Phenotypic evolutionary models have been used with great success in many areas of biology, but thus far have not been applied to the study of stem cells except for investigations of cancer. We develop a framework that allows such modeling techniques to be applied to stem cells more generally. The fundamental modeling structure is the stochastic kinetics of stem cells in their niche and of transit amplifying and fully differentiated cells elsewhere in the organism, with positive and negative feedback. This formulation allows graded signals to be turned into all or nothing responses, and shows the importance of looking beyond the niche for understanding how stem cells behave. Using the deterministic version of this framework, we show how competition between different stem cell lines can be analyzed, and under what circumstances stem cells in a niche will be replaced by other stem cells with different phenotypic characteristics. Using the stochastic version of our framework and state dependent life history theory, we show that the optimal behavior of a focal stem cell will involve long periods of quiescence and that a population of identical stem cells will show great variability in the times at which activity occurs; we compare our results with classic ones on quiescence and variability in the hematopoietic system. How exactly to model this is unclear, with the consequence that most models of stem cells focus on a single stem cell  in a single niche. We show how to make operational the recommendation in Theoretical approaches to stem cell biology have usually involved typological thinking September 2007 was the 50th anniversary of the publication of the first successful demonstration of the intravenous infusion of bone marrow in patients who had experienced radiation or chemotherapy that destroyed bone marrow In its niche, a stem cell responds to its local environment  and its global environment  but we assume that the behavior of the focal stem cell does not affect the behavior of the other stem or more fully differentiated cells. We are thus able to avoid, at least initially, game theoretical aspects of stem cell biology; ultimately they must be considered. Even with this simplified framework, we are able to understand how natural selection will act on competition between stem cells with different phenotypic characteristics (explained below) and how natural selection can lead to both quiescence and great variation in the behavior of stem cells.piwi); self-feedback, autocrine and paracrine influences; hormones, growth factors, cytokines, and short-range cell to cell signalling pathways ; hormones from remote sources; positional cues from chemical gradients as in classical Turing systems Stem cells have the ability to both self-renew and to differentiate. The niche of the stem cell is the local microenvironment that supports the maintenance, renewal, and differentiation of stem cells. In the case of the villi of the small intestine or bone marrow it is easy to conceive of the niche as a defined physical environment. In other cases, such as the epidermis, the niche will be determined by the spatial extent of signaling molecules. Signals and feedback controls (both negative and positive) affect the pattern of behavior of the stem cells in their niche. These signals include cell-autonomous promotors (such as S(t), transit amplifying progenitor A(t), and fully differentiated D(t) cells at time t associated with the stem cells in the niche being modeled. In doing so, for simplicity we compress the differentiation stages of a transit amplifying cell, but they could be unpacked should one wish to do so. In addition, there will be other transit amplifying and fully differentiated cells, produced by other stem cells, that create an additional signaling background. We denote those cells by B. For simplicity we consider only one kind of progenitor cell and only one kind of differentiated cell, but extensions are clear and obvious. The positive and negative feedback links affecting stem cell fate are illustrated in The simplest population biology that one can imagine is to focus on the number of stem Most simply, stem cells divide symmetrically – doubling and producing two stem cells, divide asymmetrically – doubling and producing one stem cell and one cell that will be a progenitor of a fully differentiated cell, undergo apoptosis  a stem cell divides symmetrically, 2) a stem cell divides asymmetrically, 3) a stem cell divides and differentiates symmetrically producing two transit amplifying cells, 4) a stem cell under goes apoptosis or migrates out of the niche, 5) a transit amplifying cell divides, 6) a transit amplifying cell undergoes apoptosis, 7) a transit amplifying cell fully differentiates, 8) a fully differentiated cell undergoes apoptosis. To fully understand the evolutionary ecology of stem cells we need to consider the entire range of possibilities of transitions and both positive and negative feedback. This is implicit in The transitions of stem, transit amplifying, and differentiated cells are characterized by a number of parameters. Biological realities will constrain some choices for parameter values. For example, we may anticipate that the probability that all stem cells in the niche either undergo apoptosis or migrate out of the niche is not too high since if it were the system could not persist. That is, if the niche empties (either through apoptosis or migration) it has essentially gone extinct. Similarly, in most situations, the population of transit amplifying cells is much more active than the population of stem cells (with consequence that the number of amplifying cells will thus be larger than the number of stem cells) and we require parameter values that make this so. We also require that when there is just one stem cell in the niche, the numbers of stem cells increase.In addition, we may expect there to exist links and trade-offs between the different parameters. For example, since symmetric renewal, asymmetric renewal and symmetric differentiation involve many of the same signals and processes, we may expect that their rates are correlated. Cells that undergo high rates of replication are more likely to have errors in them and thus are likely to undergo apoptosis. When transit amplifying cells disappear at a higher rate because of replication error, fewer of them will become fully differentiated cells and this too can be captured by a trade-off.In summary, one may conceive of the parameter set describing the transitions of the stochastic kinetics as the phenotype of a stem cell and its descendants, or if all of the stem cells in a niche are identical descendants of a single cell, then these parameters are the phenotype of the stem cell clone. These trade-offs are fully described in F to be the maximum expected accumulated fitness through differentiated cells, given that the focal stem cell has accumulated y resources towards the next division, that there are s stem cells in the niche and that there are a and d transit amplifying and differentiated cells sending signals back to the focal stem cell. State dependent life history theory, as implemented through stochastic dynamic programming To understand fully how natural selection acts on a stem cell, we need to consider the fitness of a focal stem cell. Stem cells (and transit amplifying cells) do not by themselves achieve fitness. Rather, they support the organism – which can be viewed as an organized collection of fully differentiated cells. Fitness of the focal stem cell depends upon what it does, what the other stem cells in the niche do, and how many transit amplifying and differentiated cells are present. We thus define a fitness function T).A variety of parameters are used in this section; they are explained in We begin with another look at Till et al 1 and with decreasing values of μ4 . Thus, the solution of the set of differential equations will move from S = 1  towards K, determined by the balance of μ4 and λ1. In the full stochastic simulation, all the same properties hold, as can be seen from the equations characterizing the transition rates. One important consequence of S = 0 being dynamically unstable is that when a niche goes extinct, it will do so quickly 6−λ5 and φ. The number of fully differentiated cells is determined by μ8, φ, and μ6 – thus a variety of simple mechanisms can create tissues with different structures.As described in the In 1 exceeds 3 varies between the resident and invading stem cells. Here the region of coexistence is greater and the boundary curves, while still symmetrical are more complicated.In dy = 5.5, but in the simulated stem cell population the average value of resources at the time of transition is 7.1, and the variance of those resources is 0.84. The average number of transit amplifying and background cells at the time of transition is 11.1, and the variance in those numbers is 53.5.In in vivo and show, for example, that not all manipulations will be successful (the invading stem cell line cannot persist) or perhaps may be too successful (the invading stem cells fully replace the resident ones). We have not allowed mutations in the phenotypic parameters of the stem cell clones, but this is possible too with our methods Phenotypic models of evolutionary processes have made important and long-lasting contributions in ecology and evolutionary biology An empirical goal suggested by our work would be to more carefully identify these trade-offs and the feedback functions  in the absence of feedback from the amplifying transit cells and the fully differentiated cells. Indeed, they are crucial to what happens in the niche. We have also shown that a single framework can account for the observed ratio of fully differentiated to stem cells Many questions remain and there is much work to be done, ranging from the development of systems biology models for the phenotypic parameters, to the evolution of the signalling functions and their parameters, to the development of more complicated game theoretical aspects. Each will provide new insight into the evolutionary ecology of stem cells and their niches.dt is λΔt+o(Δt), where o(Δt) indicates times that are higher powers of Δt. An improvement upon their approach is the following. It is known that there is variation in cell cycle times T = 1/λ, it is then an elementary calculation to determine the frequency distribution of cell cycle time, given the inferred gamma density.Till et al. s, a, d for particular values of the number of stem, transit amplifying, or fully differentiated cells. We characterize the feedback network in S, ΔA and ΔD denote the change in the number of stem, amplifying, or differentiated cells in the interval of time Δt, which is assumed to be small. As noted before, o(Δt) denotes terms that are higher powers of Δt .We use 1 = 1/T1 is the cell cycle time for stem cells that renew symmetrically. We assume no variation in this rate. The function Φ1(s) characterizes the inhibition of stem cells upon each other 1(s) decreases as s increases (which captures the inhibition); for computations we use Φ1(s) = ln(K)−ln(s).In this equation, λ2 is the cell cycle rate when a stem cell renews asymmetrically, with associated cell cycle time T2. Inhibition of symmetric renewal occurs for two reasons. First, negative feedback from stem cells on stem cells decreases overall activity. Second, negative feedback from existing transit amplifying and fully differentiated cells inhibits the production of transit amplifying cells. Thus Φ2 characterizes the signal from the transit amplifying and fully differentiated cells to the stem cells. Once again, this function will decrease as the number of transit amplifying and fully differentiated cells increases. For computations or analysis we use Φ2 = es(a+d+b)− which is equivalent to a Hill function when the argument is small. ε measures the sensitivity of the stem cell population to what is happening in the population of transit amplifying and fully differentiated cells. Stem cells convert graded stimuli into all or nothing responses 3 and Φ3 are analogous to those above but can be understood as a ‘more desperate’ signal. For computations or analysis we use Φ3 = ea+d+b)−γ is immaterial – the consequence is the same, a reduction in the number of stem cells in this niche. Thus we combine them in5 is the cell cycle rate for transit amplifying cells. As mentioned above, we have compressed the n stages of development from transit amplifying to fully differentiated cell; unpacking this assumption can be done through a linear chain Transit cells amplify and since they send a signal back to the niche, their dynamics must be included. Hence6 denote the total per capita rate at which transit amplifying cells disappear; a fraction φ of them are converted to differentiated cells and the remainder experience apoptosis. Thus the mortality of transit amplifying cells isTransit amplifying cells will disappear because of apoptosis (e.g. because they have too many DNA errors) and because they fully differentiate. We let μ6+ρ7 = μ6a, the total rate at which transit amplifying cells disappear.Transit amplifying cells that disappear but do not die are converted into fully differentiated cells as in8 accounts for all sources of mortality of fully differentiated cells.Finally, fully differentiated cells die, so thatt to the dynamics of the populations of cells. We assume that the time to the next transition of the system, τ, is a random variable with exponential distribution and mean time determined by the rate of all processes. The overall rate isS(t), A(t) and D(t) we first compute the time of the next transition by comparing the right hand side of Eqn 21 with a uniformly distributed random variable.We use the Gillespie τ-method if are the probability density function for the transitions of the system. It is also useful in numerical implementation to deal with the cumulative distribution function iF defined byNext, given that a transition has occurred we computeThe numbers of stem, transit amplifying, and fully differentiated cells are random variables. The rates of transitions of stem cells are nonlinear functions of the states, thus it is a nontrivial matter to connect the transition rates to ordinary differential equations for the states The transitions of stem, transit amplifying, and differentiated cells are characterized by a number of parameters. Biological realities will constrain some choices for parameter values. For example, we may anticipate that the probability that all stem cells in the niche either undergo apoptosis or migrate out of the niche is not too high since if it were the system could not persist. That is, if the niche empties (either through apoptosis or migration), it has essentially gone extinct. In computation, we choose parameters so that the probability of extinction of the niche is less than 20 percent.5>4ρ1.Similarly, in most situations, the population of transit amplifying cells is much more active than the population of stem cells (with consequence that the number of amplifying cells will thus be much larger than the number of stem cells). For purposes of computation we use the condition ρS = 1. Since Φ3 will be a decreasing function of A, D and B, it follows from Eqn 25 that we require1, λ2, and λ3 are correlated. Thus, for computations, we use λ2 = 2·λ1 and λ3 = 0.05·λ2. Similarly, cells that undergo high rates of replication are more likely to have errors in them and thus be more likely to undergo apoptosis. The simplest version of that trade-off is a linear one in which40) and mortality due to apoptosis. The second equation above allows us to separate the disappearance of transit amplifying cells into those potentially converting to differentiated cells and those suffering apoptosis.We require that the number of support cells is sufficient such that when there is just one stem cell in the niche, the numbers of stem cells increase. In terms of the associated ordinary differential equations, this is equivalent to 1 as in1,λ2,λ3,μ40,μ41,λ5,μ60,μ61,φ0,φ1,μ8] as the phenotype of a stem cell and its descendants, or if all of the stem cells in a niche are identical descendants of a single cell, then these parameters are the phenotype of the stem cell clone. But, of course, all the stem cells in a niche need not be conspecifics. We thus turn to the competition between different stem cell phenotypes.When transit amplifying cells disappear at a higher rate because of replication error, fewer of them will become fully differentiated cells. This can be captured by a trade-off between φ and λi and j to index the stem cell clone with different phenotypic parameters, let γji denote the influence of transit amplifying and fully differentiated cells of type j on stem cell clone i so thatWe can use evolutionary replacement analysis In these equations, we assume that all stem cells share the functional forms for the influence of transit amplifying and fully differentiated cells on asymmetric renewal and symmetric differentiation. After specifying the set of phenotypic parameters, we can integrate the coupled differential equations to determine the outcome of competition between any number of stem cell clones that differ in their phenotypic parameters.3 and of the invader is 3 .However, to illustrate the ideas and explain them in as simple a manner as possible, consider the situation in which we wish to manipulate some of the phenotypic parameters in a niche by inserting a new stem cell with different parameters into an existing clone. In that case, we need to consider the stem cells present (the ‘resident’ stem cells) and the new one (the ‘invading’ stem cell). For definiteness, let us assume that the rate of symmetric differentiation of the residents is λTo make the notation simpler let us return to Eqn 25 and rewrite it asFirst, it is possible that Second, it is possible that We now describe how one may characterize the behavior of a focal stem cell in its niche. To be sure, thinking of a single stem cell having behavior while others cells in the niche (and the transit amplifying and fully differentiated cells) have a fixed behavior is a simplification. Ultimately, one should view stem cells in their niche as playing a dynamic cooperative game, but we reserve that for subsequent work. For the time being, we focus on the state dependent life history of a focal stem cell, determining the optimal life history through stochastic dynamic programming Y(t) the resources available for cell division at time t, assuming that a minimum resource level dy. We assume that the inhibition of the focal stem cell by other stem cells also affects the flow of resources to the focal stem cell and has the same functional form as used in the inhibition of cell cycling. Thusq can be scaled out by interpreting t as multiples of 1/q, so we set it equal to 1.Like all cells, the focal stem cell must accumulate resources for division. We denote by FR(d) when there are d fully differentiated cells sending signals to the focal stem cell. We assume that the decision of the focal stem cell is made at the time of the next transition of the stem, amplifying or differentiated cells. Using the methods describe above, we compute τ, which is the random variable describing the time to the next transition, and the vector Δ =  that describes the changes in stem, transit amplifying and differentiated cells when a transition occurs.We assume that the organism gains fitness from these fully differentiated cells at rate y. If y is less than the resources available for differentiation, then the focal stem cell cannot do anything other than remain quiescent. In that caseV0, the fitness value of remaining quiescent. The right hand side of this equation represents the future accumulation of fitness and is averaged over the time of the next transition and the change in the cell population when that transition occurs.When this transition occurs, the focal stem cell has level of resources y exceeds dy, then in the simplest case the focal stem cell may remain quiescent , renew symmetrically  or renew asymmetrically . Allowing for full differentiation of the focal stem cell, or migration out of the niche requires certain complications which we will postpone for a later paper. After a division resources y−dy remain for correcting errors in the daughter cells. We thus let Ψ(y−dy) denote the probability that a daughter cell is error free (i.e. does not undergo apoptosis), given that y−dy resources are used for correcting errors. For computations we use Ψ(z) = 1−exp(−γcz) where γc is a parameter.If y exceeds dy we haveF by value iteration  of the focal stem cell when its accumulated resources for division are y, there are s other stem cells in the niche and the number of transit amplifying and differentiated cells sending signals to the niche are a and d respectively. We then use the Markov simulation described previously to create the environment of the focal stem cell and follow its subsequent behavior  as the number of stem cells in the niche and the number of amplifying and differentiated cells change through the stochastic processes described previously. In this way, we are able to construct the times between transitions of the focal stem cell, and thus the analogue of the distribution of Till et al. If the stem cell renews symmetrically, we assume that the focal stem cell remains error-free and a daughter stem cell joins the other stem cells in the niche. Assuming segregation of errors upon division


For these tests the entire microarray or some pre-specified subset of genes is the unit of analysis. The pairwise distances only have to be computed once so the procedure is not computationally intensive despite the high dimensionality of the data. An R software package, permtest, implementing the method is freely available from the Comprehensive R Archive Network at  We propose permutation tests based on the matrix of pairwise distances between microarrays to compare location, variability, or equivalence of gene expression between two populations. These tests can be applied to the entire genome or any subset of genes of interest. Thus they can be used as global tests of difference in gene expression or as a testing method applicable to gene set analysis.These tests have several advantages over permutation tests based directly on the gene expression data. First, they are not computationally intensive because they reduce the high dimensional expression data to the low dimensional distance matrix and this only has to be done once. Second, this same reduction in dimensionality results in a reduction in the dimensionality of the potential nuisance parameters. Thus, the assumption of exchangeability between groups, which ensures the validity of the permutation test, only has to apply to the pairwise distances not to the entire microarray.Many gene expression studies are designed to detect differential gene expression in two clinical or biological populations. Interest may focus on individual genes, specified groups of genes, or all genes sampled by a particular microarray. Early efforts were devoted to individual gene analysis In order to incorporate existing biological information into the analysis of differential gene expression more recent work has been devoted to methods that treat sets of related genes as the unit of analysis A number of self-contained, multivariate, tests have been proposed et al.2Kong et al.2 under the assumption of equal variance and zero covariance across all genes. They present the statistic's asymptotic distribution but suggest a permutation test for inference in small samples.Goeman et al.2 under the assumption of equal variance and zero covariance across all genes. Note that two statistics are permutationally equivalent Hummel In this paper we propose methods to test for a significant difference in gene expression between two study groups based on the matrix of pairwise distances between microarrays. In the following sections we present the test statistics given the assumption of Euclidean distance, additive errors, and a completely randomized design. We also present an application to respiratory recovery in trauma patients and extensions to paired and blocked designs as well as the use of a distance measure based on correlation.Consider an experiment comparing gene expression in two populations. The gene expression values, or some function of the gene expression values which measures the biological signal for each gene, can be represented by two groups of column vectors. We will represent these column vectors by Inference concerning the location and variability of groups 1 and 2 can be based on the three means: μ1 and μ2 are the respective mean vectors of the microarrays in groups 1 and 2 and 1,iε and 2,iε are random error vectors with expected value 0 and variance covariance matrices Σ1 and Σ2 respectively. The errors are assumed to be independent across microarrays.In this section we derive the expected value of i,jX and k,lX is The squared Euclidean distance between any two microarrays, We can now define a test statistic to compare the location or variability of groups 1 and 2.To compare location let: To compare variability let: v and Δl can be made using a permutation test. Each permutation consists of assigning v and Δl have to be recalculated based on the re-indexing.Inference concerning the magnitude of Δv and Δl and let Let e can also be made using a permutation test. It should be noted that the statistics Δv, Δl, and Δe are all special cases of Mantel's U statistic l is similar to a special case of the MRPP statistic Sometimes investigators design an experiment to compare a new technical method, such as sample preparation or target hybridization, to a proven “gold standard”. In such an experiment interest centers on showing that the new method is equivalent to the gold standard. To be equivalent it should not differ in mean and not exhibit greater variability. Assuming that the microarrays in group 1 were prepared using the gold standard, a summary statistic which can be used to reject the null hypothesis of equivalence is given by: In this section we consider modifications to the test statistics and permutation test to account for pairing in the experimental design. We still assume an additive error model and squared Euclidean distance.iB is a random vector, independent of all other quantities, with expected value 0 and variance covariance matrix ΣB. The quantities μ1, 2μ, 1,iε, and 2,iε are again defined as in Section.Suppose the experimental design is paired so that each microarray in group 1 is paired with a microarray in group 2. In this case v and Δl still have the same expected values given in Equations 13 and 12.To account for the pairing we drop the within pair distance in our test statistics since it will be systematically less than the between pair distances. Thus we re-define For this paired design, only paired microarrays are exchangeable and the permutation test must respect this structure. Thus, a permutation consists of exchanging microarrays across groups 1 and 2 within each pair of an arbitrary subset of paired microarrays.In this section we consider modifications to the test statistics and permutation test to account for blocking in the experimental design. We still assume an additive error model and squared Euclidean distance.kB is a random vector , independent of all other quantities, with expected value 0 and variance covariance matrix ΣB. The quantities μ1, μ1, 1,iε, and 2,iε are again defined as in Section.Suppose the experiment consists of K blocks of related microarrays. In this case For this blocked design we need to modify the mean distances The permutation test for this design must respect the blocked structure. Microarrays are exchanged across groups only within each block.2 is the variance, and E[δ] = 0.There are various possible measures of distance between microarrays besides squared Euclidean distance. In this section we consider a measure based on the Pearson product moment correlation E[2δ] is negligibly small then, if we define the distance between Z1 and Z2 by 1, Z2] is approximated by If the correlation is positive and under the assumption that v and Δl as above in Equations 11 and 10, these test statistics have expected values v estimates the log of the between group reliability ratio and Δl estimates the negative log of the correlation between the group means.Defining Δv and Δl as presented in Equations 38 and 39. In the paired design it is convenient to think of the random pair effect, iB , as being absorbed in the error term. In the blocked design it is convenient to think of the random block effect, kB , as being absorbed in the mean.Using the negative log correlation distance the three designs considered above need no modification except in the interpretation of the statistics Δμ, and the error, ε. Thus a significant difference in the within group distance as measured by Δv can be due to a difference in either the mean or the error.Note that the reliability given by Equation 34, which can be interpreted as a signal to noise ratio, is a reasonable measure of the within group distance or variability, but it depends on both the mean, l, does not have the correct expected value and does not estimate the error free distance between the groups.A commonly used distance measure is one minus the correlation between microarrays. This distance measure is nearly equivalent to the negative log correlation distance for values of the correlation near one. However, by using this distance, the corrected distance between groups, ΔF1 is the data generating distribution for group 1 and F2 is the data generating distribution for group 2. In particular they cannot test that a specific parameter of the data generating distribution differs between two groups without highly restrictive assumptions about the remaining (nuisance) parameters.Permutation tests may place weaker distributional assumptions on the data than do parametric tests but they are not totally free of assumptions. In general they only test the global null hypothesis, They do have the desirable property of being exact tests provided that, under the null hypothesis, the observations are exchangeable so that the joint distribution of the combined data set is invariant under permutations of the observation labels. The permutation tests we propose first reduce the extremely high dimensionality of the data  to the low dimensional matrix of pairwise distances. Since inference is based solely on the distribution of these distances the global null hypothesis becomes the much less restrictive assumption that the pairwise distances are identically distributed. Thus, only pairwise distances need be exchangeable by permutation of observation labels.In particular, under Euclidean distance and the additive error model proposed above, the location permutation test, which tests the null hypothesis, http://www.affymetrix.com/index.affx).As an application of our proposed method we analyzed time to respiratory recovery in ventilated trauma patients in a data set previously described by Rajicic As can be seen in the figure, it is not readily apparent that the clustering separated the early and late recovery groups. However, the highest split separated the patients into one group (right hand side of figure) having 20/38 (53%) early recovery patients while the remaining group (left hand side off figure) had only 2/10 (20%) early recovery patients. This imbalance may be suggestive of a difference in gene expression between the early and late recovery groups.To formally test for a group effect we applied our proposed permutation test for a Euclidean distance location difference between the early and late recovery groups and obtained a one sided p-value of 0.0168, indicating a significant difference in gene expression.As a check on this result, in the spirit of Kong's use of Hotelling's T-square in the principal component space et al.et al.Similar results were obtained using the globaltest R-package of Goeman Finally it may be of interest to note that, for the remaining 54230 probesets not associated with inflammation, the group difference was marginally significant with a p-value of 0.0462 by our method and p-values of 0.0508 and 0.0480 by the methods of Goeman and Hummel respectively. This suggests that factors other than inflammation may also be associated with time to ventilator recovery.Our proposed test statistics attempt to divide differences in gene expression into differences in location and differences in variation. The hope is to find tests for one parameter when the other parameter differs. This is a common statistical problem that only has a solution under limited circumstances. For instance, if data are normally distributed it is possible to find an exact test for variability differences when there are differences in location but testing for differences in location when there are differences in variability is the famous Behrens-Fisher problem which has no optimal small sample solution 2 under the assumption of equal variance and zero covariance across all genes. This holds true for our location test as well, using Euclidean distance in a balanced completely randomized design. Since permutational equivalence holds regardless of the distribution of the data, in the case of balanced completely randomized designs all three tests have equal power as permutation tests. In the moderately unbalanced case, with similar within group variability, simulations (not shown) show that our test is still nearly permutationally equivalent to the other two tests so that the power will be similar.Since the distribution functions of gene expression are unknown it is not possible to calculate the power of the location test under specified alternative hypotheses. However, as pointed out in the Introduction, for a two group comparison, both Goeman's and Hummel's methods are permutationally equivalent to Hotelling's TOur proposed location test is based on the matrix of pairwise distances between microarrays and is therefore related in a natural way to cluster analysis which applies an algorithm to the distance matrix to find clusters in the data. In experimental designs involving two predetermined groups of microarrays cluster analysis can be used as a graphical technique to see if the clustering algorithm “finds” the predetermined groups. Our proposed method can be thought of as a formal significance test of whether two predetermined groups form two distinct clusters. Thus our test is consistent with an intuitive visual display of the data. Of course applying our location test, or any significance test, to two groups that were discovered using a clustering algorithm would be circular and therefore invalid.


Escherichia coli. Here, we present the crystal structure of the E. coli RNase E catalytic domain in the apo-state at 3.3 Å. This structure indicates that, upon catalytic activation, RNase E undergoes a marked conformational change characterized by the coupled movement of two RNA-binding domains to organize the active site. The structural data suggest a mechanism of RNA recognition and cleavage that explains the enzyme's preference for substrates possessing a 5′-monophosphate and accounts for the protective effect of a triphosphate cap for most transcripts. Internal flexibility within the quaternary structure is also observed, a finding that has implications for recognition of structured RNA substrates and for the mechanism of internal entry for a subset of substrates that are cleaved without 5′-end requirements.RNase E is an essential bacterial endoribonuclease involved in the turnover of messenger RNA and the maturation of structured RNA precursors in  E. coli . A bent tetramer may explain this observed 1:1 ratio of stoichiometry of tetramer to RNA. Further experiments are needed to determine the role of RNase E tetramer organization in the maturation of structured RNA.In each of the four apoprotein protomers, the combined S1 subdomain and 5′ sensor (from here on referred to collectively as the “5/S1 subdomain”) has moved as a single unit through an angle of about 60° relative to the holoprotein configuration B. This cIn their report of the holoprotein structure of RNase E, The holoprotein model identified a magnesium ion in the catalytic site coordinated by D303 and D346 with the support of N305. Each of these residues has been implicated from mutation studies as required for catalytic activity . In the We favor this model because it accounts for the preference of the enzyme for substrates with a 5′-monophosphate terminus over those with either a triphosphate or hydroxy cap. It predicts that with only a terminal hydroxy cap, binding of an RNA substrate by RNase E will be substantially weaker and cleavage will be impeded under nonsaturating conditions. If a triphosphate cap is present on substrates, then the 5′ sensing site may still be able to bind them with moderate affinity because there is sufficient space in the open configuration of the enzyme to accommodate the three phosphate groups. However, RNA cleavage will again be greatly impeded because the extra phosphates are likely to sterically clash with the rest of the structure during the transformation to the closed configuration, acting as a wedge at the base of the fulcrum. Furthermore, the movement of these charged groups into a hydrophobic environment represents an additional thermodynamic barrier to domain closure.Our crystal structure suggests that the enzyme will bind isolated phosphate and sulfate groups within the 5′ terminal recognition site. Thus, it seems likely that the recognition site residues make a key contribution to the RNA binding affinity. In this respect, our model conflicts with reports that the 5′ end of the substrate provides no preference for RNA binding affinity and that the effect of the 5′-monophosphate is primarily due to its role in catalytic activation . HoweverAlthough the model for substrate interaction effectively describes global, nonspecific RNA degradation by RNase E, the mechanism used to process complex substrates through restricted cleavage at only specific sites is still unknown. The open configuration of the apoprotein may allow these more complex RNAs to be accommodated within the active site and bound at the 5′ end without the requirement for the same structural change that maneuvers small single-stranded RNA into the cleavage orientation. We would envisage that a single-stranded segment would be accommodated into the shallow channel that leads to the active site. The secondary structure of complex substrates may be sufficient to bring a defined RNA segment into close proximity to the catalytic site. In such cases, the substrates may not depend on RNase E undergoing a conformational change and may be effectively cleaved in the presence of either a 5′ mono or triphosphate cap. Likewise, it is also possible that the observed open configuration may allow a complex substrate to bind to a single protomer while positioning another segment of the RNA to be cleaved by a second protomer. Such a model would help to explain the functional value of the enzyme's tetrameric organization and is particularly attractive because of the observed flexibility of the structure.A recent report indicates that the conversion of a 5′-triphosphate cap to a monophosphate represents the rate-limiting step in bacterial mRNA decay . Recent 2BX2)  over their equivalents in the first apoprotein model, running MOLREP rigid-body refinement on these, and manual rebuilding using Coot. The model was then refined again using TLS plus restrained refinement and structure idealization in Refmac. The positions of the zinc ions were again confirmed by an anomalous Fourier synthesis. This later model has electron density and stereochemistry significantly improved over the first and is presented here. The structure was examined and validated using PROCHECK   were use2BX2)  and mult2BX2) . The “ZnPROCHECK  and SFCHPROCHECK  and was PROCHECK .


A number of genetic mechanisms have been suggested for driving anti-pathogen genes into natural populations. Each of these mechanisms requires complex genetic engineering, and most are theoretically expected to permanently spread throughout the target species' geographical range. In the near term, risk issues and technical limits of molecular methods could delay the development and use of these mechanisms. We propose a gene-drive mechanism that can be self-limiting over time and space, and is simpler to build. This mechanism involves one gene that codes for toxicity (killer) and a second that confers immunity to the toxic effects (rescue). We use population-genetic models to explore cases with one or two independent insertions of the killer gene and one insertion of the rescue gene. We vary the dominance and penetrance of gene action, as well as the magnitude of fitness costs. Even with the fitness costs of 10 per cent for each gene, the proportion of mosquitoes expected to transmit the pathogen decreases below 5 per cent for over 40 generations after one 2 : 1 release (engineered : wild) or after four 1 : 2 releases. Both the killer and rescue genes will be lost from the population over time, if the rescue construct has any associated fitness cost. Molecular approaches for constructing strains are discussed. Plasmodium is an appealing concept  gene and a gene that we will call the ‘rescue’ gene, when these two genes are inserted at independently segregating loci. The killer gene could code for a toxic protein or RNA, and the rescue gene could code for an enzyme or an RNA that neutralizes the impact of the killer gene. Any individual that receives one or more copies of the killer gene from its parents but no copies of the rescue gene would die. If the transgenic construct with the rescue gene was also engineered to include an anti-pathogen gene, the frequency of the anti-pathogen gene in a population would always be exactly the same as the frequency of the rescue gene.This system has the benefit of being less complex at the molecular level than other gene-drive mechanisms, so its construction in non-model organisms is likely to be more feasible with currently available tools. The killer–rescue system is predicted to locally spread an anti-pathogen gene to high frequency for only a limited period of time, so it presents less of a risk than other gene-drive systems. The self-limiting properties of this system derive from fitness costs associated with the transgenic constructs as well as general population-genetic properties of the interaction between killer and rescue genes. In most discussions of gene drive e.g. , emphasiHere, we present simple two- and three-locus population-genetic models that describe the properties of this system when one rescue gene and one or two copies of a killer gene are each engineered into independently assorting loci of the transgenic strain. We also discuss some potential genes and genetic tools that could be used to build the system.In the simplest form of our model, it is assumed that one copy of the killer gene is sufficient to kill any individual that does not have the rescue gene. It is also assumed that carrying one copy of the rescue gene is sufficient to completely suppress the action of the killer gene, no matter how many copies of the killer gene are present. These assumptions are based on the idea that the killing is caused by the expression of a gene that produces a toxic protein or RNA, and that the rescue is caused by a gene that renders individuals immune to the action of the killer alleles, no matter what the ratio is of the killer to rescue alleles. It is assumed that one copy of the anti-pathogen gene completely interferes with pathogen transmission. In the simple form of the model, it is always assumed that homozygous engineered individuals are released at a 1 : 1 sex ratio.Throughout this paper, we refer to the killer allele as ‘K’ and the rescue allele as ‘R’. The alternate null alleles are referred to as ‘k’ and ‘r’, respectively. When there are two insertions of the killer alleles, the second insertion will be referred to as K′. Each locus at which a K or R allele is inserted has independent recombination with all other loci with insertions. The model is completely deterministic, and it assumes infinite population size and random mating. In §3, we consider an allele to be lost from the population if it is below a frequency of 0.001 and decreasing.If there are no fitness costs associated with carrying either R or K (when it is suppressed), then for the two-locus case the fitness of seven of the nine genotypes is simply 1.0 . By contrast, the fitness of genotypes Kkrr and KKrr is 0 because they have one or two copies of the K allele and no R alleles.cK2 and cR2, respectively. We assume that the fitness costs are additive both at a locus and between loci, i.e. the costs to heterozygotes, cK1 and cR1, are half of the homozygote costs, and, when an individual has both the killer and rescue alleles, the total cost is the sum of the costs due to the killer and rescue alleles. The iterative equations that describe the dynamics of this simple model with one K and one R insertion are given in the electronic supplementary material, §IV.The fitness costs associated with being homozygous for the killer and rescue genes are The simple model is modified by adding female-specific and partial killing. Other modifications to the model including variation in the dominance of fitness costs, and male-only releases, are presented and discussed in the electronic supplementary material. The C++ computer code is available from the authors.Detailed mathematical analysis of the dynamics, equilibria and introduction thresholds are provided in the electronic supplementary material, §§V, VI and VII. Here, we present time-series dynamics for a set of potentially realistic cases.A. aegypti in some tropical climates). Figure 1a shows the results of a single release of a homozygous engineered strain with one K insertion and one R insertion, and a 1 : 1 sex ratio. The release ratio is two engineered insects for each native insect, so the initial frequency of homozygous engineered insects is approximately 0.66. As shown in figure 1a, the frequency of K drops slightly over the 120 generations, while the frequency of R increases to approximately 0.99. The frequency of insects within the population that can transmit the pathogen is equal to the frequency of kkrr (i.e. the wild-type) because only rr individuals can transmit the pathogen, and any individual that is homozygous for rr but carries a K allele (KKrr or Kkrr) will die.b. Here, the K allele is nearly lost from the population while the R allele increases to 0.86 due to mortality of rr genotypes caused by the K allele. If the small release is repeated four times over the first four generations (in a population that maintains a stable density) so that the total number of released insects is equal to the number released in a single-generation release depicted in figure 1a, then the frequency of K after 120 generations is substantially higher, as can be seen by comparing figure 1a,c.The result of a similar release with one-quarter the number of engineered insects  is shown in figure 1d depicts the dynamics of a single 1 : 2 release of a homozygous strain that has two independently assorting loci with insertions of the same killer gene (K and K′) and one locus with a rescue gene insertion. Even though the release frequency is the same as in figure 1b, the frequency of R after 120 generations is substantially higher, and the frequency of individuals that can transmit the pathogen is less than 1 in 1000. Results in Figure 1a–d depict the relative fitness of the populations after the release of the killer–rescue strain. In all cases, the fitness of the population decreases somewhat in the generations immediately after the release due to the impact of the killer allele, but over time, the relative fitness approaches 1.0 as either the killer allele is lost from the population or the rescue allele approaches fixation.The solid lines at the top of figure 1a,b.)It would be impractical to present individual time-series figures for all possible release ratios. figure 1In the long term, each trajectory approaches an equilibrium. Depending on the release fraction, this equilibrium will either include (green line) or not include (red line) any K alleles. In all cases where some K alleles are present at equilibrium, the R allele is fixed, and no individuals can transmit the pathogen. The trajectory that separates these two outcomes is close to  a straight line, which allows an approximate value of the release threshold that delineates these two qualitatively different outcomes to be calculated. Details of this calculation appear in the electronic supplementary material, §VII, and give the release threshold as approximately 0.354. Numerical simulation can determine the precise value of the threshold and we find it to lie between 0.350 and 0.351. cK1=0.5 cK2. Similarly, cR1=0.5 cR2.  Results of the model when cK2 and cR2 are each equal to 0.1 are shown in figure 3a–d, where the types of releases are identical to those in figure 1a–d. With this 0.1 fitness cost, the K and R alleles are lost (or are at extremely low frequency) by generation 120 if there is a 1 : 2 release , even when there are two insertions of K. When there is a single 2 : 1 release or four 1 : 2 releases , the frequency of pathogen transmitting individuals is reduced below 5 per cent for 50 and 70 generations, respectively. However, in both cases, K is lost before generation 120 and R is less than 0.30 by generation 120.In figures cK2 and cR2 are set lower, at 0.05, and the release is large enough to result in an initial frequency of 0.95 of R and K (19 : 1 release), and there are two insertions of K, the R alleles are lost by generation 600 (not shown). This loss of R occurs because both Ks are almost lost by generation 300 due to selection and fitness costs.Even if To further understand the selection factors that impact the pattern of allele frequency changes, we have plotted the fitness of each allele over time in the electronic supplementary material, §III. A mathematical analysis of possible equilibria under conditions where both K and R have fitness costs is given in the electronic supplementary material, §V.The fitness costs associated with the K and R constructs could be very different. Both could have costs associated with the insertion site, but costs to the K construct could also come from low levels of toxin production even when the R allele is homozygous. A fitness cost to the R construct could occur if the expression of the rescue substance (protein or RNA) had effects other than interfering with the action of the construct K. Because the rescue gene and the anti-pathogen gene are part of the same construct, any negative effects of the anti-pathogen gene on the insect's fitness would be part of the fitness cost associated with the R allele.a–d), where both cK2 and cR2 were 0.10 to the extreme case where either cK2 or cR2 was set at 0.20 while the other cost was set at 0.It is difficult to predict just what the fitness cost differences would be between R and K, but to provide insight into how such differences would affect the dynamics of the gene-drive system we compared the cases above a–d, whera). By contrast, the R allele frequency reaches a high equilibrium level when the 0.20 fitness cost is associated with K and there is no fitness cost for R (figure 4b). If four 1 : 2 releases are conducted and the 0.20 cost is associated with K, the results (figure 4d) are generally similar to those with the single 2 : 1 release. However, when the 0.20 cost is associated with R, the four small releases result in a more sustained impact on pathogen transmission (figure 4c) than the single larger release (figure 4a). This occurs because in the single-generation intervals between each of the four releases, K and R are increasing due to selection, and the final frequencies of K and R after the four releases are therefore higher than those after the single 2 : 1 release. This result contrasts with the behaviour of engineered underdominance constructs where fewer total insects need to be released in a single release than in multiple releases (Results shown in figure 4releases .cR2 is only 0.05 (resulting in heterozygotes with a 0.975 relative fitness), the R allele can remain in the population for over 1000 generations if the release ratio is high. However, both the K and R alleles are lost by generation 1000 if the release results in an initial frequency below approximately 0.50. Thus, even with no immigrants from beyond the spatial range of the release, the killer–rescue system can be self-limiting over time with fitness costs that are typically too small to be detected experimentally.When the fitness costs associated with the R insertion are large, it is not surprising to find that the R allele is eventually lost. When there is no fitness cost associated with K, and All potential equilibria when only R or only K has associated fitness costs are explored in the electronic supplementary material, §V.cK2=cR2=0.10 and the cost is additive.In all of the results that follow we use a 2 : 1 release and assume that a). Under both of these conditions, pathogen transmission is not reduced as much as when there is a similar release with 100 per cent killing (shown in a).Partial killing can be viewed as killing less than 100 per cent of rr individuals when K is homozygous. We compare the two cases: (i) 50 per cent of males and females are killed and (ii) 0 per cent of males and 100 per cent of females are killed. When the killing by K is dominant, results are identical if only rr females die or 50 per cent of all rr individuals die a. Under shown in a.a therefore also represent this case).If all KKrr individuals die, but only 50 per cent of Kkrr individuals die, then killing is considered additive. It is interesting that under this condition the results of the model are identical to those in which there is complete dominance but only 50 per cent mortality  for almost 200 generations (not shown). This result seems counter-intuitive, but it makes sense because at frequencies above 0.90 the additive genetic variance for fitness associated with R is much greater when R is additive than when it is dominant in effect. This could be important to consider when building the rescue constructs.If the killing impact of K is completely dominant and rescuing by R is additive, there is a very different result, with R approaching fixation b for almPlasmodium transgenes, as well as strong gene drivers, may be engineered into mosquitoes within the next 5–10 years, it is not expected that the first field releases of transgenic mosquito strains will carry a strong gene driver that could presumably spread throughout a species' range based on the release of a small number of individuals .Our contention is that at some time between the release of a mosquito strain with a non-driven refractory gene and the release of a strain with a strong drive mechanism such as a It has generally been assumed that one goal of laboratories that are engineering mosquitoes for gene drive should be production of strains with the lowest possible fitness costs, and that lines with large fitness costs should be discarded. However, it is reasonable to expect that regulatory agencies will want the first released strains to be lost from native populations in a relatively short period of time. In anticipation of the possibility of such requests or requirements, it might be a good idea to save some lines that have substantial fitness costs due to defined mechanistic causes.In addition to the killer–rescue genetic drive system limiting itself over time, this drive mechanism has the advantage of being relatively simple to construct compared with already-proposed gene-drive mechanisms  a genetic construct with a gene coding for a miRNA or RNAi sequence that silences any single gene in the insect that is critical to survival, and (ii) a construct with a synthetic sequence of the critical gene that is not recognized by the interfering RNA fragments (see d).Instead of the killer gene being a sequence for the expression of miRNA or RNAi, it could actually be a gene that codes for a toxic protein. In this case, the rescue sequence would code for a product that detoxified the protein or silenced the expression of the killer gene. It is expected (but not certain) that both the systems described here would show dominant effects of the killer and rescue genes. If two copies of the rescue allele were needed to completely overcome the effects of the killer gene, then the rescue trait might be inherited in a more additive fashion that could, somewhat counter-intuitively, lead to more sustained high frequencies of the rescue gene construct .Here, we theoretically explored only cases with one or two killer genes and one rescue gene. Our results provide a general prediction of the usefulness of adding more killer and rescue genes that are inherited independently. It might be useful to insert killer and rescue genes at positions on the same chromosome that would lead to limited recombination. The model framework used here assumed infinite population size and random mating. There was no age structure to the populations, and there were no stochastic processes in the model. This type of simple model has been successfully used to gain insights into many population-genetic processes. However, it will be critical to further examine the dynamics of the killer–rescue system with more detailed computer simulation models that include the specific biology of the target mosquito populations (e.g. Legros


Statistical density modification can make use of local patterns of density found in protein structures to improve crystallographic phases. A method for improving crystallographic phases is presented that is based on the preferential occurrence of certain local patterns of electron density in macromolecular electron-density maps. The method focuses on the relationship between the value of electron density at a point in the map and the pattern of density surrounding this point. Patterns of density that can be superimposed by rotation about the central point are considered equivalent. Standard templates are created from experimental or model electron-density maps by clustering and averaging local patterns of electron density. The clustering is based on correlation coefficients after rotation to maximize the correlation. Experimental or model maps are also used to create histograms relating the value of electron density at the central point to the correlation coefficient of the density surrounding this point with each member of the set of standard patterns. These histograms are then used to estimate the electron density at each point in a new experimental electron-density map using the pattern of electron density at points surrounding that point and the correlation coefficient of this density to each of the set of standard templates, again after rotation to maximize the correlation. The method is strengthened by excluding any information from the point in question from both the templates and the local pattern of density in the calculation. A function based on the origin of the Patterson function is used to remove information about the electron density at the point in question from nearby electron density. This allows an estimation of the electron density at each point in a map, using only information from other points in the process. The resulting estimates of electron density are shown to have errors that are nearly independent of the errors in the original map using model data and templates calculated at a resolution of 2.6 Å. Owing to this independence of errors, information from the new map can be combined in a simple fashion with information from the original map to create an improved map. An iterative phase-improvement process using this approach and other applications of the image-reconstruction method are described and applied to experimental data at resolutions ranging from 2.4 to 2.8 Å. Terwilliger & Berendzen, 1999et al., 1997et al., 2002et al., 1997Electron-density maps corresponding to macromolecules such as proteins have features that differ in fundamental ways from those found in maps calculated with random phases. These differences have been used in many ways, ranging from improving the accuracy of crystallographic phases to evaluating the quality of electron-density maps. For example, maps corresponding to proteins often have large regions of relatively featureless solvent and large regions containing of polypeptide chains, while a map calculated with random phases has similar fluctuations in density everywhere  will depend on the coordinate difference Δx between each point near x and x. The function g               x(Δx) is a function of both x and Δx and therefore must be calculated separately for each point x and offset Δx in the map. We would like the value of the function g               x(Δx) to be generally similar to the value of the electron density at x + Δx, which we will represent by ρ(x + Δx). As Δx is increased, we would like g               x(Δx) to become very close to ρ(x + Δx). That is, we would likeg               x(Δx) to be uncorrelated everywhere with the value of the electron density at x, given by ρ(x). The function g               x(Δx) gives modified values of the density at x + Δx. We would like to be able to say that if we compare the modified density at x + Δx [given by g               x(Δx)] with the density at x [given by ρ(x)], these quantities should be unrelated . One way to specify this is to require that for any offset Δx, if we go through the entire map and calculate g               x(Δx) for each point x, then g               x(Δx) and ρ(x) are to be uncorrelated,g               x(Δx) for the current purpose is to have its value at Δx = 0 be equal to the mean value of g               x(Δx) for nearby points Δx. The reason this is desirable is that we would like to compare local patterns to a template based on the correlation of densities and have no contribution from the mean value of local density. Setting the value of g               x(Δx) to any fixed value (e.g. zero) at Δx = 0 would introduce a contribution that comes from the mean value of local density ρ(x) to the correlation between g               x(Δx) and a template. A way to remove information about the mean value of local density is to specify the requirement thatx in the region to be used later in calculations of correlations of densities are considered in the averaging.An alternative method is to create a local density function for points near g               x(Δx) that has all these properties isW(Δx) is given byU(Δx) is the normalized value of the Patterson function near the origin, calculated from the electron-density map itself using the relation g               x(Δx) is equal to the value of the electron density at x + Δx, after correction for the difference between ρ(x), the value of the electron density at x, and 〈ρ(x + Δx)〉xΔ, the mean of nearby values, all using the weighting function W(Δx). It can be verified by substitution that both . The approach used here to obtain templates is hierarchical. First, three separate sets of N               max  templates are generated using only points in an electron-density map that have low, medium or high electron density. A subset  of these templates that have low mutual correlation is then selected. Finally, an even smaller subset of N               final  templates is chosen from this group in order to maximize the predictive power of the templates while maintaining a fixed number of total templates.The first step in the procedure for density modification by pattern matching is to obtain templates that correspond to common patterns of local electron density. These templates are generated using the local electron density near each point x), the modified local electron density g               x(Δx) is calculated for all neighboring points within a radius r               max . This modified electron density is compared with all existing templates using the correlation coefficient of density in the template with the modified local density as a measure of similarity. The grid used is normally the same grid as is used for all FFT, NCS-averaging and other density calculations and is typically between 1/6 and 1/4 of the resolution of the map. The number of points typically used in a template is approximately 100. For each existing template, N               rot different rotations of the template are considered so as to attempt to match the modified local density in any orientation and the highest correlation coefficient of the match for all rotations of the template is noted. In the examples considered here, we use a total of N               rot = 158 rotations to sample the possible three-dimensional rotations of an object with a rotation of about 50° relating neighboring orientations. If the correlation coefficient of the local modified electron density at this point x with an existing template k is greater than CCmin , then the local modified density at this point is included in the definition of template k by rotating the density to match the current template k and including the rotated local modified density in the average density for this template. If the local modified electron density does not have a correlation with any existing template greater than CCmin, then the local modified density is used to start a new template. Once N               max templates have been created , then the local modified density at each subsequent point is included in whichever template it matches most closely.To generate a set of templates, each grid point in an electron-density map is considered, one at a time, only including points that are associated with either low  of these is chosen based on mutual correlation coefficients in order to obtain a set of templates with the minimum possible similarity to each other. To do this, the correlation coefficients of all pairs of templates are calculated and the template with the highest correlation to another template is eliminated. The process is repeated until the desired number of templates is obtained. The final selection of templates based on predictive power is carried out after analyzing the statistics associated with each of the 2.4.x. This is peformed for experimental maps both with and without added errors. There are many possible ways to describe these relationships, but a simple approach used here is to break it down into two parts.The second overall step in this process is to identify the relationships between the correlation of each template with local modified density in a map and the value of the electron density at x in a map is quite strongly dependent on the two templates k and l that have the highest (k) and next-highest (l) correlation coefficients with the local modified density at x. That is, for electron-density maps of proteins, the probability distribution p can be very informative about the electron density ρ at x.The first part consists of an examination of the statistics of high-quality experimental maps. We have found that the electron density at a point x to a map that contains errors and then to use these in a calculation of the probability that a particular pair of templates k and l would have the highest two correlation coefficients in the corresponding high-quality map. In this case, the statistics of density for the high-quality maps p obtained above can then be applied.The second part is to consider the relationship between maps with and without added errors. The approach is to begin with the observed correlation coefficients of all the templates at a point p that the correlation coefficient for template k to a point x in a high-quality map would have the value CCk, given the observation that this template has a correlation coefficient of CCkobs, to the same point in a map with additional errors. To account for differing levels of error in the experimental map, these probabilities are tabulated as a function of the overall figure of merit of the map with errors.To carry this process out, a second set of probabilities are needed. These are the probabilities x in a new (‘observed’) electron-density map, the correlation coefficient of each template k to the local modified density near x is first determined . This set of correlation coefficients {CCobs} and the two probability distributions p and p can then be combined as follows to obtain an estimate of the electron density ρ at x in a high-quality version of the same map.To apply these probability distributions to data near the point k and l have the highest correlation coefficients to the local modified density near x in a high-quality version of the new ‘observed’ map, then we could use our probability distribution p directly to estimate the probability distribution for ρ. We do not know the identity of k and l, but suppose instead that we had probabilities, p, for each possible pair k and l based on the correlation coefficients observed for the ‘observed’ map. Combining these, we could write thatk and l. An estimate of the electron density at x can then be obtained from the weighted meanp, that the pair k and l have the highest correlation coefficients to the local modified density near x in a high-quality version of the ‘observed’ map can in turn be estimated from the observed correlation coefficients of all the templates to this map, {CCobs}, in several steps. We separate the probability into two parts, one for the probability that template k has the highest correlation and one for the probability that template l has the next highest, given that template k has the highest correlation, k has the highest correlation with the (non-existent) high-quality version of the ‘observed’ map. We will integrate over all possible values of CCk, the correlation of template k with the high-quality map. For each value of CCk, we will calculate the probability that this is indeed the value of the correlation of template k, given by p(CCk) = p, and the probability that all other templates have a correlation coefficient less than CCk,k. The probability that template l has the next-highest correlation is given byIf we somehow knew which two templates 2.5.x as a function of the correlation coefficients of the N               max templates with the local modified density at x. Each of the N               max templates is compared with the modified local density at all points in a set of high-quality maps. At each point x, the two templates k and l that have the highest and next-highest correlation coefficients, respectively, with the local modified density at x are identified . The value of the (unmodified) electron density ρ(x) is then tabulated as a function of k and l. These histograms are then normalized to yield an estimate of the probability distribution, p.An important part of this step consists of generating histograms of values for the electron density at p, relating the correlation coefficient value, CCkobs,, observed for a particular template at a point x in a map that contains added errors to the correlation coefficient, CCk, that would be observed for the identical template at the identical point x in the corresponding map without any added errors. These probability distributions are calculated by using paired sets of high-quality experimental maps with and without added errors. At each point in a map, the correlation coefficient of each template k to the map without added errors, CCk, and the correlation to the map with added errors, CCkobs,, are noted. Normalization of the resulting histograms leads to an estimate of the probability, p, that CCk is the correlation to the map without added errors if the value CCkobs, is observed in the map with added errors. This calculation is repeated for maps with varying levels of additional errors by creating simulated phase sets with Gaussian distributions of phase errors with varying overall values of the cosine of phase error, 〈cosΔϕ〉, ranging typically from 0.5 to 0.8. In application to new ‘observed’ map, the probability distribution obtained using data with added phase errors with a mean cosine 〈cosΔϕ〉 similar to the figure of merit of the experimental map is used.The second part of this step is to obtain probability distributions, 2.6.N               final templates is based on predictive power. A subset of N               final templates is selected from the N               max templates obtained earlier using high-quality electron-density maps. The subset is selected to maximize the correlation between the electron density calculated using (9) and the electron density in the maps. The histograms that form the basis of (9) are calculated from experimental density for one set of proteins and the correlation is calculated for another. The pair of templates that yields the highest correlation is first identified. Then, one by one, the template that increases this correlation by the largest amount is added to the group, until N               final templates are chosen.The final selection of 2.7.N               final templates, considering as many as 158 rotations of each template  for each point. We have developed a simple indexing system that reduces the number of rotations that need to be considered for each template. The index for a point x is based on the density at M points near x . Point m is given an local index i               m from 0 to 3, based on the local density at that point , where σ is the r.m.s. of the entire map. An overall index I is then calculated for the local density from the relation M nearby points. Next, the relationship between the index I and the best rotation is tabulated for each of the templates using high-quality experimental maps containing added errors. For each point in each map used above to calculate statistics of the correlation of templates with local modified density, the index I is calculated and the optimal rotation is noted for each template. An indexing table is then constructed in which each index I is associated with a list of preferred rotations for each template. The table is constructed so that about 95% of the time the optimal rotation for a given template is contained in the list. This indexing procedure reduces the number of rotations that need to be considered by about a factor of five. Other indexing methods could be applied that might further reduce the number of rotations to be considered  to produce a probability distribution, p(ρ|{CCobs}), for the electron density at x. The estimate from (9) of density at x, ρest , can then be used to construct a new estimate of the electron density in the map. This ‘recovered image’ of the electron-density map can be visualized with or without smoothing, can be used as a target for statistical density modification  to obtain the best possible electron-density map without using any pattern-based information. This density-modified map is then analyzed for local patterns and an image of the map is recovered. Thirdly, the density in the recovered image is used all by itself to estimate phase probabilities. This third step is carried out here using statistical density modification , then this estimate can be used as prior information about the map. In this procedure, observed values of the amplitudes of structure factors are used and an estimate of uncertainty in the electron density is required. This procedure is used to estimate phase probabilities from a recovered image, where the expected electron density is simply the best estimate from  shows a section of a density-modified MAD electron-density map for initiation factor 5A . Note that the density at x is positive in this case. In Fig. 2b), the density is adjusted to remove the information about the density at x from x and from all neighboring points. This calculation essentially consists of subtracting the origin of a normalized Patterson function corresponding to this map, multiplied by the value of the density at x minus the mean local density, from all neighboring points, as described in §x to the mean density in the local region, setting the density very near x to intermediate values and leaving the value of points far from x unchanged.An important aspect of the pattern-matching density-modification method presented here is that it is designed to yield an estimate of the electron density that has errors uncorrelated with the errors in the original map. This is accomplished by using only information from the region around a point 3.2.1bkb; Berman et al., 2000et al., 1998x from the nearby density. The patterns are selected after considering rotations about the central point, so any rotational differences between templates are not significant in determining their features.The analysis of local patterns in electron-density maps was carried out using the density-modified MAD electron-density map from IF5A, calculated at a resolution of 2.6 Å  (open circles) shows the correlation of estimated and model density as a function of the number of templates used. Much of the information is contained in just two templates and almost all the rest is in the first 20. Based on this observation, we have used 20 templates for the remainder of this work.The final templates were chosen on the basis of their predictive power. The a) show that such an association exists and that only a small number of templates are needed to describe it. We next tested whether a similar association exists for random maps. The closed triangles in Fig. 3a) were obtained in the same way the open circles, except that all the maps were calculated after randomizing all the crystallographic phases. The closed triangles in Fig. 3a) show that there is essentially no association between local patterns of density and density at their central points for the random maps. This means that the correlations between patterns and densities at their central points is a feature of protein-like maps and not a feature of maps with random phases.The fundamental property of macromolecular electron-density maps that is used in our approach is that different local patterns of density in these maps are associated with different values of the density at their central point. The open circles in Fig. 3x in the analysis of the patterns surrounding x using  shows that this choice of methods is also important for discriminating between patterns that arise from noise and those that arise from protein-like features. Fig. 3b) was calculated in exactly the same way as Fig. 3a), except that the local density was not adjusted to remove information about the value of the density at the central point and a completely new set of templates and statistics was used, reflecting this different approach. This was accomplished by not applying (5b) show that if the local density is not adjusted to remove information about the central point, then templates can be obtained that give a very high correlation between the value of the density calculated from (9b) show that randomized maps give essentially the same correlations as protein electron-density maps when the information about the central point is not removed from the calculations.An important part of the present approach was the removal of information about the density at a point a) and 4(b) show contours of positive density corresponding to the N               max = 20 templates obtained. The templates are arranged in order of decreasing contribution to the estimates of density. The patterns are very simple, typically containing one to three spherical or extended regions of positive density and one or more rings or regions of negative density in various relations to the central point. Some of the pairs of templates are similar  and, as shown in Fig. 3α atom, the junction of three chains of atoms. This density, after removing the information about the density right at this point, is most closely similar to pattern No. 12 in Fig. 3Figs. 4x in a map is compared with the 20 templates and the two templates that match the density most closely are identified. The procedure is first performed with high-quality experimental maps to associate pairs of templates with expected density and then with an observed map to estimate the values of electron density in a high-quality version of the observed map. In order to use as much information as possible, the process is carried out in a probabilistic fashion, considering the possibility that any pair of patterns might best match the density in a high-quality version of the observed map.The core of the method described here is the association of different templates with different expected values of electron density at the point that is at the center of the templates. The electron density near a point The 20 patterns are each associated with different average values of density at their central points. For example, template No. 1 contains two spherical regions of positive density situated on opposite sides of the origin. At locations where this pattern is the one that best matches the density in model maps, the mean density at the central point is about −0.3 ± 0.6 . Template No. 12 contains a curved lobe of positive density immediately adjacent to the origin. Template No. 12 is associated with mean density of about 0.6 ± 0.9. Table 13.3.a) shows a section of model electron density with errors calculated using the structure of gene 5 protein  and a version of this density, smoothed with a radius of 1.5 Å, is shown in Fig. 5c). Finally, phases were estimated using statistical density modification based on the model structure-factor amplitudes from the reconstructed density . The reconstructed density has a correlation coefficient to the original (model) map of 0.19, the smoothed image has a correlation of 0.38 and the map calculated with phases obtained from the reconstructed density and model amplitudes has a correlation coefficient of 0.46.The templates shown in Fig. 4y               1(x) and y               2(x) have correlated errors is assumed for the present analysis. For convenience, in this analysis the maps y               1(x), y               2(x) are each normalized to an r.m.s. value of unity and a mean of zero. In this error model, each map has a component that is related to t(x), the true density in a perfect map , each map has a component c(x) that is an error term unrelated to t(x) but that is the same in the two maps and each map has an independent error term e               1(x) and e               2(x). As this is model data, we know the values of t(x) as well as the values of y               1(x) and y               2(x), 1 and α1 can be estimated from the known maps t(x), y               1(x) and y               2(x), errors with the relationb) is −0.01. The same calculation for the recovered smoothed map in Fig. 5c) leads to a correlation coefficient of the errors of −0.02. Similarly, the calculation for the map in Fig. 5d) obtained using phases calculated from the recovered image and model amplitudes lead to a correlation of errors of −0.04. This indicates that the errors in the recovered image are not correlated with the errors in the original map.As model data were used to obtain the images in Fig. 5We have found that the independence of errors is not as perfect when density-modified phases are used. To examine this, we started with model phases and amplitudes, introduced errors into the phases, leading to an electron-density map with a correlation to the perfect map of 0.6, and then carried out statistical density modification on this map , leading to a density-modified map with a correlation to the perfect map of 0.83. This density-modified map was then analyzed for local patterns as described above. In this case the smoothed recovered image had a correlation to the perfect map of 0.50. The correlation of errors with the density-modified map was 0.21, considerably higher than in the case where the map used for pattern identification had completely random errors. This suggests that the method might not be quite as effective when used on density-modified maps as on experimental maps.3.4.1vqb; Skinner et al., 1994RESOLVE electron-density map had a correlation coefficient of 0.79 to the model density calculated from PDB entry 1vqb. Fig. 6a) shows a section through this density-modified map. Local pattern analysis was applied to this map as described above. Fig. 6b) shows the image that was recovered from this map, Fig. 6c) shows a smoothed version of this image and Fig. 6d) shows the map obtained using phases calculated from the recovered image and observed structure-factor amplitudes. The recovered image in Fig. 6b) has a correlation of 0.25, the smoothed recovered image in Fig. 6c) has a correlation of 0.42 and the map calculated using phases from the recovered image in Fig. 6d) has a correlation of 0.52.The analysis described above was carried out with electron density calculated from models so that the error analysis could be performed in detail. We next applied the method to electron density obtained from a MAD experiment so that its utility with real data could be examined. The electron density obtained after applying statistical density modification . The correlation of errors between the starting RESOLVE map in Fig. 6a) with the errors in the recovered image in Fig. 6b) was 0.15 and the correlation of errors between the starting RESOLVE map with the errors in the smoothed recovered image in Fig. 6c) was 0.23. The correlation of errors in the map calculated using phases from the recovered image in Fig. 6d) with the errors in the starting RESOLVE map was 0.36. This means that the errors are not highly correlated in this analysis, but that they are also not completely independent. Part of the correlation of ‘errors’ could be because of the fact that the ‘true’ density is not known and the errors are estimated using model density for gene 5 protein. Consequently, any errors in this model density would lead to correlation of ‘errors’ in all the maps in this analysis.An approximate version of the error analysis described in the previous section for Fig. 43.5.d) shows an electron-density map calculated using observed structure-factor amplitudes for gene 5 protein and phase probabilities obtained using statistical density modification on the reconstructed image in Fig. 6b). These phase probabilities were then combined with the original phase probabilities from the three-wavelength MAD experiment to yield a set of phase probabilities and a new electron-density map. The original SOLVE electron-density map . This map has a correlation with the model gene 5 protein map of 0.56. The electron-density map calculated from combined phases is shown in Fig. 7b). This new electron-density map has a correlation to the model map of 0.65. Finally, the combined phases and the experimental structure-factor amplitudes were used in statistical density modification using the same parameters as those used to obtain the original RESOLVE phase probabilities. The resulting map is shown in Fig. 7c); it is very similar to the original RESOLVE map shown in Fig. 5a), but is slightly improved, with a correlation to the model gene 5 protein map of 0.82 .Fig. 6x from the analysis of patterns of density around of x. We tested the importance of this step by repeating the entire process of generating templates and histograms and then applying them to the gene 5 protein MAD data, but without removing this information. In this case, the recovered image had a higher correlation with the model map than in the test case described above (0.55 compared with 0.25) and the smoothed recovered image had a correlation of 0.59, compared with 0.42. On the other hand, the correlation of errors between the recovered image and the starting RESOLVE map was also much higher (0.68 compared with 0.15), as was the correlation of errors between the smoothed recovered image and the starting RESOLVE map (0.85 compared with 0.23). Finally, the resulting combined phases were used as a starting point for density modification, but in this case no improvement in the final map was obtained (correlation coefficient with the model map of 0.79 in both cases), supporting the idea that this step is an important element in the process.A key element of the process used here is to remove information about the density at each point 3.6.Thermotoga maritima  at a resolution of 2.4 Å. Fig. 8a) shows a section through the RESOLVE electron-density map obtained without using local pattern matching. Figs. 8a), 8b) and 8c) show the density-modified map after one, three and five cycles using local pattern matching. The correlation coefficient of the starting RESOLVE electron-density map with a map calculated from the refined model of nusA is 0.65; the map after five cycles has a correlation of 0.85.Fig. 1RESOLVE map had a correlation with the model map of less than 0.7, with smaller improvements obtained when the RESOLVE map was better than this. To provide a rough measure of the utility of the method, the automatic model-building capability of RESOLVE was applied to the maps obtained for each structure with and without information from local patterns , as were the correlations of the final maps density modified including the local pattern information .Although the templates used in this procedure were calculated using data to 2.6 Å, the procedure is not strongly dependent on resolution. Using the nusA data as a test case, the effect of resolution was examined by truncating the analysis at resolutions of 2.4 , 2.6, 2.8 and 3.0 Å, respectively. The correlation of the original 4.We have shown here that local features of electron-density maps can be used as an important source of information in a density-modification procedure. The improvements in map quality obtained using the information from local patterns range from none (0.87 to 0.87 for β-catenin) to small (from 0.79 to 0.82 in correlation coefficient for gene 5 protein) to very substantial (from 0.65 to 0.85 in correlation coefficient for nusA).P. aerophilum listed in Table 2The computational requirements of the methods are moderate. Carrying out a complete set of five cycles of pattern identification and density modification using local patterns takes 90 min on a Compaq 833 Mhz Alpha for the ‘hypothetical’ protein from There are many additional applications of the procedures that we have developed here. A key aspect of the methods is that the image that is recovered from an electron-density map has errors that are relatively uncorrelated with those in the original map. This allows the use of the recovered image in phase improvement in the moderate-resolution range demonstrated here. It is also possible that the same approaches could be used for low-resolution as well as very high resolution phasing and phase extension. Additionally, the independence of errors means that an image recovered from a random map will have little or no correlation to the original map, while an image recovered from a map that has protein-like features will have a correlation. Consequently, the method could be used to evaluate the quality of protein electron-density maps. Similarly, points that are in the solvent region of a crystal will have local features unlike those found in the protein region and the methods described here could be used to distinguish the protein from solvent regions.A weakness of the pattern-matching approach developed here is that it cannot readily distinguish protein-like features that are the result of systematic bias or errors in a map from those that actually reflect protein structure. This may be reflected in the small but significant correlation of errors between the density-modified model gene 5 protein map and its recovered image described above. Perhaps more importantly, it means that the method in its present form is not as well suited to improving maps that contain significant bias towards protein-like patterns of density, such as those obtained using phases from an atomic model, as it is to improving maps in which the errors are essentially random, such as those obtained by experiment.A useful extension of the methods described here will be to recalculate the templates and histograms using data in various resolution ranges and using various radii for the regions considered in obtaining templates and to apply the appropriate set to experimental data. The effects of the grid spacing used in calculations could also be investigated. The use of correlations to more than two templates could be used in (8