the two circulations, whatever the absolute values might be 4.3The effects on rate and pressure of inflow are mediated by the terminal dilations of the spiral arteries, yet the vessels undergo conversion as far as the inner third of the myometrium where the vessel does not dilate to the same extent. As discussed earlier, we speculate that the goal of this deeper invasion is to remove the smooth muscle from the highly contractile segment of the spiral artery in the junctional zone b. Doing For much of pregnancy intermittent placental perfusion may be of little consequence, as the maternal blood retained within the intervillous space will provide a supply of oxygen and nutrients to tide the placental tissues over until the circulation restarts. However, towards term, when fetal and placental oxygen extraction are at their maximum, even temporary cessation in the circulation is likely to lead to a dip in the oxygen concentration within the intervillous space. This will be exacerbated by the fact that the volume of maternal blood acting as a reservoir around each villus will decrease as the size of the intervillous pores diminishes Physiological conversion may therefore be a means by which the conflicting requirements for vasoreactivity to support menstruation, and vaso-inactivity to support placental perfusion, can be met within a single vessel.4.4Uterine artery blood flow increases dramatically during gestation in all mammalian species whether or not trophoblast invasion occurs, and so is most likely an endocrine mediated event. The dilatation of the human uterine artery initially observed by Burchell will be more than adequate to accommodate the increase in uteroplacental flow measured during pregnancy 4.5Dilation of the distal segment of a spiral artery will reduce the resistance to flow within the vessel. One of the few studies to test for correlation between Doppler assessments of uterine arterial resistance and morphological quantification of spiral arterial trophoblast invasion in first trimester pregnancies found that low-resistance was associated with a greater proportion of the arteries displaying endovascular trophoblast The recent evidence of Schaaps et al., who observed that the resistance index in the uterine artery did not change following delivery of the placenta, indicates that flow through the spiral arteries and the intervillous space has little impact on overall resistance These data suggest that uterine artery vascular resistance may be a proxy marker for trophoblast invasion, but that is not determined by it. Other factors, such as the release of vasoreactive mediators, or inward eutrophic remodelling of the radial arteries, may be more directly associated with uterine artery resistance 4.6Reviewing these data it is clear that physiological conversion of the spiral arteries has little impact on the volume of maternal blood flowing into the placenta. In all species this is likely to be mediated by endocrine factors acting on the whole vascular arcade from the uterine arteries onwards, and on maternal blood viscosity and erythrocyte deformability. However, conversion will have profound effects on the rate of delivery, and on the constancy of the maternal blood flow. Ramsey et al. put this succinctly when they proposed that the terminal dilatations formed an antechamber to the intervillous space in which the velocity and force of the maternal blood is reduced 5It is well established that defective trophoblast invasion and a failure to convert the spiral arteries is associated with early-onset preeclampsia and FGR Firstly, the maternal blood will enter the intervillous space at a greater velocity than normal, the calculations above indicating in excess of 1 m/s . In the in vivo as “jelly-like” or ‘wobbly’ In the most severe cases of deficient arterial conversion the intervillous circulation is abnormal from the beginning of the second trimester, resulting in pronounced ultrasonographic changes in texture that have led the organ to be described In addition, even if blood entering at 1.0 m/s dilutes by a factor of 10–100 as it moves through a distance of a few millimetres to a centimetres, it will traverse much of the lobule in the order of 1 s as it decelerates and mixes. As a result, materno-fetal oxygen exchange will be impaired, even though the total volume of blood being supplied to the intervillous space from the arteries is comparable to that from the dilated arteries. This could explain the higher oxygen content in the uterine venous blood reported in cases of FGR in vitro, and of many of the changes seen in early-onset preeclampsia Secondly, it is likely that defective trophoblastic invasion will lead to a greater degree of intermittent perfusion of the intervillous space as described above. There is general consensus that it is the myometrial segments of the spiral arteries that are most affected in early-onset preeclampsia and FGR Reduced trophoblast invasion and deficient conversion are associated with increased impedance within the spiral arteries in cases of early-onset preeclampsia, although interestingly the central/peripheral differential is still maintained It seems likely that failure of conversion of the spiral arteries by itself will not influence the volume of maternal blood flow to the placenta, for this will be determined by upstream changes in the rate-limiting radial and arcuate arteries. Thus, there is little basis to assume that the placenta will be hypoxic, and indeed there is no metabolic or other evidence to support this common assumption in cases of preeclampsia 6What becomes clear from this review is that flow through the spiral arteries appears to play no role in determining the uterine arterial waveform in either normal or pathological pregnancies. Why then is there usually an association between the degree of conversion and Doppler indices of resistance to blood flow such as the pulsatility index? If we accept that endocrine mediated dilation of the radial and arcuate arteries, and the formation of myometrial arterio-venous anastomoses are the major determinants of vascular resistance in the uteroplacental circulation, then it is possible to speculate as to how the two may be linked. It is becoming increasingly clear that some form of preconditioning of the spiral arteries precedes trophoblast invasion of the vessel walls, and this may well be endocrine mediated Further research is also required to determine the morphological correlates of vascular resistance within the uteroplacental vasculature, and to investigate, for example using gadolinium-enhanced MRI, if myometrial arterio-venous anastomoses are less frequent or sizeable in pathological pregnancies with abnormal uterine artery Doppler. However, it is clear that when considering the impact of deficient conversion of the spiral arteries by itself on intervillous blood flow we should be thinking in terms of the velocity, pressure and constancy of the maternal inflow, and bear in mind that there may be associated secondary changes that have a greater impact on the volume of that flow.


Analyses of similarities and changes in protein conformation can provide important information regarding protein function and evolution. Many scores, including the commonly used root mean square deviation, have therefore been developed to quantify the similarities of different protein conformations. However, instead of examining individual conformations it is in many cases more relevant to analyse ensembles of conformations that have been obtained either through experiments or from methods such as molecular dynamics simulations. We here present three approaches that can be used to compare conformational ensembles in the same way as the root mean square deviation is used to compare individual pairs of structures. The methods are based on the estimation of the probability distributions underlying the ensembles and subsequent comparison of these distributions. We first validate the methods using a synthetic example from molecular dynamics simulations. We then apply the algorithms to revisit the problem of ensemble averaging during structure determination of proteins, and find that an ensemble refinement method is able to recover the correct distribution of conformations better than standard single-molecule refinement. Protein structures play an important role in molecular biology, and are used for example in protein engineering studies, in drug design and as basis for understanding biological mechanisms at the molecular level. However, not only the average structure but also the dynamics around this structure plays a role in function The ability to compare different protein conformations is an important tool in structural biology In a pioneering study, Brüschweiler extended the RMSD measure to ensembles of conformations We here describe three alternative methods to compare ensembles of conformations. The methods are all based on the idea that two ensembles can be compared by estimating the probability densities underlying the ensembles. Using distance measures for probability densities that are based in information theory we can then provide a direct measure of the similarity between protein ensembles.We first explore the properties of the methods and validate them by examining approximately normally distributed conformations obtained from MD simulations. We then use the methods to revisit the problem of ensemble averaging during protein structure determination from NMR dataOur strategy towards a quantitative comparison of conformational ensembles is first to estimate a probability density for each ensemble and subsequently to compare these densities. We thus view a particular set of conformations as a sample from an underlying distribution and aim to model this distribution based on the sample at hand. By comparing the probability densities we are not only able to test whether the conformations in the two ensembles are similar, but also whether they occur with the same frequency. The latter point is an important criterion in many applications such as for example in the analysis of convergence of MD simulations The problem of comparing two ensembles is thus broken down into two steps A and B, and an estimate of the corresponding density functions, A and B is given as the distance between Given two ensembles, In an information theory context, J-divergence defined as the average of A noteworthy property of Another symmetrized, and ‘smoothed’, version of For probability distributions of discrete variables the integrals in the calculations of Molecular dynamics and structure determination below.We devised three complementary methods for calculating ensemble similarities. These three methods (harmonic ensemble similarity (HES), a clustering based similarity and a dimensionality based similarity) are all described in more detail in the results section and we here provide only some of the more technical details. We applied all three methods to a series of protein ensembles that were generated as described in the subsection First, the harmonic ensemble similarity was calculated using Eq. 7. In those calculations the mean positions of the Secondly, the clustering based ensemble similarities were calculated using the recently described Affinity Propagation (AP) clustering algorithm S we obtain low dimensional vectors whose pairwise distances are close to the conformational similarities. The basic step in the SPE algorithm involves an update of the positions of the vectors in the subspace. The size of this update is determined by a learning rate which we decreased from 1.0 to 0.001 over 500 steps. At each value of the learning rate we performed 6·106 update steps for the calculations using ensembles A–C  and 2·106 update steps for the calculations using the reference and NOE-derived ensembles . We verified that these values of the parameters in SPE were sufficient to find low values of S, and that the results could be reproduced using multiple independent runs.The final method for calculating ensemble similarities is based on a dimensionality reduction approach and consists of two steps. First, we use the Stochastic Proximity Embedding (SPE) algorithm In the second step of the calculations we use kernel density methods to estimate the densities in the projected subspaces l'th structure in the A ensemble, k'th structure in the B ensemble, and M (N) is the number of conformations in the A (B) ensemble.For comparison with the ensemble RMSD method we calculated the squared value of the eRMSD using its definition i'th atom, k, the force constant, chosen to be 0.1, 0.01 and 0.001 in ensemble A, B and C, respectively. The structure was first heated to 300 K over a period of 0.2 ns and then equilibrated for further 2 ns at this temperature before final sampling for 10 ns. Structures were saved every 4 ps giving rise to 2500 structures in each ensemble.In this section we describe how the different structural ensembles used in the study were generated. For the first validation part of our study we generated three ensembles (A–C) of the GB1 domain of protein G using MD simulations. The starting structure for the simulations was the first model in the PDB entry 3GB1. We used the polar hydrogen model In the second part of our study we analysed to what extent structure determination methods can recover the correct distribution of conformations. We thus generated a reference ensemble of the GB1 domain using an MD simulation with the CHARMM22 topology format and force-field For the generation of a synthetic NOE dataset we calculated the distances between all pairs of protons in each conformation, and then determined effective distances The synthetic data were subsequently used as input to a structure determination protocol. To diminish the bias introduced by using the same force field to generate the data and in the structure determination protocol we used a different solvation model All simulations were performed using the CHARMM molecular dynamics program As described in the x, is given by:n is the number of degrees of freedom, 〈x〉 are the mean coordinates, Σ is the covariance matrix that contains information about the fluctuations of the ensemble, |Σ| is its determinant and A is drawn from an underlying Gaussian distribution We model the protein ensemble as generated from a probability density function. The form of such density functions are, however, in general unknown and must either be guessed or inferred from the sample. A commonly used approximation is to assume that the ensemble is derived from a multivariate normal distribution n, and tr denotes the trace of a matrix.When J-divergence defined as the average of We have been unable to derive a closed-form expression for k = 0.1), B (k = 0.01) and C (k = 0.001). As the force constant is decreased, the amplitude of atomic fluctuations become larger and the ensembles thus become increasingly more ‘broad’. We extracted 2500 structures from each MD simulation and all of these structures were used in the calculations described below unless otherwise stated. By construction, the three ensembles were generated so that A and C would both be more similar to ensemble B than to each other. In practice we find that the three ensembles have similar average structures, with the In order to demonstrate the ability of n = 3·56 = 168 degrees of freedom  = 0.45 Å, eRMSD = 0.94 Å, and eRMSD = 0.91 Å. The eRMSD thus gives a similar ordering as the harmonic ensemble similarity score, although ensemble C is found to be only slightly more similar to B than to A. However, while the self-similarity using eRMSD = 0.26 Å, eRMSD = 0.50 Å, and eRMSD = 0.92 Å. Taken literally, this means that ensemble C is more similar to B than to itself, which makes it more complicated to use the eRMSD to rank the different similarities.In the comparison of the three ensembles we focus on the 56 A′, B′ and C′ and present the results as the average and standard deviation over 100 randomly selected subensembles. The resulting similarity scores are In order to demonstrate that the calculations of n). While we have demonstrated above that the n×n covariance matrix can be estimated from less than n conformations using the shrinkage-approach, we have until now only looked at the fluctuations of the n = 3N where N is the number of residues. However, in many cases it is of interest to include fluctuations of side chain atoms in the comparison of the ensembles. Even ignoring hydrogen atoms in the GB1 domain, n>1000 making the estimation of the covariances from small ensembles difficult.Although the calculations of k, is chosen based on the ‘preference’ for each conformation to act as the centre of a cluster. We set the preferences to a common value for all conformations so that all conformations are equally likely to become cluster centres, and vary the preference-value to obtain different levels of clustering. After the structures are clustered we take the population, X) in each cluster (i) as a probability distribution of conformations. This idea is illustrated in X, k = 12.Recently, a method has been proposed to analyse the convergence of MD simulations. The idea is to quantify the co-occurrence of conformations from the first and second half of a simulation when the two halves are clustered together k = 7500. In this limit the three ensembles do not share any clusters making At high values of the common cluster-preference value each conformation ends up in its own cluster and hence culated  . This pl50 bins.One advantage of the clustering method described above compared to the harmonic ensemble similarity score is that it does not require that the ensembles are distributed according to a normal distribution. A potential limitation of the approach is the discreteness of the clusters which means that any conformational differences between structures in the same cluster are ignored. Similarly, any similarities between structures in separate clusters are also ignored. We therefore sought an alternative approach which would provide a continuous description of the distribution of conformations. The main obstacle here is the so called ‘curse of dimensionality’ which highlights the problem of estimating densities in high dimensions without having astronomically large samples d that is much smaller than the full n-dimensional configuration space d-dimensional projection that aims to preserve these distances as well as possible K-means clustering problem Dimensionality reduction methods such as principal components analysis (PCA) and multi-dimensional scaling (MDS) have a long tradition in structural biology, and can be used to represent protein conformations in a subspace of dimension An important underlying assumption of methods such as PCA and MDS is that the conformational space is linear. In practical terms this means, for example, that the algorithms give equal weight to an RMSD of 1 Å and 10 Å between two conformations when the projection is constructed, whereas it is clear that large RMSD values are not very useful measures of the similarity between conformations Other non-linear projection methods exist, and we have here chosen Stochastic Proximity Embedding (SPE) d-dimensional vector and finds a collection of vectors so that the Euclidian distance between two vectors in the d-dimensional projection is close to the RMSD between those two conformations if they are neighbours of each other (i.e. the RMSD is within a specified cut-off). Only for large values of d can all the restraints be fulfilled perfectly. For smaller d we calculate the remaining ‘stress’, S, as a measure of how well the restraints are fulfilled:n = 168. In order to illustrate the low dimensional projections we show in The SPE algorithm represents each conformation as a n-dimensional conformation space, they will be distributions over the generalized coordinates in the d-dimensional subspace. As we do not want to restrict ourselves to particular distributions of conformations we use non-parametric density estimation to quantify the similarities between the ensembles d-dimensional samples. In The reduction of the dimensionality of the data makes it feasible to estimate the probability density of each conformational ensemble. Instead of being distributions over the full After validating the methods using the ensembles described above, we then applied the ensemble similarity calculations to an important problem in structural biology. We wanted to examine to what extent it is possible to recover the correct distribution of conformations based only on a set of experimental data, and whether different structure determination approaches differ in their ability to recover the distributions. In particular, we revisit the problem of whether ensemble (multi-conformer) refinement using only NOE distance restraints can be used to recover the conformational distribution of a protein Most standard NOE-based structure determination protocols apply the available distance restraints on to a single conformation. That is, an optimization algorithm  is used to find individual conformations that satisfy as many of the restraints as possible. Through multiple rounds of simulated annealing one then obtains an ensemble of conformations. It is important to note that in this approach it is attempted that all conformations should individually satisfy the restraints. However, these conformations may still differ because of the stochastic nature of the optimization algorithm and because the distance restraints typically do not uniquely identify a single conformation It is clear, however, that in reality — as well as with the synthetic data generated here — the distance restraints reflect a distribution of conformations, and that no single conformation needs to agree fully with the restraints. In order to reflect this ensemble view of the experimental data, it is possible to use the data in so-called ensemble refinement protocols A practical problem with ensemble simulations is that as We used the global ensemble similarity scores described above to quantify whether the ensembles obtained from ensemble simulations  RMSD between all 2307 conformations (reference ensemble and five ensembles determined using NOE restraints). These distances were then used as input to the calculations of ensemble similarities using both the clustering and the projection approach.We first applied the clustering method to the six ensembles. We repeated the calculations using a range of preference values and calculated To analyse further the distributions of conformations in the six ensembles we used SPE to project the 2307 conformations in to low dimensional subspaces. In these calculations we defined neighbouring structures as those that had an all-atom (excluding hydrogens) RMSD less than 1.75 Å, but validated that the conclusions below are the same using other definitions of neighbours (not shown). In We have presented three different algorithms to compare two or more ensembles of protein conformations. We first tested the methods using three ensembles, A–C, obtained by MD simulations with additional harmonic restraints. We then proceeded to use the methods to examine how well an ensemble refinement method is able to recover an ensemble of conformations using (synthetic) experimental data.The first measure we present is the harmonic ensemble similarity. It is based on the assumption of a multivariate normal distribution whose mean and covariance are estimated from the ensembles at hand. Two ensembles are then compared using an analytical expression of the Kullback-Leibler divergence between two normal distributions. The approach is therefore related in spirit to the allosteric potential recently introduced The harmonic ensemble similarity method is based on the assumption of a normally distributed ensemble, and it may not always be clear how to interpret A potential limitation in the calculation of In an attempt to evaluate some of the potential practical problems with the harmonic ensemble similarity we devised and tested a similarity score based on clustering the conformations in the ensembles. In particular we used the co-occurrence in conformational clustering as an of estimate for the similarity of two ensembles. In the original implementation of this idea the differences in cluster populations were used as a measure of the similarity of two ensembles at the chosen clustering level. The limitations of the clustering method arise because of the discrete nature of the clusters over which the ensembles are compared, so that conformations in the same cluster are considered equivalent irrespectively of how similar they are.The idea of analysing co-occurrence in clusters was originally designed to examine the convergence of molecular simulations, and hence mainly to answer the question of whether two sub-ensembles are likely to originate from the same distribution In order to overcome these potential problems we devised an approach to estimate the density function over a set of continuous variables. In this way we hoped to solve the problems associated with the discreteness of the clusters. Because of the high dimensionality of conformational space it is very difficult to estimate the densities directly Finally we applied the similarity measures to reexamine the problem of ensemble averaging in structure determination. This is an important problem as it is well known that a wrong interpretation of the dynamical contribution to distance restraints may cause errors in structure determination An important and general method to compare different structure determination protocols is to generate synthetic data and use this as input to the different protocols. The idea is then to compare the ensembles obtained from structure determination with that used to generate the synthetic data. Until now, such comparison have mainly been carried using validation with independently determined data, cross-validation, measures of the local structural similarity We here use the ensemble similarity scores as a direct method for quantifying how well the reference ensemble can be recovered. The results show clearly that not taking averaging in to account, i.e. having As standard structure determination protocols correspond to The three ensemble comparison methods that we present each have different strengths. The harmonic ensemble similarity is simple to calculate and is based on an analytical relationship (Eq. 7). The equation clearly highlights the fact that for two ensembles to be similar both their ‘average’ structure as well as the fluctuations away from this should be similar. Importantly, the computational complexity in the calculation of the harmonic ensemble similarity is only linear in the number of conformations. In contrast, in the current implementations of the two other similarity scores the algorithms require all pairwise RMSDs, the calculations of which may become prohibitively expensive for very large ensembles. For these reasons we suggest that the harmonic ensemble similarity should be used as a starting point for comparing protein ensembles. In cases where significant deviations from the a normal distribution are expected (or found) we suggest to supplement the calculations of We hope that the methods we present will be of use in many areas of structural biology. For example, the idea of using reference ensembles to validate structure determination protocols has recently been applied to non-native states of proteins


Vapor diffusion is the most widely used technique for protein crystallization and the rate of water evaporation plays a key role on the quality of the crystals. Attempts have been made in the past to solve the mass transfer problem governing the evaporation process, either analytically or by employing numerical methods. Despite these efforts, the methods used for protein crystallization remain based on trial and error techniques rather than on fundamental principles.Here we present a new theoretical model which describes the hanging drop method as a function of the different variables that are known to influence the evaporation process. The model is extensively tested against experimental data published by other authors and considering different crystallizing conditions. Aspects responsible for the discrepancies between the existing theories and the measured evaporation kinetics are especially discussed; they include the characterization of vapor-liquid equilibrium, the role of mass transfer within the evaporating droplet, and the influence of the droplet-reservoir distance.The validation tests show that the proposed model can be used to predict the water evaporation rates under a wide range of experimental conditions used in the hanging drop vapor-diffusion method, with no parameter fitting or computational requirements. This model combined with protein solubility data is expected to become a useful tool for a priori screening of crystallization conditions. Obtaining good quality crystals is a critical step for protein structure determination by X-ray crystallography. The most commonly employed techniques to grow crystals of biological macromolecules are by vapor diffusion As a consequence of the key role of the solvent evaporation kinetics on the vapor diffusion technique, a comprehensive mathematical treatment was proposed for the first time in 1988 to describe the hanging drop method The aim of this paper is to provide the mathematical description of the hanging drop vapor-diffusion method, so that it can be used to predict the water evaporation rates under a wide range of experimental conditions, without the limitations of the existing models. The new model shall be useful for obtaining good protein crystals regarding their quality, size and number, and should be combined in the future with complementary kinetic theories of crystal nucleation and growth R, and with a radius of curvature R that varies with time. The inner diameter of the reservoir is given by 2a, and the vertical distance from the point O to the cover slip corresponds to R sin αR. aR is the radius of the sphere centered on O that intercepts the reservoir walls at the level of the cover slide, and bR is the radius of the concentric sphere that is tangent to the surface of the solution in the reservoir:R = 0°, and therefore aR = a and bR corresponds to the vertical distance from the cover slip to the solution in the reservoir, b. Different implications arise from the simplifying hypothesis adopted on the derivation of the model. This will be gradually illustrated by interpreting the limitations of the existing theories and by proposing new ways to overcome those limitations. We start by considering the simplest case of aR≥bR, and the evaporation kinetics is assumed to be exclusively determined by the solvent diffusion through the gas phase. Then, the more common case of aR<bR will be analyzed, and finally, the importance of the diffusion step within the droplet will be assessed. The model validation will be presented using experimental data reported in literature by different authors on the equilibration rates of different water-precipitant systems, under varied experimental conditions.As represented in aR≥bR) and that the evaporation of a droplet containing the solvent (water) and the precipitant occurs at a given rate that is exclusively determined by the vapor diffusion across the air space between the droplet and the reservoir. The latter assumption was initially justified in the derivation of the FM R0 gets progressively smaller, the geometrical center O moves toward the coverslip so as to provide that the droplet contact angle αR remains constant. This implies a small decrease of the radius bR (and aR) with time, which will be ignored since the droplet size is generally much smaller than b (and a) Let us start by considering the simplest case of small droplet-to-reservoir distances  results from solving the continuity equation for steady-state:p|rR =  = Rp and R = 0°.The mass transfer area (A) for vapor diffusion over the air space is of a spherical cap centered on O with radius r comprised between I1) is evaluated at r = R according to Fick's first law:A is the surface area shape factor of the droplet (2π(1−sinαR)), D is the diffusion coefficient of water vapor in air, ℜ is the gas constant and T is the absolute temperature. Hereafter, the subscript 1 stands for water and the subscript 2 for the crystallizing agent. Substituting Equation 6 into the previous equation, and lettingR′b reduces to bR when αR = 0°:Rp decreases in the direct proportion to the water molar fraction in the droplet x1, following Raoult's Law for vapor-liquid equilibrium:1 is the activity coefficient and p* is the vapor pressure of pure water. The presence of protein is not considered to significantly affect Rp, nor should it affect the water equilibration rates x2; in the reservoir, this value is assumed to remain equal to xb2, while in the droplet, it will increase as the droplet volume decreases. Accordingly, Equation 9 is rewritten asx2,0 and V0 are the initial precipitant molar fraction in the droplet and the initial droplet volume, respectively. The molar rate of water vapor leaving the droplet can also be expressed as a function of the droplet change of volume with time and of the molar volume of pure water (V̅1):by corresponds to y evaluated at R = R′b, y∞ is the dimensionless radius of the droplet achieved at the equilibrium, when dy/dt = 0:y) and the time elapsed since the beginning of the evaporation (t) by integrating Equation 16, subject to the initial condition y|t = 0 = 1:R = 0°, and will not be discussed here. Concerning the VLE, while we propose Raoult's Law to relate the vapor pressure at the droplet surface and the droplet composition is given in terms of molar fractions (Equation 11), in the FM (as well as in the subsequent models SM and LM) that relationship is given as a function of the number of moles of water and salt in solution , and of the vapor pressure lowering coefficient w:y∞). As we have previously defined (Equation 17), y∞ is a function of the 1/3 power of the relative molar fractions in the droplet and in the reservoir (dilution factor), while in the FM y∞ is a function of the molar volumes of water and precipitant, and of the respective number of moles in the droplet and in the reservoir 3/mol y3∞ expected from Equation 17.The molar rate of water vapor leaving the droplet . Now, one finds from trigonometric transformations that:r = aR and r = bR will be called αa and αb, respectively. The geometrical shape of the vapor space should therefore be divided in two zones separated at r = aR. For r < aR, the vapor pressure profile is again obtained from Equation 5, subject to the boundary conditions p|rR =  = Rp and ap is to be determined from the condition of continuous flux at the boundary . The same procedure was adopted, and applied to Equation 19, which predicts that the NaCl concentration should be independent of the droplet-to-reservoir distance after the considered evaporation period has elapsed. The agreement of the proposed model (Equation 27) with the collected data is remarkably good for low, intermediate, and large values of b (R = 0°) that remains unchanged during the evaporation period. The four points of As demonstrated in the fundamental analysis presented so far, the applicability of theoretical models of the hanging drop method is in a good deal determined by the value of the droplet-to-reservoir distance (and its relation with the reservoir diameter). The same conclusion was drawn by measuring the dependence of water equilibration kinetics on the droplet-to-reservoir distance ues of b , especiar<aR, is now subject to the boundary conditions p|rR =  = ip and ip is the effective interfacial vapor pressure. On the other hand, the interfacial vapor pressure is in equilibrium with the interfacial composition in the liquid phase represented in xi1 . Apart from the alteration in the boundary condition, the boundary value problem is solved as in the previous section to obtain a new equation for I1 that is analogous to Equation 24:xi2 and the precipitant molar fraction in the bulk of the droplet, xd2, is given as a function of the mass transfer coefficient of water in solution, ck, using the following expression for the current I1:In the theoretical study done by Fowlis et al. on the several resistances that determine the evaporation rate during the hanging drop method, the diffusion of water molecules from the interior of the droplet to the surface was estimated to be of higher magnitude than the diffusion of vapor across the air space I1 given by Equations 13, 28 and 29 can be combined to give the ODE of the variation of the droplet radius with time. Accordingly, after expressing xd2 as a function of the initial concentration in the droplet, x2,0, and using the dimensionless constants τ, y∞ and cy, one obtainsy|t = 0 = 1) provides the most general form of the equilibration curve, contemplating liquid and vapor phase diffusion, and the cases where aR≤bR:ck) are considered in Equation 31. In this case, a good agreement with the experimental data was found for β = 2.06, which corresponds to ck = 1.5×10−5 cm/s. This value is close to a rough estimation of the mass transfer coefficient of 4×10−5 cm/s, obtained using a value of the liquid water diffusion coefficient of 10−5 cm2/s ck obtained from the results shown in 1/3 was applied to that value to account for the new diffusion path in the smaller drop, so that in the simulation of ck = 2.38×10−5 cm/s. As it arises from the definition of β, the same value of 2.06 is obtained for this parameter after the droplet radius correction. Accordingly, the equilibration rates of the 8 µl droplets were simulated and the predicted profiles are again in good agreement with the experimental data , the solution thermodynamics is also altered according to the solubility curve of the system. We envisage that future work will address the incorporation of nucleation and crystal growth models on the full characterization of the crystallization technique. This is important, but not straightforward since several parameters need to be studied. For example, depending on the hydrophobicity of the coverslip material, different evaporation kinetics are expected (due to different contact angles and shapes of the droplet) but also different nucleation kinetics will occur The applicability of Equation 32 can also be extended in the future to vapor diffusion techniques in the presence of oils A new equilibration rate model was presented describing the hanging drop method for protein crystallization. The presentation of the model was progressively made with increasing degrees of complexity as a consequence of the model assumptions that were being considered. This also allowed emphasizing the differences between the present theory and existing models, namely the distinct ways of characterizing the vapor-liquid equilibrium, the geometrical assumptions involved in the different models, and the role allocated to water diffusion within the droplet. Several measurements of water equilibration rates using different precipitants at the different experimental conditions were taken from literature and compared with the predictions of the proposed model. The experimental curves described the variation of the droplet volume  with the evaporation time, and with the droplet-to-reservoir distance, at a fixed evaporation time. Good validation results were obtained in all cases. In the performed simulations, only the experimental parameters and physicochemical constants reported in the analyzed literature data were used. Precipitants which are expected to significantly increase the water diffusion resistance within the droplet were indentified. The clearest example was MPD, for which water mass transfer coefficients were possible to be estimated for the liquid phase. The obtained coefficient was then used to successfully predict the water equilibration rates in the presence of MPD, at different conditions. The proposed model will be used to predict the water evaporation rates in a variety of different conditions used in the hanging-drop vapor diffusion technique for protein crystallization. In fact, this model is a consistent step for a rational protein crystallization experimental set up.


F               o − F               c)exp(iϕc) difference density from macromolecular structures in the Protein Data Bank.An automated ligand-fitting procedure has been developed and tested on 9327 ligands and ( F               o − F               c)exp(iϕc) difference density calculated using entries from the PDB without these ligands. The procedure was able to place 58% of these 9327 ligands within 2 Å (r.m.s.d.) of the coordinates of the atoms in the original PDB entry for that ligand. The success of the fitting procedure was relatively insensitive to the size of the ligand in the range 10–100 non-H atoms and was only moderately sensitive to resolution, with the percentage of ligands placed near the coordinates of the original PDB entry for fits in the range 58–73% over all resolution ranges tested.A procedure for fitting of ligands to electron-density maps by first fitting a core fragment of the ligand to density and then extending the remainder of the ligand into density is presented. The approach was tested by fitting 9327 ligands over a wide range of resolutions (most are in the range 0.8–4.8 Å) from the Protein Data Bank (PDB) into ( Tickle et al., 2004Fitting of ligand density is an important step in the completion of macromolecular structures. It is often carried out as one of the very last steps in structure determination, after essentially the entire macromolecule and most solvent molecules have been fitted and refined. In pharmaceutical settings, many hundreds of structures may be solved in which the principal difference between these structures is the ligand  and then to build the remainder of the ligand from this core by following the density, keeping in mind the stereochemical constraints of the ligand. This approach is attractive because it is similar to the approach that an experienced crystallographer would use. More importantly, it is suitable for large flexible ligands because the process is sequential and scales relatively linearly with the size of the ligand. We describe here the method and its application to over 9000 ligands from the PDB.2.2.1.We assume that the conformation of the ligand in the structure to be modeled can be generated from another conformation of the same ligand by simple rotations around bonds. A few simple rules are used to decompose the ligand into a set of overlapping fragments, each of which has no internal rotatable bonds but which is connected to at least one other fragment through a rotatable bond. Only non-H atoms are considered in the analysis of a ligand. These rules are, owing to their simplicity, quite incomplete; they are intended to give a first approximation to the geometrical features of the ligand.(i) Any two atoms separated by less than the sum of their ‘maximum half-bond lengths’ (defined below) are bonded.(ii) Any set of atoms in a ring or set of rings with 20 or fewer atoms are in a fixed arrangement.A and C bonded to a central atom B are in a fixed arrangement (i.e. the angle A—B—C is fixed).(iii) Any two atoms A—B—C—D that are coplanar are in a fixed arrangement .(iv) Any set of four bonded atoms A—B—C—D that are not specified as having a fixed arrangement can have any rotation about the B—C bond that does not place A—B—C—D within a specified tolerance  of being coplanar and that does not place any atoms separated by two or more bonds closer than a specified tolerance.(v) All sets of four bonded atoms In this analysis, a group of atoms that is always in a fixed arrangement is considered a fixed ‘fragment’ of the ligand. Pairs of fragments that are connected (through a rotatable bond) will always have the two atoms that form this bond in common and will share no other atoms.Our rules are as follows.Once a ligand has been broken down into a set of rigid fragments connected by rotatable bonds using these rules, it is simple to construct possible conformations of the ligand. Firstly, the location and orientation of any rigid fragment is fixed. Any one of the fragments that are connected to this fixed fragment is then placed. The placement of this second fragment is determined by the bond that connects the two fragments and by the rotations allowed around this bond by rule (v) above. This process is repeated until all fragments are placed. The procedure can start with any of the rigid fragments and any order of addition of fragments connected to each other can be followed.A limited default set of maximum half-bond lengths is used to identify commonly bonded atoms. These are essentially the half-bond lengths for these atoms plus a tolerance of about 0.1–0.3 Å. The maximum half-bond-lengths used are C, N, O, 0.8 Å; S, Br, I, 1.5 Å; P, F, 1.0 Å. The algorithm is not very sensitive to these values because the van der Waals radii of most atoms are far greater than their half-bond lengths so that there is usually little question as to whether two atoms are bonded. Maximum half-bond lengths for atoms that are not in the default set are estimated from the distance to the nearest other atom in the ligand and the half-bond length of that atom.sp               2-hybridized bond are treated as fixed, while they could have either of two possible configurations. These limitations mean that a complete search for the conformation of a ligand that has more than one possible conformation of atoms in a ring or about an sp               2-bonded pair of atoms needs to be carried out more than once, beginning with examples of the ligand that have each of these conformations. In essence, the same ligand with different conformations of these types needs to be treated as two different ligands in our approach. We use antibumping constraints to ensure that atoms in the ligand avoid serious overlap with other atoms in the ligand and with other atoms in the structure of the macromolecule. If any two atoms in the ligand are not bonded to each other, they must be separated by at least the sum of their maximum half-bond lengths plus 1 Å or by the distance that they are separated by in the starting ligand conformation minus 0.5 Å, whichever is smaller.Our rules have limitations, but they are sufficient in many cases to identify most or all of the rotatable bonds. A general limitation of the approach is that the bond angles and lengths are all assumed to be identical in the ligand to be fitted and the ligand used to generate it. While significant, this limitation can in principle be overcome by subsequent refinement of the ligand structure. Another significant limitation is that the conformations of all the atoms in a ring are assumed to be fixed. While nearly true for aromatic rings, rings such as those in sugars can have alternative conformations that are quite different from each other. Similarly, two groups of atoms connected by an 2.2.et al., 2004P1 with the same grid spacing as the original map and all the searches are performed within this small map.Our approach to modeling a ligand begins with finding plausible placements of core fragments of the ligand in density. We use a simple approach to limit the search for the locations of these fragments to a small region within the unit cell by only searching within and near the largest and highest contiguous region of density in the map. This approach is similar to that used in several other ligand-fitting algorithms  and used in a convolution search for similarly shaped density in the map. The orientation and positions  yielding the highest overlap between the core fragment and the map are then refined based on correlation coefficient of density calculated from the core fragment and density in the map and the top refined placements  are saved. These top placements of each of several rigid fragments of the ligand form the starting points for ligand-building trials.The positions and orientations of core fragments of the ligand are identified using an FFT-based convolution search (2.3.i.e. more complete) ligands as long as the density is positive. The procedure terminates when all the top-scoring ligands have served as templates for addition of further fragments and no new top-scoring ligands are found.Once a rigid core fragment has been placed at a particular location and in a particular orientation in the unit cell, building the remainder of the ligand consists of an iterative procedure. A fragment that can be connected to the already built part of the ligand but which is not yet placed is picked, a placement for this fragment is chosen and the new partial or complete ligand is scored as described below. Typically, possible orientations for the additional fragment are sampled at intervals of 20°. At any point, a list of top-scoring partial and complete ligands is maintained . The scoring procedure is designed to favor larger (2.4.N atoms in the ligand (ρi) and on the atomic number of each atom in the ligand (iZ). The score Q is given byIn this ligand-fitting procedure, the central criteria for choosing a particular configuration of a ligand or partially built ligand is the fit of the ligand to the density. Aside from the initial fitting of core fragments to density described above, we use a simple score for this fit that is based on the density at the coordinates of the An additional criterion is used to help ensure that all atoms in the ligand are above a threshold of minimum density. Typically, the minimum allowed density for any atom in the ligand is −1.0 times the r.m.s. of the map.Configurations of partially built cyclic ligands that cannot possibly be made to cyclize are also eliminated. These are identified as configurations in which any two fragments are so far apart that no arrangement of the fragments linking them can possibly connect.2.5.O for each crossover given typically byThe procedure described above produces a list of partial and complete fitted ligands ranked by their fit to the electron-density map. These fitted ligands will have been generated starting from different core fragments and therefore will have in general been traced beginning from different parts of the density for the ligand. We developed a procedure for recombination among the fitted ligands to increase the quality of the fit to density followed by further addition of fragments to increase the completeness of the ligand. In this step the requirement that all atoms be above a minimum threshold is removed so as to create essentially fully complete ligands even if some atoms do not match the density. The recombination is carried out among pairs of ligands in the top group of saved ligands . A new ligand is built from fragments of the two existing ligands, beginning with a fragment from one ligand, adding fragments connected to the original fragment one at a time, then at selected points crossing over to the corresponding fragment from the other ligand. Crossovers of this kind are made only between two copies of a fragment where the coordinates of atoms in the two copies match within a specified tolerance . During the creation of crossovers, the score of a ligand is modified based on the r.m.s.d. between atoms in the fragments, with the offset in score 3.3.1.et al., 2000et al., 2002CCP4 program cif2mtz exp(iϕc) difference map was calculated. The correlation coefficient of the original ligand with this F               o − F               c difference density (ccorig) was noted and used as a basis for classifying the quality of the difference density. Ligand fitting was then carried out using a second ligand file generated from another copy of the same ligand. In most cases, the second ligand file was derived from a different PDB entry (6930 ligands). If this was not possible, the second ligand file was derived from a second instance of the ligand in the same PDB entry (862 ligands) or, if there were no other instances of the unique ligand (1535 ligands), the original ligand was used but oriented arbitrarily so that it could not be simply replaced.The 9327 ligands were fitted in the following way. For each ligand, two separate PDB files were generated: one containing non-H-atom records for the ligand only (ligand file) and one containing all other non-H-atom records in the same PDB entry (minus file). These files were generated from an Oracle database that had been populated using v.1.5.1 of the The difference map, the second copy of the ligand and a mask calculated from the coordinates without the ligand were used as the inputs to the ligand-building procedure described in §F               o − F               c difference maps were classified according to the correlation of difference density and density calculated from the original ligand in the PDB entry (ccorig). There were 6590 ligand–PDB entry combinations with clear density for the ligand as found in the PDB entries (ccorig ≥ 0.75). For these cases, the mean correlation after fitting was lower (0.76) than the mean correlation for the original ligands (0.85). However, 41% of the fitted ligands had r.m.s. coordinate differences relative to the original ligands of less than 1.0 Å and 71% had an r.m.s.d. of 2.0 Å or less from the coordinates of the original ligands. Only 310 (5%) were placed very differently from the original ligands . For the 2737 ligand–PDB entry combinations with weak density for the ligand as found in the PDB entries (ccorig < 0.75), 41% had an r.m.s.d. of more than 10 Å from of the coordinates of the original ligands; however, the mean correlation of the fitted ligands (0.60) was nearly the same on average as that of the original ligands as found in the PDB (0.61), suggesting that equally well fitting density was found in most cases.Table 1For the entire set of 9327 ligand–PDB entry combinations, 5421 (58%) were rebuilt with an r.m.s.d. of 2.0 Å or less from of the coordinates of the original ligands and 1425 (15%) had an r.m.s.d. more than 10 Å from of the coordinates of the original ligands.In an actual fitting experiment, water molecules would normally either already be placed (including placements in the ligand-binding site) or would never be placed at all. Furthermore, the structure would normally be refined without the ligand present, leading potentially to weaker density for the ligand. We did not test the effect of refinement, but we did test the effect of including the solvent. We expected that solvent molecules in the PDB entries could have an effect on the ligand-fitting procedure both through exclusion of some locations from being considered as a place for a ligand to be located and through contributions to structure factors. We tested this effect by carrying out a matched pair of fitting experiments which differed in that all water molecules were removed prior to map calculation and fitting for one experiment. In these experiments, ligand fitting was carried out beginning with the original ligand from the PDB entry under consideration. We tested an arbitrarily chosen set of 2641 ligand–protein pairs (including multiple instances of a ligand from the same PDB entry) fitted with and without solvent molecules and the two approaches were found to yield similar but not identical results. The mean of the correlation coefficient between matched ligand–PDB entry pairs was 0.02 higher with solvent. The fraction of ligands rebuilt with an r.m.s.d. of 2.0 Å or less from of the coordinates of the original ligands was 67% for this set of ligand–PDB entries including water molecules and 58% after removing them. These results suggest that the fitting obtained with the inclusion of water molecules as detailed in Table 13.2.a) illustrates the fit of a single copy of this ligand and Fig. 1b) the fits of five copies. Fig. 1c) shows the range of r.m.s.d. values of 74 fits from the coordinates in the original ligand in the PDB entry and the correlation coefficients of each fitted ligand to the F               o − F               c density map and Fig. 1d) shows histograms of the correlation coefficients. Most of the 74 fits yield correlations between 0.7 and 0.8 and r.m.s.d. values less than 1.5 Å. This indicates that the procedure, while not giving precisely the same conformation in every trial beginning with a different conformation of the ligand, gives a relatively reproducible fit of the ligand to density.We next examined the reproducibility of the procedure and sensitivity to the starting ligand conformation by fitting a NAG-NAG-NAM ligand 74 different times, each time using a different conformation of the ligand from an entry in the PDB. Fig. 13.3.F               o − F               c density from the PDB according to the resolution of the data used. In order to focus on the resolution of the data and minimize effects of differing qualities of maps at different resolutions, only the 6590 ligand–PDB combinations for which the correlation of the original ligand to the F               o − F               c map is at least 0.75 are included. Fig. 2Fig. 2F               o − F               c difference maps, though the precision with which the resulting model can be defined is clearly much better for the higher resolution maps than those at lower resolution.Fig. 3F               o − F               c map is at least 0.75. It indicates that very small ligands are fitted relatively poorly; just 44% are fitted within an r.m.s.d. of 2 Å of the original ligand in the PDB. In contrast, the percentages of ligands with 10–90 atoms fitting within an r.m.s.d. of 2 Å of the original ligand in the PDB is high and relatively constant, with a mean value of 74 ± 8%.Fig. 44.We have developed a procedure for fitting flexible ligands that is useful over a wide range of resolutions and that works well for ligands with over 90 non-H atoms. There remain significant improvements that could be made to the procedure. In particular, our simplistic method for analysis of allowable torsion angles and for ligand geometry could be replaced with results from semi-empirical or quantum-mechanical calculations. This would remove the limitations on the method arising from assuming that atoms in rings have fixed relative positions, for example. Other significant improvements that could be made might include a more detailed analysis of alternative choices for the location of the ligand, optimization of the scoring function used and optimization of the choices of the numbers of partially built ligands to keep at each stage. Additionally, subsequent to building of ligand models, the refinement of these models would be expected to improve their geometries and fit to the density. We expect that this procedure may prove useful as one of the tools that can be routinely applied during the final stages of model building for macromolecular crystallography to assist in model completion. Additionally, non-bonded interactions among atoms in the ligand may be useful in optimizing the conformation of the ligand.


Hueter-Volkmann's law regarding growth modulation suggests that increased pressure on the end plate of bone retards the growth (Hueter) and conversely, reduced pressure accelerates the growth (Volkmann). Literature described the same principle in Rat-tail model. Human spine and its deformity i.e. scoliosis has also same kind of pattern during the growth period which causes wedging in disc or vertebral body.This cross sectional study in 150 patients of adolescent idiopathic scoliosis was done to evaluate vertebral body and disc wedging in scoliosis and to compare the extent of differential wedging of body and disc, in thoracic and lumbar area. We measured wedging of vertebral bodies and discs, along with two adjacent vertebrae and disc, above and below the apex and evaluated them according to severity of curve (curve < 30° and curve > 30°) to find the relationship of vertebral body or disc wedging with scoliosis in thoracic and lumbar spine. We also compared the wedging and rotations of vertebrae.In both thoracic and lumbar curves, we found that greater the degree of scoliosis, greater the wedging in both disc and body and the degree of wedging was more at apex supporting the theory of growth retardation in stress concentration area. However, the degree of wedging in vertebral body is more than the disc in thoracic spine while the wedging was more in disc than body in lumbar spine. On comparing the wedging with the rotation, we did not find any significant relationship suggesting that it has no relation with rotation.From our study, we can conclude that wedging in disc and body are increasing with progression on scoliosis and maximum at apex; however there is differential wedging of body and disc, in thoracic and lumbar area, that is vertebral body wedging is more profound in thoracic area while disc wedging is more profound in lumbar area which possibly form 'vicious cycle' by asymmetric loading to spine for the progression of curve. Scoliosis is a three-dimensional deformity involving coronal, sagittal and axial angulations. Extent of the disease progress, decision to change the treatment strategy and efficacy of the treatment are mainly dependent on the severity in coronal plane angle, commonly known as Cobb's angle as a primary interest. In idiopathic scoliosis, factors responsible for the progression of curve are still unclear and authors have their own point of view in the literature. It has been suggested by Taylor et al.  that the150 patients with adolescent idiopathic scoliosis were studied in this cross sectional analysis. Their Cobb's angle was 10°~60° in 141 thoracic and 120 lumbar curves from 21 thoracic, 4 lumbar and 116 with thoracic and lumbar curves, in 122 females and 28 males with their average age 14.2  years. All patients underwent for radiogram in form of antero-posterior view of whole spine  for thoracic and 25.13°  for lumbar curves. Vertebral body and disc wedging  test. Unpaired t-test was used to compare the apical thoracic body wedging between curves less than 30° and curves more than 30°. In lumbar spine, comparison of the apical disc wedging with the wedging of four discs, above and below the apex, was done using ANOVA test and similarly, the apical lumbar discs wedging were compared according to severity of curves using unpaired t-test. Additionally we have used analysis of covariance to find out relationship between Cobb angle and wedging in the apical disc and body. For all these statistical methods, 'P' value less than 0.05 was considered significant.Our results indicate that in thoracic spine, vertebral body wedging (mean 2.7° with curves less than 30° and mean 4.1° with greater than 30°) in the apical vertebrae is significantly greater than the disc wedging   Figure . The disOur result showed that in thoracic spine average wedging at apex at vertebral body was 2.70° and at disc was 2.13° with curve less than 30° while 4.08° and 3.60° at body and disc respectively with curve more than 30°. The apical vertebral body wedging was found to be significantly more in patients having curve more than 30° as compared those having curve less than 30° . In lumbar spine, the apical wedging at body was 1.47° and at disc were 4.43° with curves less than 30° while 2.67° and 3.72° at body and disc respectively with curves more than 30°. The apical disc wedging was found to be significantly greater  in the patients with curves more than 30° as compared to those with curves less than 30° . Figure  AlthoughComparing the thoracic vertebral wedging with different grades of rotation, we found that most of the vertebrae had grade 1 rotation and statistically we could not analyze them according to severity of curves.Our study showed that the wedging of vertebral body and intervertebral disc was consistently present in both the thoracic spine and in lumbar spine. In the thoracic spine, the vertebral body wedging was more distinct as compared to the thoracic disc. However, in the lumbar spine the intervertebral disc wedging was more evident. The progress of the scoliotic curves is commonly thought to as per the Hueter-Volkmann's law . As per In the thoracic spine, the wedging pattern of growth modulation was thus according to the Hueter-Volkmann's law . It was In contrast to the thoracic spine, the lumbar spine showed no significant difference in vertebral body wedging between the apical and other vertebrae; however, it is significant finding that the intervertebral discs follow the same nature of wedging, as it is body in thoracic spine. Actually, in lumbar spine discs are more flexible than thoracic discs; so more stress concentration in disc explain more wedging than the vertebral bodies. According to Stokes and Aronsson , among tGrivas et al  has suggIn present paper we, however, could not study the effect of vertebral rotation according to severity of curve because we have measured the rotations Nash and Moe method which is not in measurement but in grades. Therefore we were not able to analyse statistically with severity of curve as most of the rotations were grade 1. We think that may be the weak point in our study that we could not analyze this ordinal data statistically.From this study we can, therefore, conclude that the vertebral wedging is the most important vertebral deformation of the thoracic spine, and the deformation while disc wedging is most deformed structure in the lumbar spine in idiopathic scoliosis and probably, important factor for different disease nature in same group which possibly could produce 'vicious cycle' theory by asymmetric loading for the progression of curve.The authors declare that they have no competing interests.HNM has contributed in conception and design and acquisition of data, analysis and interpretation of data, drafting the manuscript and revising it critically; SWS has contributed in conception and design of data, drafting the manuscript and given the final approval of manuscript; HRS has contributed in acquisition of data, revising the manuscript critically and given the final approval; JHY has contributed in acquisition of data and analysis and interpretation of data; HJK has contributed in acquisition of data and analysis and interpretation of data; and CHM has contributed in drafting the manuscript and designing of data and revising it critically. All authors read and approved the final manuscript.All authors give their consent to publish this study and accompanying images.


Time series gene expression data analysis is used widely to study the dynamics of various cell processes. Most of the time series data available today consist of few time points only, thus making the application of standard clustering techniques difficult.ASTRO and MiMeSR, are inspired by the rank order preserving framework and the minimum mean squared residue approach, respectively. However, ASTRO and MiMeSR differ from previous approaches in that they take advantage of the relatively few number of time points in order to reduce the problem from NP-hard to linear. Tested on well-defined short time expression data, we found that our approaches are robust to noise, as well as to random patterns, and that they can correctly detect the temporal expression profile of relevant functional categories. Evaluation of our methods was performed using Gene Ontology (GO) annotations and chromatin immunoprecipitation (ChIP-chip) data.We developed two new algorithms that are capable of extracting biological patterns from short time point series gene expression data. The two algorithms, .Our approaches generally outperform both standard clustering algorithms and algorithms designed specifically for clustering of short time series gene expression data. Both algorithms are available at  Then Z2 is calculated as Z2 = A - Z1. Then, MiMeSR identifies the submatrix with constant values on rows across the whole time points in Z2. This step is easily performed by identifying the set of rows of Z2 such that 2 - min) <εmax in A. For simplicity and without loss of generality, let us consider an example of the synthetic gene expression matrix A, with coherent values cluster in it, corresponding to rows r1, r3 and r4 ), a new matrix Z2 is generated whose rows r1, r3, and r4 correspond to the submatrix with constant values on rows. Note that, the same cluster will be constructed by using any of the rows r1, r3, or r4. Therefore, after a cluster has been identified, its rows are not further considered in the construction of new Z1 matrices. This approach is guaranteed to identify all submatrices with minimum mean squared residue across all time point experiments. Note that, since the operation Z2 = A - Z1 is performed using all the rows of A during each iteration, and since we are seeking for the set of rows of Z2 such that 2 - min) < εmax to belong to more than one cluster. The biological equivalent of this notion is that genes may be involved in more than one genetic pathway or to be regulated by more than one transcription factors.ε parameter on the clusters that are identified by MiMeSR. This can be done by sensitivity analysis in which the parameter ε is perturbed and the results are compared. For this analysis, it is usually sufficient to consider one or two values above and below the originally selected value of ε. Only clusters that are consistently identified by MiMeSR as ε varies should be retained for further examination. Note that the number of genes in these clusters may also change. The user therefore needs to determine a rule for dealing with genes that may be dropped from the clusters as ε changes. The most conservative approach would be to retain only the genes that remain in the clusters for all values of ε around its selected value. It can be easily shown that the overall complexity of MiMeSR is ~O(NMK), where K is the number of minimum mean squared residue clusters in A. Note that K corresponds to the maximum number of constant columns matrices that can be constructed using the rows of A without identifying redundant clusters.For practical reasons, it is important to assess the effects of the PVB and ABT designed the study, analyzed the results and wrote the paper. KVB and TM designed the web server and contributed to the writing of the paper.Supplementary materials. The following additional data are available with the online version of this paper. Additional file ASTRO and MiMeSR.Click here for file


Experiments are described here in which a murine squamous carcinoma has been used to obtain a survival curve for cells of tumours treated in vivo with single doses of bleomycin alone or in combination with radiation given before or after the drug. The survival curve for drug alone was a multicomponent curve with D0 values of 0·1, 0·75 and 2·7 mg/kg. However, the results of the experiments with combined treatment showed no evidence of potentiation.Many previous reports testify to the effectiveness of bleomycin as a drug for the clinical treatment of squamous carcinomata, and some radiotherapists have considered the possibility that an improved therapeutic ratio might be obtained by combined treatment with bleomycin and radiation. This consideration has been stimulated by recent reports of the effects of combined treatment with bleomycin using mammalian cells


In a recent work Levine et al. (Ann Henri Poincaré 17:1677–1711,  V and edge set E. By E and we write “s(x) indicates a mass of particles at site x. Note that here, unlike the ASP, s(x) is a real-valued (possibly negative) number. If a vertex The concept of self-organized criticality was introduced in Bak et al.  as a latGiven x before time n to any of its neighbours. If odometer of s. We have the following dichotomy: either s(x) is assumed to be i.i.d. on an infinite graph which is vertex transitive, and if s does not stabilize, while stabilization occurs for s almost surely does not stabilize.Let e et al.  that if V has distributionV starting at x and For a finite connected graph, one can give quantitive estimates and representations for d. They write (after the statement of Proposition 1.3):“We believe that if Note that, although they work with Gaussian weights in the proof of Proposition 1.3, their comment comprises also the case when random distribution living in an appropriate Sobolev space on The interplay between the odometer of the sandpile and the bilaplacian becomes more evident in the observation made by Levine et al. on the odometer in mentclass2pt{minimRelated work Scaling limits for sandpiles have already been investigated: in the ASP literature limits for stable configurations have been studied, for example, in Levine et al. [e et al.  and Pegde et al. . Their wand Jara , the autmembrane) model was introduced in Sakagawa [The discrete bilaplacian  Let The reason to impose mentclasspt{minimaWe can now show the next Theorem, which generalises the previous one to the case in which the weights have an arbitrary distribution with mean zero and finite variance. We keep the proof separate from the Gaussian one, as the latter will allow us to obtain precise results on the kernel of the bilaplacian, and has also a different flavor. Moreover, the more general proof relies on estimates we obtain in the Gaussian case. With a slight abuse of notation, we will define a field mentclass2pt{minim Assume Ξn as in  for suchmentclass2pt{minimu, we always mean We now give an explicit description of the covariance structure of d such that(Kernel of the biharmonic operator in higher dimensions) Let d interchanging sum and integrals is possible, so that we can write(Kernel of the biharmonic operator in lower dimensions) The convergence result of Theorem Outline of the articleThe necessary theoretical background is given in Sect. In this section we review the basics of the spectral theory of the Laplacian on the discrete torus from Levine et al. . We alsoilvestri . We alsomentclass2pt{minimilvestri  and for ilvestri .L is independent of x). Finally, we recall Levine et al. [We now recall a few facts about the eigenvalues of the Laplacian from Levine et al.  for compecalling , we use Since our conjectured scaling limit is a random distribution, we think it is important to keep the article self-contained and give a brief overview of analytic definitions needed to construct the limit in an appropriate functional space. Our presentation is based on Sheffield [abstract Wiener space (AWS) is a triple H with respect to the measurable norm H, equipped with the Borel B, then H such that We remark that, in order to construct a measurable norm H, it suffices to find a Hilbert–Schmidt operator T on H, and set An u with Fourier series Let us construct then an appropriate AWS. Choose mentclass2pt{minimGaussian Hilbert spaces [m. Indeed, by the properties of AWS, the mapping S. Taking There is a perhaps more explicit description of Firstly, we show that Y with variance L is as in (Y is independent from There exist a centered Gaussian field y) as in  and a ceis as in ), such tFrom the above tightness result, there exists a subsequential scaling limit This will conclude the proof. Since the “finite dimensional” convergence is somewhat more interesting, we will defer the tightness proof to Sect. This decomposition is similar in spirit to the one in the proof of Levine et al. [A note on test functions By the above construction, the set of test functions we will consider is the set of smooth functions not correspond to the class of continuous functions on In this section we provide a proof of Proposition g in terms of the simple random walk x by translation invariance. Hence we get that the first term in the left-hand side of  and (P2) to achieve the Theorem. We first use to our advantage the fact that the test functions we consider have zero average, hence we can get rid of the minimum term which appears in the definition of the odometer. Let us recall the field in \documentOverview of the proof We have just seen thatIn the notation of this Section,The second Proposition says the remainder term is small.In the notations of this Section, Then an application of the Cauchy-Schwarz inequality will allow us to deduce thatBefore we begin our proof we would like to prove a bound which would be crucial in estimating the eigenvalues of the Laplacian on the discrete torus. This lemma will be used later for other parts of the proof too.There exists We consider side of . Moreove>0, thus  is proveThe equation  is not eWe begin with the proof of Proposition bound of  immediatbound in  we get\dThe case n, and The case gence of  is equivWe owe the reader now the last proofs on }Rn (see ). First There exists a constant Using the mean value theorem as We reprise now the proof on the limit of We first compute We are then done with the proof of (P2) on page 7.We proceed to prove tightness. Before that, we must introduce a fundamental result: Rellich’s theorem.(Rellich’s theorem) If Sketch of the proof  The proof is readily adapted from the one in Roe [B be the unit ball of B in the We take n, being a finite combination of Gaussian variables and their minimum. Hence Choose Assume we can proveThere exists Using the above Claim and d2, from  we get\dWe are then left to show the claim we have made:C possibly depends on First we use the bound  and the ee below ). We rew)¯.Using  we obtai.Finally  tells usgging in  the exprThis gives a proof of P1) on page 7 and completes the proof of Theorem  on page Strategy of the proof We will argue as in Theorem Reduction to a bounded field We first recall some facts from Levine et al. [y. We can then say thate et al. . Note the et al.  that thedσ(z).By  it folloThe goal of this Subsection is to determine the scaling limit for bounded weights, namely to prove Assume Ξn as in  for suchmentclass2pt{minimn on the Fourier series of the mean of u in a small ball.Before showing this result, we must prove an auxiliary Lemma. It gives us a uniform estimate in n large enough we can find a constant Fix w and plug it in  from ):\documeP and as P, let us consider in the product over m will give a contribution of order unity to the sum. Since, for Having concluded the case {0} (see ) we obtamentclasspt{minimaTightness The proof of tightness is, not suprisingly, a re-run of that in the Gaussian case. In fact tightness depends on the covariance structure of the field we are examining; since both the Gaussian functional At the moment we are able to determine the scaling limit when the weights are bounded almost surely. To lift this condition to zero mean and finite variance only, we begin by defining a truncated field and show it will determine the scaling limit of the global field. Fix an arbitrarily large (but finite) constant S be a metric space with metric (Billingsley [For a constant As a consequence we will obtain that Following this Theorem, we need to show two steps:We notice thation into , equal t at hand , which wu with zero average. Hence the field Our idea is to use the computations we did for the case in which entclass1pt{minimaPreliminaries We must conclude with the proof of Theorem N as Grafakos , we havent}Gd in . In partt}d≥5 by . Hence \Note thathanks to  and the mentclass2pt{minimTherefore, rather than working on , we willn, being a finite sum. As for the first term in (n in each occurence above. These inequalities plugged into (Following the proof of Proposition o derive . \docume