Affinity Conversion Model (ACM). This model postulates that IL2R receptor association is sequential and dependent on ligand binding. Most likely free IL2 binds first to IL2Rα, and then this complex binds to IL2Rβ, and finally to IL2Rγ (γc). However, in previous mathematical models representing this process, the binding of γc has not been taken into account. In this work, the quantitative contribution of the number of IL2Rγ chain to the IL2-IL2R apparent binding affinity and signaling is studied. A mathematical model of the affinity conversion process including the γ chain in the dynamic, has been formulated. The model was calibrated by fitting it to experimental data, specifically, Scatchard plots obtained using human cell lines. This paper demonstrates how the model correctly explains available experimental observations. It was estimated, for the first time, the value of the kinetic coefficients of IL2-IL2R complexes interaction in the cell membrane. Moreover, the number of IL2R components in different cell lines was also estimated. It was obtained a variable distribution in the number of IL2R components depending on the cell type and the activation state. Of most significance, the study predicts that not only the number of IL2Rα and IL2Rβ, but also the number of γc determine the capacity of the cell to capture and retain IL2 in signalling complexes. Moreover, it is also showed that different cells might use different pathways to bind IL2 as consequence of its IL2R components distribution in the membrane.Interleukin-2 (IL2) is a growth factor for several immune cells and its function depends on its binding to IL2Rs in the cell membrane. The most accepted model for the assembling of IL2-IL2R complexes in the cell membrane is the  Interleukin-2 (IL2) is a protein initially identified as a T cell growth factor . IL2 is Kd~10-8M [Kd~10-7M [Kd~10-11M) [Kd~10-10M) [Kd~10-9M) [Interleukin 2 mediates its functions in the target cell through the interleukin 2 receptor (IL2R). IL2R is a multimeric functional protein consisting of three different chains: IL2Rα , IL2Rβ  and γc . The firKd~10-8M  and Kd~1Kd~10-7M . Simulta~10-11M) , the pse~10-10M)  and the d~10-9M) . The sigd~10-9M) .IL2 activity in different cell types is given by the differential distribution of the IL2R chains on the cell membrane. The distribution of IL2Rα in different cells has been studied by several techniques, including Scatchard method. These studies have revealed that the expression of IL2Rα highly increases during the activation of T cells. For this reason, IL2Rα expression has been established as a marker of activated T cells. On the other hand, IL2Rβ is highly expressed in NK and CD8+ T cells. The differences on IL2Rα and IL2Rβ expression leads to differences in the capacity of cells to capture and retain IL2 ,12,14,16The γc was the last component of IL2R to be discovered . Now it The proportion of the γc expressed on the cell membrane, in relation to IL2Rβ or other cytokine receptors, might determine the competition for using this common chain by different cytokines. There are two possibilities: a) γc is in excess with respect to IL2Rβ and other cytokines receptors ensuring its accessibility to all cytokine receptors; or b) γc is in defect regulating its availability to different cytokines receptors and the overlap of different signaling cascades in the cell.Affinity Conversion Model (ACM). It postulates that IL2R association is sequential and dependent on ligand binding [The assembling mechanism of IL2-IL2R complexes on the cell membrane remains to be elucidated. The most accepted model is the  binding . This is binding ,21. Rece binding ,23. More binding  did not  binding , and IL2 binding , are cloIn this work, it is studied how γc chain quantitatively contributes to the IL2-IL2R apparent binding affinity and signaling. This matter passes through the understanding of the IL2-IL2R assembling mechanism. Taking all that into account a mathematical model was formulated including the minimal biology known for the IL2-IL2R system. It is considered the existence of three different chains that coexist in the cell membrane and dynamically interact depending on previous ligand binding. The model we proposed is based in the affinity conversion model for the IL2-IL2R assembling and explicitly includes the γc in the dynamic.It can be observed in a further reading, that the model properly explains the available experimental observations. It was estimated, for the first time, the kinetic coefficients of IL2-IL2R complexes interaction in the cell membrane. Moreover, the number of IL2R components in different cell lines is also forecasted. It was obtained a variable distribution in the number of IL2R components depending on the cell type and the activation state. It is shown that the IL2Rs expression determines how different cells might use different pathway of IL2-IL2R assembling. Of most significance this study predicts that not only the number of IL2Rα and IL2Rβ, but also the number of γc determine the capacity of the cell to capture and retain IL2 in signalling complexes.The presented model considers that IL2R components interact sequentially, upon IL2 binding to the cell. Furthermore, differently from previous works, it explicitly includes the binding of γ chain in the IL2-IL2R complex assembling. Three different pathways see  are inclThe following 9 ordinary differential equations and an algebraic relation represent these processes. In Tables Eqs  describe describe describekint) and the internalization of signalling complexes . The values for kαL, and k−αL could not be set to the values estimated by SPR, since it will prevent the proper fitting of second phase Scartchard plot data in cell lines where IL2Rα is in excess. Therefore the value of kαL was fixed in the estimated by SPR and the value of k−αL was slightly corrected from the one obtained by SPR, following fitting of the second phase of several independent Scartchard plots. With this new value the resultant affinity for IL2 binding to IL2Rα (51 nM) is close to the range estimated by Robb [L2Rβ see . The val by Robb  . It was also assumed equal kinetic rates for the interaction of γc with IL2-IL2Rβ and IL2-IL2Rα-IL2Rβ complexes . These assumptions are based in the fact that extracellular domains of IL2Rα does no interact with γc.rane see . The valAi). These parameters are treated in the model as unknown parameters to be fitted. Although, for the number of IL2Rs, a rough estimation exists from the intercept analysis of some Scarchard plots (see column 3 on Ai) is always referred in relative terms to the one of stimulated blast cells (cell A2).The second group of parameters corresponds to: the equilibrium number of IL2 receptors components per cells in the absence of IL2  was set to the theoretical diffusion limits for the IL2R chains moving in the cell membrane  were set respectively to the dissociation rates of IL2 from IL2Rα and IL2Rβ in solution. The lower limit of these parameters was set in two order of magnitude below. The numbers of IL2R per cell were explored in wider ranges than those originally estimated in the literature (from Scatchard plot intercepts). The relative cell membrane area was explored between 1 and 10.In model fittings, unknown parameters were constrained within a reasonable, but wide, range of values see . The upprane see ; and theIn order to calibrate the model it was used the results of IL2 binding assays to different human cell or cell lines. In these assays, the IL2 is added to the cells and after some time (mostly 20 min) the experiment is stopped and the number of IL2 associated to the cells is estimated.In the case of our system, this type of data contains information regarding the dynamic of IL2-IL2R complexes assembling in the cell membrane, which depends on the density of IL2Rs in the cell surface and the kinetic rates describing the reaction network. This information is related to the apparent affinity corresponding to different phases of Scatchard plot. On the other hand, this data contains information about the IL2 capture from solution, which only depends on the number of IL2Rs and is related with the intercept of Scatchard plot phases with the x axis. This infomation decouples the parameters of the number of IL2Rs with those parameters characterizing the dynamic of IL2-IL2Rs interaction in the cell membrane. Additionally, the assumption of membrane area equal to 1 for A2 cells should decouple the information about kinetic rates and the membrane areas, allowing their independent estimation. Because of the explained above, it was considered that this is the best data for the calibration of the unknown parameters of the model.6 in all cases. All the studied cells are of human origin. The reported temperature of the assays is 37°C.The IL2 binding assay corresponding to the used data for the model fitting takes 20 min after IL2 addition to the media (except for B1 and B4 that take 10 and 90 minutes respectively). According to the authors, this time period is sufficient to reach equilibrium, a precondition for a proper Scatchard plot analysis ,30. The i-cell membrane is initially in equilibrium, when the specified amount of free IL2 (t = 0. The j index refers to the specific point in the i-th Scatchard plot.The predicted number of IL2 molecules bound (or associated) to the cell i is then calculated as:tα refers to the time consumed by the binding experiment (10 and 90 min for B1 and B4 cells respectively and 20 min in other case).In order to simulate the typical binding assay, the system of eqs –10 is nuThe least square method was applied to fit the model to the experimental data. In this method the following chi-square function is defined for the simultaneous fitting of available Scatchard plots.N is the total number of experimental data sets, Mi is the total number of points in the ith set and σij is the standard deviation of jth experimental point in ith set. Standard deviation was taken as 10% of the mean experimental value, as reported by Wang [Ai) and the set of kinetic coefficients for the interactions of different complexes in the cell membrane  respectively. The total number of C1 group of parameters is 31 (considering the 8 studied cells and that the cell membrane area of A2 cells is fixed to 1). Adding the 6 C2 parameters, the number of degrees of freedom is 37. The total number of observable point in the experimental data is 144.In , N is th by Wang . The varLocal minimums of chi-square function were numerically searched, using FindMinimum routine of version 9.0 of Mathematica, starting from 1000 randomly generated initial sets of the C1 and C2 parameters values into the ranges explained above. The sets of optimal parameters C1m and C2m that minimize the chi-square function and properly explaining the experimental data were initially selected. For this aim, the relative errors between each theoretically predicted Scatchard plot (for each minimum) and the experimental one were computed by the following formula:Eij in the range , were selected .Those sets of C1m and C2m predicting more that 90% of the kαLβ, k−βLα} or {kβLα, k−βLα} or {kβLγ, k−βLγ}). This perturbation was performed 50000 times for each pair of C2 parameters and each optimal C1m and C2m selected above. Those perturbed sets of parameters, which predict a low relative error were added to our set of potential good solutions .To further increase the explored space of good fittings of the model, we apply the Monte-Carlo method for the values of C2 parameters around the good solutions . This is, we fixed the selected C1 parameters to C1m and randomly vary the value of C2 parameters by pairs ({b/cell(20min)/IL2b/cell(40min). For all the explored solutions, this kinetics is far from the equilibrium state  in cells with low number of IL2Rα because of the low association rate of IL2 to IL2Rβ. Therefore, only the cells with considerable expression of IL2Rα were considered in the equilibrium criteria. We wipe out those fitting solutions were simulations of IL2 binding to these cells predicts a value of pr lower than 0.85.Last, but not least, the set of potential good solutions were filter out with an equilibrium criteria. This is, the kinetic of IL2 binding to the cells reach or get close to an apparent equilibrium in the total number of IL2 bound at the end of experiment. To quantify the proximity to the equilibrium state, the kinetic of IL2 binding to the cells  were simulated and computed the ratio pr = IL2From the set of good solutions the C1 and C2 parameters corresponding to the lowest value of chi-square function was selected as the best fit. A confidence interval for each individual parameter was estimated as the range covered from its minimum and maximum values among the sets of all good fitting solutions founded.To evaluate the predictive power of the model we used the data obtained by Cotari and coworkers in . This isNβγL) plus the number of IL2-IL2Rα-IL2Rβ-γc (NαβγL)). To this aim it was assumed a linear relation of pSTAT5 level and the number of signaling complexes. This is based in the result of this same group that the number of IL2 molecules associated to the cells linearly correlates with the level of pSTAT5 [To explain Cotari’s data, the model was extended to simulate how pSTAT5 level varies with the total number of signaling complexes (the number of IL2-IL2Rβ-γc (f pSTAT5 . A new epSTAT5 represents the number of pSTAT5 molecules into the cell due to the interaction with the IL2. The kinetic rate of STAT5 phosphorylation (parameter κ) is undetermined but its value is not required to compute the EC50 value.The variable We simulate the experiment in which the cells remains in equilibrium in IL2 absence and in the initial time the concentration of the ligand is increased to certain value . TherefpSTAT5 for different concentrations of IL2. The EC50 value is determined as the concentration at pSTAT5 is the half of the value obtained at the higher concentration by fitting the generated data to a sigmoid function. The number of IL2Rβ and γc were fixed in the values measured in [To compute the EC50 value we generate sets of sured in . As in t-0.1, 0.1), confirming the high quality of the fitting.We fit our extended affinity conversion model, to the available Scatchard plots data from 8 different cell lines . The modThe latter result demonstrates that our model is able to explain the experimental results of IL2 binding to different cells with a diverse distribution of IL2Rs. Furthermore, we proved with several models obtained by simplifying our model and have no success in fitting individual data sets see , indicatFurthermore, this result supports the hypothesis of identical kinetic coefficients for all cells. Note that this is not an obvious result, since the movement of IL2R components may vary depending on the composition and structure of the cell membrane, affecting the value of the kinetic coefficients. For instance, it is expected that the differences in the relative presence of IL2Rs components on the lipids rafts also affect the homogeneity in the IL2Rs dynamic between different cells.Ai) for each cell line after the selection according to the equilibrium criteria explained in the previous section. In most cases, and as expected, the estimated number of high affinity IL2 receptors closely corresponds to the smallest number of model predicted IL2R chain component, either IL2Rα, IL2Rβ or γc. In other words, the IL2R chain, which is in defect at the cell membrane, limits the dynamical formation of the high affinity IL2 receptor. Following the same line of reasoning, the number of estimated intermediate affinity receptors seems to match closely the lowest number between the predicted number of IL2Rβ and γc; and the number of low affinity IL2 receptor is mostly similar to the number of IL2Rα chains when they are in a significant excess. However it must be noted that in some cases there is no clear correspondences between the estimated values of the number of low, intermediate and high affinity IL2 receptors estimated from intercepts on Scatchard plots and the IL2R values predicted by model fittings. For instances in the cell A4.The predicted distribution of IL2Rs shows a wide variety of cases, regarding IL2R composition, among the 8 studied cell lines. For the cells A1 to A4, the number of IL2Rα is larger than the number of IL2Rβ, but for cells B1 to B4 the number of IL2Rβ is larger. There is also a significant diversity regarding the predicted number of γc. Cells A1, A2 and B2 are predicted to have a clear excess of γc in respect to the number of IL2Rβ. But cells A3, B1, B3 and B4 are predicted to have less γc than IL2Rβ at the cell membrane. In some cells the number of γc is comparable to the number of IL2Rβ (A4 cells).The estimation of cell membrane areas also shows significant differences among the studied cells. In general, the model predicts a relative cell membrane area greater than 1 for all cells see . Thus, iOverall the large variation in the number of IL2R components, even in the number of γc, and the cells surface area suggests a fine control of IL2 binding and signaling dynamics on different cells or cell activation states. An interesting case in our data comes from comparing the IL2R and cell surface area between the cells B2 and B3. Cell B3 is obtained by activation of B2 with forskolin. Therefore according to our fitting, the activation of these cells, leads to an expected increase in the number of IL2Rα and IL2Rβ and the cell surface area, but a curious decrease in the number γc. Whether or not the latter is a general phenomenon will require the study of other experimental cases of cell activation.The selected solutions from model fitting show variations of around 2 orders of magnitude in the value of the kinetic coefficients, indicating that they can be considered as practical no identifiable according .kαLβ, k−βLα} or {kβLα, k−βLα} or {kβLγ, k−βLγ}). Each dark point in the graph corresponds to a good fitting solution, including the equilibrium criteria. Each gray point corresponds to a good fitting solution but without applying the last equilibrium filter. As can be seen for all kinetic parameter (kon and koff) a large range of good fitting values is obtained. The koff values span through the whole range in the allowed searched space for the fittings procedure. The kon values start from some minimum, which is larger than the lowest value in searched interval, but goes up to the highest allowed value in the searched interval. Interestingly the good fittings, in equilibrium, lay around straight line in the graph. This indicates that the fitting tries to fix the ratio kon/koff, which by definition correspond to the affinity of the process.The estimated values for the kinetic coefficients for IL2R formation at the cell membrane are shown in kon and koff. Obtaining an independent measurement of the number of IL2R components in the cell (for instance by flow cytometry based techniques), could also improve the fittings precision. In particular it could help to reduce the range of possible values for the affinities, which so far is constrained inside one order of magnitude.Overall our results show that affinities rather than kinetic coefficients seems to be constrained in the study process. This is an expected result since Scatchard plots are obtained from a typical experiment on equilibrium conditions. It must be expected that the experimental data is less informative regarding the kinetics to reach such equilibrium. In order to get an estimation of the kinetic coefficients a different types of experiment must be designed. For instance our simulations with the model suggests that performing the same binding assay experiment but for two or three different time points , will provide separate Scatchards plots, whose simultaneous fitting would significantly improve the estimations of individuals kint and In the model it has been taken into account the process of IL2R internalization, both constitutive and induced by the ligand. However previous mathematical models ,21 have Interestingly, we obtain no differences on the fittings predictions for IL2R binding kinetics, affinities at the cell membrane and surface areas. But a slight overestimation of the number of all IL2R components for each cell was observed see . For insOn the other hand, internalized complex account in the measurement as a receptor that does not dissociate from the ligand, which is equivalent to a higher affinity complex. This phenomena may affect the estimation of kinetic coefficients. In the case of IL2, only intermediate and high affinity receptors exhibit a high internalization rate. The internalization process does not significantly influence in the apparent affinity due to it is already high.in vitro experiments during 72 hours.Additionally, we simulate the possible effect of IL2Rα-mediated recycling in the parameters estimation. We compare the model considering or not the recycling process in the estimated number of IL2Rs per cell and the kinetic coefficients , obtainiWe evaluate whether or not our model of IL2-IL2R binding, once calibrated, can be used to study a completely different experimental context. To this aim we use our model to explain the data obtained by Cotari in  as explaWe obtained that our model qualitatively and quantitatively agree with Cotari’s data. This result is shown in Overall this result highlights the validity of our calibrated model. Interestingly, in  several kon for the second step of each pathway is set to zero: i.e kαLβ = 0 for the pathway i, kβLα = 0 for the pathway ii; and kβLγ = 0 for the pathway iii. Note that in these simulations, the capture of IL2 from solution is not affected in principle. The contribution of each particular pathway to the IL2 binding kinetics is evaluated, by the ability of the remaining ones to compensate its absences.In this section the relevance of the three different pathways of IL2R assembling included in the ACM  is exploAs IL2Rα is the higher affinity IL2R component and it is highly expressed in many cell types, it is logical that the pathway i is the most accepted pathway for the IL2-IL2R assembling. Here we obtained that the capture of IL2 by IL2Rβ can be relevant for cells with higher expression of IL2Rβ than IL2Rα. In this case the number of γc determines not only the use of ii and iii but also the use of i pathway (corresponding to the capture of free IL2 by IL2Rα), indicating that γc controls the assembling mechanism despite it does not participate in the free IL2 capturing.The quantitative influence of IL2Rα and IL2Rβ in cell signaling capacity and sensitivity have been previously studied ,31. TheyNγ0), but constant values of IL2Rα (Nα0) and IL2Rβ (Nβ0), at constant concentration of 1 pM. κ = 1; panel C) after 20 min of IL2 stimulation.We simulate the IL2 binding to cells with different expression of γc  corresponding with the doubling time of T cells after antigenic stimulation [In order to simulate a scenario more like mulation .The IL2 binding and cumulative signaling at 8 hours is greater than obtained at 20 min but with the similar dependency on the number of γc. This result is shown for both types of cells in 3 copies of γc per cell where measured by Cotari [5, indicating that in T cells this number may variates in around two orders of magnitude during activation. On the other hand, the results of Voss [3 copies of γc per cell, and we obtain that this number in an NK cells lines (YT) can reach to 105, which also suggest variations in around two orders of magnitude.The biological relevance of γc in IL2/IL2R interaction is also given by the variations in the γc expression that may occur in the cells in different scenarios. This aspect has been poorly studied. However, in the lymphocytes population around 10y Cotari . Our res of Voss  indicateFeinerman and coworkers  measuredWe developed a model for the assembling of high affinity IL2-IL2R complex aiming to understand how IL2 interacts with the cells. The model includes the minimal aspects of the known biology of this system, and it was demonstrated that this is the minimal model that explains the available experimental data of IL2 binding to the cells. The model was calibrated by fitting to experimental data, which allow to determine the number of IL2Rs, the membrane area and the kinetic coefficients describing the reactions of intermediate IL2-IL2R complexes. Our main prediction is that the interaction of IL2 with the cells depends on the three chains of IL2R.It should be noted the role of γc in this interaction since mostly of the publications around the topic refers to IL2Rα and IL2Rβ as the principal mediators of the IL2 biological functions  and the The calibrated model can be used for modeling other phenomena regarding the IL2 interaction with the cells. An interesting example is the IL2 derived muteins that have being designed to increase or decrease the affinity for some IL2R component Click here for additional data file.S2 File(DOCX)Click here for additional data file.S3 File(DOCX)Click here for additional data file.S1 Fig(TIF)Click here for additional data file.S2 Fig(TIF)Click here for additional data file.S3 Fig(TIF)Click here for additional data file.S4 Fig(TIF)Click here for additional data file.S5 Fig(TIF)Click here for additional data file.S6 Fig(TIF)Click here for additional data file.


In the paper, two inaccuracies have been found. of immune rejection might lead to recognition by cytotoxic T lymphocytes and immune rejection.”The first inaccuracy is in Section 2.2 on page 3 (in the pdf) in the following sentence: “These are classical MHC I that induce inflammation and their expressionIt should be without “of immune rejection.” external antigens and to support trophoblast growth by secreting IL-8, which promotes trophoblast invasion.”The second inaccuracy is in Section 3.4.1 on page 7 in the following sentence: “These lymphocytes are thought to protect the pregnancy againstIt is not “external antigens” but “pathogens.”We apologize for this oversight and for any confusion that it has caused.


Escherichia coli and Saccharomyces cerevisiae clamp loader specificity toward 3′DNA, fluorescent β and PCNA clamps were used to measure clamp closing triggered by DNA substrates of differing polarity, testing the role of both the 5′phosphate (5′P) and the presence of single-stranded binding proteins (SSBs). SSBs inhibit clamp loading by both clamp loaders on the incorrect polarity of DNA (5′DNA). The 5′P groups contribute selectivity to differing degrees for the two clamp loaders, suggesting variations in the mechanism by which clamp loaders target 3′DNA. Interestingly, the χ subunit of the E. coli clamp loader is not required for SSB to inhibit clamp loading on phosphorylated 5′DNA, showing that χ·SSB interactions are dispensable. These studies highlight a common role for SSBs in directing clamp loaders to 3′DNA, as well as uncover nuances in the mechanisms by which SSBs perform this vital role.Sliding clamps are loaded onto DNA by clamp loaders to serve the critical role of coordinating various enzymes on DNA. Clamp loaders must quickly and efficiently load clamps at primer/template (p/t) junctions containing a duplex region with a free 3′OH (3′DNA), but it is unclear how clamp loaders target these sites. To measure the  Sliding clamps and clamp loaders play a crucial role in a variety of processes in DNA replication and repair. Because of their vital nature, sliding clamps and clamp loaders are present in all kingdoms of life . SlidingEscherichia coli and Saccharomyces cerevisiae sliding clamps have similar structural characteristics despite having little sequence similarity. Both clamps make a ring shape large enough to encircle DNA, and have similar features such as positively charged α-helical regions on the interior of the clamp that interact with DNA, and β-sheets on the outer surface , is a pentamer composed of Rfc1-5 subunits, which form a cap-like structure similar to the γ complex core ..E. coli + ATPase family, characterized by the use of ATP binding and hydrolysis to drive conformational changes that rearrange macromolecules  junctions with recessed 3′-ends for both DNA replication and repair. If the clamp is not at this location, or is facing the wrong direction, the polymerase will not bind in an effective manner required for DNA synthesis. At a replication fork, a new sliding clamp is needed every 2–3 s on the lagging strand for each Okazaki fragment. Because of this pressing timetable, the clamp loader must be able to quickly target the p/t junction and load a clamp.E. coli and T4 bacteriophage clamp loaders in complex with p/t DNA give insight into DNA binding by the clamp loaders .,15.E. cos Figure .One possible method to target clamp loaders to the correct polarity of DNA is through interactions with single-stranded DNA binding proteins (SSBs) present at the p/t junction. These proteins may sense the orientation of DNA and confer this information to the clamp loaders. Another possible mechanism is that DNA specificity is mediated by sensing the charge of the p/t junction inserted in the clamp loader cap. If 5′DNA is present, a phosphate or triphosphate group will press up against the inside the cap, and this may be inhibitory to clamp loading. The clamp loaders must have a mechanism for discriminating between different DNA structures, because if clamp loading occurs on the incorrect polarity, the sliding clamp will not be available for interaction with downstream proteins.Pre-steady state fluorescence-based clamp closing assays were used to examine how different DNA polarities affect clamp loading by γ complex and RFC. Two questions were asked for each clamp loader: first, are SSBs required to target clamp loaders to the correct polarity of DNA? The second question asked is: does the 5′phosphate (5′P) group at the p/t junction prevent clamp loading on the incorrect DNA polarity? These questions were addressed by measuring clamp loading on DNA of different polarities with and without SSBs. Additionally, DNA structures that lack a 5′P were also used. Results from both γ complex and RFC are compared to determine if there is a similar mechanism between the two clamp loaders, or if this step marks a divergence in function.2) (3δδ′ψ) was provided by the M. O'Donnell laboratory.β was purified as described previously and stored in a buffer containing 20 mM Tris pH 7.5, 10% glycerol, 0.5 mM Ethylenediaminetetraacetic acid (EDTA) and 2 mM dithiothreitol (DTT) ,31. For 2) . The ind2) ,33–36. T2) (Proliferating cell nuclear antigen (PCNA) was purified as described previously ,38. PCNA2) . To meas2)  and MarzE. coli BL21 (DE3) Rosetta cells (Millipore) for 8 h at 25°C (RFC with a full-length Rfc1 subunit was expressed from a single plasmid (provided by the M. Hingorani laboratory) in  at 25°C . Cells w at 25°C .SSB was purified as described previously , and stoACGACCAGTAATAAAAGGGACATTCTGGCC and CTTTCAGGTC AGAAGGGTTC TATCTCTGTT GGCCAGAATGTCCCTTTTATTACTGGTCGT. Oligomer sequences used for 5′DNA were: CTTTCAGGTCAGAAGGGTTCTATCTCTGTT GGCCAGAATG TCCCTTTTAT TACTGGTCGT and AACAGAGATAGAACCCTTCTGACCTGAAAG CGTAAGAATA CGTGGCACAG ACAATAGTCA. The DNA sequence regions shown in bold represent the complementary regions that form the duplex, while the sequence regions in normal type represent the single-stranded overhangs. These symmetrical DNA overhangs were designed to simplify kinetics by presenting the clamp loader with only ss/ds junctions, one on each side, as opposed to a structure in which one end is blunted while the other end is a ss/ds junction. Because the clamp·clamp loader has a footprint of ∼20 bp on p/t DNA (DNA oligomers (Integrated DNA Technology) were purified using 10% denaturing polyacrylamide gels. Oligomers were designed such that annealing resulted in p/t DNA structures with a 30-nucleotide duplex region, and symmetrical 30-nucleotide overhangs of either the 3′ or 5′ polarity Figure . Oligome p/t DNA ,43,44, i2, 4% glycerol, 0.5 mM EDTA and 2 mM DTT. Reactions with RFC and PCNA had final buffer concentrations of: 30 mM HEPES pH 7.5, 150 mM NaCl, 10 mM MgCl2, 10% glycerol, 100 mM maltose, 0.5 mM EDTA and 2 mM DTT.Pre-steady state clamp closing reactions were measured using an Applied Photophysics SX20 MV stopped-flow. Reactions were performed at 20°C with a 3.72-nm band pass. A sequential mixing scheme was used in which the clamp, clamp loader and ATP were mixed and incubated for 4 s before mixing with a solution of ATP, DNA and excess unlabeled clamp to initiate the reaction Figure . In reac2 closing and PCNA-MDCC release were fit to a single exponential decay . The decrease in fluorescence of AF488 as the clamp closes was monitored as a function of time. Final concentrations of the components are 20 nM γ complex, 20 nM β-AF4882, 0.5 mM ATP, 40 nM DNA, 200 nM unlabeled β and 400 nM SSB (when present). The unlabeled clamp serves as a chase to limit the fluorescent closing reaction to a single turnover. The four different DNA structures used throughout this study are shown in Figure In order to be effective for DNA replication and repair, β must be loaded onto p/t junctions with a recessed 3′-end (3′DNA). Therefore, γ complex must have a mechanism to preferentially target this p/t junction instead of a p/t junction with a recessed 5′-end (5′DNA). Additionally, the single-stranded (ss) regions of these structures are coated with SSB g events . Briefly complex . Figure −1 compared to 1.9 ± 0.3 s−1. This small difference in rates of clamp closing indicates that there is not strong discrimination between the two polarities of DNA at this step in the clamp loading reaction. Because clamp closing is a downstream step in the clamp loading cycle, it is possible that 5′DNA has a larger effect on the rate of an upstream step, such as DNA binding or ATP hydrolysis. Nevertheless, 5′DNA still triggers an overall clamp loading reaction that is only 2-fold slower than the reaction triggered by 3′DNA.Representative β closing traces with 3′DNA (dark blue) and 5′DNA (green) are shown in Figure −1 in the absence of SSB, and 5.7 ± 0.3 s−1 in the presence of SSB. However, SSB greatly reduces the observed clamp closing rate on 5′DNA by 30-fold compared to naked 5′DNA, and by 100-fold compared to 3′DNA·SSB. This strong, 100-fold difference, between 3′ and 5′DNA supports the hypothesis that SSB confers DNA specificity to γ complex, by inhibiting clamp loading on 5′DNA.One reason why there is not a large difference observed between 3′ and 5′ DNA may be that naked DNA is not sufficient to confer polarity information to the γ complex; the γ complex may receive additional polarity cues from SSB. To test this hypothesis, the same fluorescent assay was used to measure β closing on 3′ and 5′DNA, but this time with the addition of 400 nM SSB to the reactions. Figure Clamp closing with 5′DNA·SSB is so slow, that it behaves as if ATPγS is present  or no DNOn a technical note, the first 50–100 ms of each β closing time course contain a small increase in signal Figure , inset, E. coli. In eukaryotes, the RFC clamp loader must load PCNA at 3′DNA for various DNA replication and repair functions. To determine the contribution of DNA structure and SSB (RPA) to RFC specificity, the closing assay described above for β was adapted for PCNA to measure productive PCNA loading by RFC. PCNA was selectively labeled with AF488 on either side of the interface so that when PCNA-AF4882 is open, the fluorophores are highly fluorescent, but when PCNA-AF4882 is closed, the fluorophores self-quench, leading to a decrease in signal. Control experiments with the PCNA-AF4882 clamp indicate that the fluorophores do not affect interactions with RFC  would represent the decrease in fluorescence as the PCNA-AF4882 clamp closes, while the second, slower phase (kobs2), would represent the decrease in fluorescence as the PCNA-AF4882 clamp is released from RFC. This second hypothesis is supported by directly measuring rates of PCNA release using an independent PCNA binding assay. Briefly, PCNA was labeled with MDCC on the surface to which RFC binds, and the fluorescence of the RFC·PCNA-MDCC complex is greater than the fluorescence of unbound PCNA-MDCC ((kobs2) measured in closing reactions for each of the DNA substrates with and without bound SSB .The decreases in PCNA closing time courses were biphasic and were fit to a double exponential decay Eq.  to estimNA-MDCC  and 5′DNA (green) structures with observed rates reported in Table kobs1) is referenced in the text, because this phase of the reaction that likely represents PCNA-AF4882 closing, however, both rates are reported in Table −1 compared to 1.3 ± 0.08 s−1, respectively. This small difference in rates indicates that, like γ complex, RFC cannot markedly discriminate between the two polarities of naked DNA at this step in the clamp loading reaction.Figure 2 closing assays were performed in the presence of 400 nM RPA. The addition of RPA to 3′DNA did not change the observed rates of time courses significantly  indicating that PCNA is closing as it is released by RFC. Overall, RPA decreased the PCNA closing rate on 5′DNA by 20-fold compared to naked 5′DNA, and 50-fold compared to 3′DNA·RPA. A longer time scale for closing with 5′DNA·RPA is shown in Figure To determine whether RPA enhances specificity of RFC for 3′DNA, PCNA-AF488ly Table igure 3A;The p/t junction sits inside the cap of the clamp loader  and 5′DNA (green) without 5′P groups are shown in Figure 2 closing remained about 2-fold faster on 3′ than 5′DNA. SSB was added to these DNA structures to determine if 5′ phosphorylation status contributes to inhibition of β loading on 5′DNA by SSB. Figure β-AF4882 closing time courses for 3′DNA (dark blue) and 5′DNA (green) with observed rates reported in Table 2 loading on 5′DNA whether or not the 5-end was phosphorylated.The contribution of 5′ phosphate groups to DNA substrate specificity was also addressed for PCNA loading by RFC. Figure 3δδ′ψ); average observed rates calculated from three independent experiments are shown in Table Based on the data above, SSBs enhance the specificity of clamp loaders for 3′DNA over 5′DNA. Given that the χ subunit of the γ complex has been implicated in mediating interactions with SSB ,47, the 3δδ′ψχ) whether or not the DNA is phosphorylated on 5'-ends  is about 13-fold slower than clamp closing on 3′DNA·RPA (kobs = 1.6 s−1), showing that RPA enhances the specificity of γ complex for 3′-recessed ends over naked DNA. However, the magnitude of the difference is not as great as for SSB-bound DNA where the rate of clamp loading on 3′DNA·SSB (kobs = 5.7 s−1) is nearly two orders of magnitude faster than the rate of clamp loading on 5′DNA·SSB (kobs = 0.06 s−1). On the correct polarity DNA, clamp closing by γ complex is slower on 3′DNA·RPA than 3′DNA·SSB, and the magnitude of this difference, about a 3-fold decrease, is similar to the decrease in β closing rates on 3′DNA·SSB for the χ-less γ complex. This suggests that species-specific interactions between γ complex and SSB mediated by the χ subunit enhance clamp loading on DNA of the correct polarity, but are not required.The presence of SSB and RPA confers DNA specificity to γ complex and RFC, respectively. However, is it possible for RPA to confer specificity to γ complex or SSB to confer specificity to RFC? To test this, clamp closing reactions were performed for each clamp loader in the presence of the non-cognate SSBs. Figure E. coli clamp loader, there was about a 10-fold difference in rates of clamp closing on 3′ and 5′DNA with the non-cognate SSB. The rate of PCNA closing on 5′DNA·SSB was 0.45 s−1 compared to 4.6 s−1 on 3′DNA·SBB. In contrast to the E. coli clamp loader, non-cognate SSB binding to 3′DNA had no effect on the observed rates of clamp loading, with a kobs1 of 4.4 s−1 for 3′DNA·RPA and 4.6 s−1 for 3′DNA·SSB. This shows that unlike γ complex, species-specific interactions between RFC and single stranded binding proteins do not enhance the rate of clamp loading on 3′DNA; RFC can load clamps equally as well on 3′DNA in the presence of SSB or RPA. Although SSB decreased the PCNA closing rate on 5'DNA by an order of magnitude, it did not inhibit clamp loading as strongly as RPA did  and 5′DNA (gray) in the presence of SSB Figure . As withd Figure , therefoClamp loaders are required to load sliding clamps at nucleic acid ss/ds junctions to be used by DNA polymerases and other enzymes required for DNA replication and repair. The sliding clamps serve as a mobile platform to anchor or recruit these enzymes to DNA. Given that sliding clamps have two distinct faces, one of which is responsible for most protein·protein interactions, the orientation of the clamp at the ss/ds junction is important. Clamps are oriented appropriately when loaded at ss/ds junctions with 3′-recessed ends (3′DNA), thus clamp loaders must have a method for targeting 3′DNA over a junction with a 5′-recessed end (5′DNA). When sliding clamps are needed for DNA replication, there is an additional demand to find these 3′DNA junctions quickly, as a new sliding clamp is needed every 2–3 s for each Okazaki fragment. Clamp loaders must be able to bind and open the sliding clamp, find the correct DNA substrate, and load the clamp around DNA all within a few seconds. In this study, the question of how clamp loaders target the ss/ds junctions with 3′-recessed ends was addressed.Results presented in this work show that SSBs play an important role in clamp loader discrimination between the correct (3′DNA) and the incorrect (5′DNA) polarity by preferentially inhibiting clamp loading on 5′DNA. In the absence of the SSBs, neither γ complex nor RFC showed a strong preference for loading on 3′DNA over 5′DNA Figures . To detePhosphorylation of the 5′-recessed end at ss/ds junctions plays a small role in SSB-mediated inhibition of clamp loading by γ complex. When a 5′P group is present, SSB inhibits clamp closing on 5′DNA such that closing likely occurs via a passive dissociation mechanism. When the 5′P was removed, the rate of clamp closing on 5′DNA increased by a factor of three Figure , gray, bE. coli SSB is tetramer of identical subunits, the polarity with which ssDNA binds each monomer is such that different residues in the DNA binding domains are present at the 3′- and 5′-ends of ssDNA to create different interaction sites for proteins such as the clamp loader. However, a previous study showed when reverse polarity linkages were made to change the polarity of the ssDNA relative to the ss/ds DNA junctions, the E. coli clamp loader still showed a preference for 3′-recessed ends  but not on the correct polarity DNA (3′DNA), and that specific protein·protein interactions between the clamp loaders and SSBs make a relatively small contribution to this specificity. Instead, the structure and dynamics of SSB·DNA complexes may make a larger contribution to specificity and this may be influenced by 5′phosphorylation. We propose a model by which SSB confers specificity for clamp loading at 3′-recessed ends in which the clamp loader·clamp complex initially binds the dsDNA near an ss/ds junction and slides to the junction displacing SSB from the junction. When the junction contains a 3′-recessed end, either the conformation of the SSB·DNA complex or the position and mobility of SSB on DNA is such that the clamp loader can more easily access the junction than when a 5′-recessed end is present. For the Our model for the clamp loader displacing SSB agrees with the model proposed by Downey and McHenry to explain the effects of SSB on initiation complex formation , and extS. cerevisiae, the primed template junction is composed of a RNA:DNA duplex at the 5′-end, and a DNA:DNA duplex at the 3′-end, while in E. coli, the p/t junction is an RNA:DNA duplex. These nucleic acid structures differ from the DNA:DNA structures used in this study in two main ways. First, an RNA:DNA duplex typically forms an A-DNA structure as opposed to a B-DNA structure for the DNA:DNA duplexes , it In vivo, the SSBs oligomerize on ssDNA and additional molecules of SSB could have further effects on the structure and dynamics of SSB·DNA at the ss/ds junction. These effects could potentially magnify the difference in clamp loading on 3′ versus 5′DNA, but to do so clamp loading on 3′DNA·SSB would have to be enhanced because clamp loading on 5′DNA·SSB is already as slow as the passive clamp loader·clamp dissociation reaction.The DNA structures used in these experiments were designed so that only a single SSB tetramer or RPA trimer could bind each ssDNA overhang based on the binding site sizes of SSB and RPA ,69. In vE. coli and S. cerevisiae clamp loaders.Clamp loaders perform the vital task of loading sliding clamps onto DNA for both replication and repair functions. These molecular machines must be able to bind the sliding clamp, find the correct position on DNA, and load the clamp for use by other proteins quickly and efficiently. A sliding clamp is required at ss/ds junctions with 3′-recessed ends in order to be an effective platform for DNA polymerases and other interacting enzymes. Therefore, clamp loaders must have a mechanism to target this location while simultaneously disregarding other DNA present in the cell. This work measures the DNA substrate specificity of γ complex and RFC and shows that SSBs prevent clamp loading on the wrong polarity of DNA by both clamp loaders. This SSB-mediated specificity is either partially, or entirely dependent on the presence of a 5′P group at the ss/ds junction. This study demonstrates a common general mechanism while uncovering subtle differences for DNA specificity of the Supplementary Data are available at NAR Online.


The fundamental parameters approach to line profile fitting uses physically based models to generate the line profile shapes. Fundamental parameters profile fitting (FPPF) has been used to synthesize and fit data from both parallel beam and divergent beam diffractometers. The refined parameters are determined by the diffractometer configuration. In a divergent beam diffractometer these include the angular aperture of the divergence slit, the width and axial length of the receiving slit, the angular apertures of the axial Soller slits, the length and projected width of the x-ray source, the absorption coefficient and axial length of the sample. In a parallel beam system the principal parameters are the angular aperture of the equatorial analyser/Soller slits and the angular apertures of the axial Soller slits. The presence of a monochromator in the beam path is normally accommodated by modifying the wavelength spectrum and/or by changing one or more of the axial divergence parameters. Flat analyzer crystals have been incorporated into FPPF as a Lorentzian shaped angular acceptance function. One of the intrinsic benefits of the fundamental parameters approach is its adaptability any laboratory diffractometer. Good fits can normally be obtained over the whole 20 range without refinement using the known properties of the diffractometer, such as the slit sizes and diffractometer radius, and emission profile. K(2θ) is first synthesised by convoluting together the geometrical instrument function J(2θ) with the wavelength profile W(2θ) at the Bragg angle 2θB of the peak,J(2θ) itself is a convolution of the various instrument aberration functions associated with the diffractometer, ie.,I(2θ) by convoluting the broadening function B(2θ) into the instrument profile function as shown The fundamental parameters approach to line profile fitting uses physically based models to generate the line profile shapes. The instrument profile shape 2θ)⊗J(2θ)where theθ)⊗B(2θ).This techlexander . More relexander ,7 have bFPPF has been used to synthesise and fit data from both parallel beam and divergent beam diffractometers. The refined parameters are determined by the diffractometer configuration. In a divergent beam diffractometer these include the angular aperture of the divergence slit, the width and axial length of the receiving slit, the angular apertures of the axial Soller slits, the length and projected width of the x-ray source, the absorption coefficient and axial length of the sample. In a parallel beam system the principal parameters are the angular aperture of the equatorial analyser/Soller slits and the angular apertures of the axial Soller slits. The presence of a monochromator in the beam path is normally accommodated by modifying the wavelength spectrum and/or by changing one or more of the axial divergence parameters. Flat analyser crystals have been incorporated into FPPF as a Lorentzian shaped angular acceptance function.θ range without refinement using the known properties of the diffractometer, such as the slit sizes and diffractometer radius, and the emission profile. Fine tuning is sometimes necessary to accommodate a monochromator or to compensate for the fact that certain aberrations are not completely independent [θ range. Such correlation occurs between the axial divergence parameters and absorption as both of these aberrations can produce similar forms of asymmetric profiles between 2θ = 50° and 100° in diverging beam diffractometers. Correlation is minimised by using data with a large 2θ range so that the unique angular dependence of individual aberrations becomes evident. When a set of instrument profiles cannot be fitted by FPPF, this is usually an indication that either the model used is invalid , the instrument is misaligned, there is an overlapping impurity line or, the specimen is generating crystallite size broadening or is inhomogeneously strained.One of the intrinsic benefits of the fundamental parameters approach is its adaptability to any laboratory diffractometer. Good fits can normally be obtained over the whole 2ependent . Under t6 SRM 660a (µpowder ≈ 500 cm−1) is used as a reference, compensation can be made for differences in the absorption of the sample and LaB6. In the latest version of the commercial software package (TOPAS)FPPF was designed originally as a tool for analysing diffraction line broadening. Fitting is done by convolution and corrections for instrument broadening and peak shift are intrinsic to the refinement. When an instrument is well characterised, line broadening can be analysed without a reference specimen. Moreover, when a reference standard is used, which has different properties from the specimen with line broadening, some compensation can be made for these differences. For example, when LaBIn this paper we will discuss the physical origin of the instrumental profile shapes for various laboratory diffractometer configurations including both divergent beam and parallel beam instruments. This will include a description of the geometrical aberrations as well as discussion on the nature of the wavelength distribution and the influence of monochromators on this distribution. Some discussion is also presented to demonstrate how the FPPF may be fitted to experimental data from a material with a low attenuation coefficient.One of the basic objectives of the FPPF technique is to be able to fit any powder diffraction profile using a physically based model to describe both the instrument profile and any diffraction broadening generated by the specimen. In principle, therefore, the technique should be adaptable to any powder diffractometer and fit profiles of widely differing shapes, such as those in θ ranges can be very time intensive and it is of central importance to minimise this time to enable Rietveld refinement to be completed in “seconds” without loss of accuracy within the profile function synthesis procedures. Various procedures have been implemented, some of which are described by Cheary and Coelho [Although most applications of FPPF have focussed on the conventional diffractometer it has also been utilised for analysing neutron diffraction data  and syncd Coelho ,3,4, butUp until the mid-1990s most of the laboratory powder diffractometers in use were divergent beam instruments with a narrow receiving slit, diffracted beam monochromator and a simple proportional/scintillation counter detector as shown in α1 and Kα2 components of the K spectrum in the graphite monochromator case and only the Kα1 with the Ge monochromator [The most widely used laboratory diffractometer in use today is still the divergent beam diffractometer with either a bent graphite monochromator in the diffracted beam see  or, a grhromator . Furtherthe finite width of the x-ray source,a divergent incident beam on to a flat specimen (flat specimen error),the finite width of the receiving slit,the beam penetration into the specimen (specimen transparency),the deviation of the beam from the equatorial plane .These aberrations all produce some degree of line broadening and, in the case of flat specimen error, specimen transparency and axial divergence, some asymmetry is also introduced. Zero 2θ and specimen surface displacement errors may also be present in a diffractometer, but these only affect the 2θ position of a profile and not its shape. In both configurations the monochromators not only determine the wavelength distribution, but they also act to reduce the axial divergence and it is often considered unnecessary to include Soller slits between the sample and the monochromator.The principal geometric aberrations contributing to profiles from the above diffractometers are,Divergent beam diffractometers used under symmetric conditions only measure diffraction from planes parallel to the specimen surface. To measure diffraction from planes angled relative to the specimen surface it is necessary to operate under asymmetric conditions as illustrated in ω about the diffractometer axis so that the angle of incidence on to the specimen varies between θ + δω and θ − δω. This oscillation moves the focus of the diffracted beam continuously back and forth in front of and behind the receiving slit,the sample is oscillated ±δthe receiving slit is replaced by a position sensitive detector (PSD). The only position on the detector that is normally in focus is its centre see ; all difDefocussing also occurs in diffractometer configurations where,There are two common forms of the parallel beam powder diffractometer which are illustrated in the angular acceptance function of the analyser foils or analysing crystal,deviation of the beam from the equatorial plane .In most laboratory diffractometers, the parallel beam is produced by using a parabolic graded multilayer mirror with the line x-ray source positioned at the focus of the mirror [Parallel beam diffractometers have emerged as one of the most popular forms of laboratory diffractometer over the past ten years and now constitute more than 30 % of the new diffractometer purchases. There are fewer geometric aberrations contributing to the profile and systematic errors arising from specimen displacement, specimen transparency and surface roughness are not significant. There are two geometric aberrations contributing to a parallel diffractometer,e mirror . Althougθ angles . At high 2θ angles (2θ > 100°), the profile conforms primarily to the shape of the wavelength distribution in the beam. With the exception of the aberrations associated with “receiving system” of the diffractometer and the x-ray source, all of the geometric instrument aberration profiles vary with 2θ. In the following sections the shapes of the major instrument aberrations used in FPPF analysis to describe the various laboratory diffractometer configurations are discussed for conditions that are typical of those encountered in practice. The aberration functions generated by mis-setting a diverging beam diffractometer or using it under asymmetric conditions are also discussed. Most of the results quoted here are for a diffractometer radius R = 215 mm.The geometric instrument aberrations tend to determine the shape of a diffractometer profile at low 2ϕ refers to the continuously variable angle measured on the diffractometer whereas 2θ or 2θB refers to the Bragg angle of the diffraction line. The angle ε refers to the difference between the measured angle 2ϕ and 2θ,The convention adopted here for describing the angular variables is that 2xθ as shown xw ≈ 0.04 mm and the aberration profile has a width Δ2xθ ≈ 0.01° and does not contribute significantly to the overall width of the instrument profile.The profile shape of this aberration is generally expressed as an impulse function of width Δ2xw ≈ 0.2 mm at 6° take-off so that the aberration profile width Δ2xθ ≈ 0.056°. At this level the source width makes a much bigger contribution to the overall width of the instrument profile and a more accurate form for the aberration profile shape is necessary. A good approximation under these circumstances is a Gaussian shape rather than an impulse function. In diffractometers with curved crystal incident beam monochromators the source width can also have a greater contribution because of the magnification effect introduced by the monochromator. This occurs with asymmetrically-cut Johansson incident beam monochromators where the source-crystal and crystal-focal point distances are typically ≈120 mm and ≈230 mm, respectively. A fine focus tube with a projected width of 0.04 mm is then effectively magnified to ≈0.08 mm. Under these conditions the effective source width can be trimmed down by reducing the width of the focal line slit.In broad focus tubes, the target width ≈2 mm and Z1 and Z2, and the intensity of the tail f relative to the intensity at the tube focus. In most instances the intensity of the tails is ≈0.1 % of the peak intensity and is only significant when analysing intense lines. The tails themselves are not necessarily symmetric with respect to the tube focus and can extend over a 2θ range up to 0.6°.For accurate line profile analysis it is also necessary to modify the simple impulse model even with long fine focus tubes. Bergmann  has showθ.In diverging beam diffractometers the receiving slit is placed at the focus of the diffracted beam and for perfect focussing should have an infinitely small width. Owing to the many aberrations present, focussing is never perfect and the count rate incident on the receiving slit tends to increase with increasing slit width, but at the expense of resolution. In parallel beam diffractometers the receiving system is based on using either the Hart-Parrish system of analyser slits  or, a flθr given by,wr is the width of the receiving slit. The angular width Δ2θr subtended by the receiving slit relative to the diffractometer axis is therefore normally between 0.013° (0.05 mm) and 0.08° (0.3 mm). When the slit size is larger than 0.15 mm, the receiving slit aberration is often the dominant aberration in a diffractometer over the angular range 2θ = 15° to 60°.Most commercial diffractometers have a selection of receiving slits ranging in width from 0.05 mm up to 0.3 mm although occasionally larger slit sizes up to 0.6 mm are used to measure integrated intensity rapidly. The aberration function for a perfectly aligned receiving slit is an impulse function of width Δ2θr is given by the angular aperture Δ of the slits.Analyser slits act as an angular filter in the diffracted beam. The aberration function or transmission function for these slits is a triangle function, as shown in θ aperture Δ ≈ 0.06°. In laboratory diffractometers the angular aperture Δ is typically ≈0.1°. A problem often encountered with analyser slits is specular x-ray reflection from the analyser foils [6, SRM 660a. An alternative approach to determining the aberration profile of an analyser slit system, without the effects of the wavelength profile distorting the result, is to simply carry out a 2θ scan across the incident beam. Provided the axial divergence of the incident beam is kept small, by including axial Soller slits, and the equatorial divergence is negligible then the incident beam scan will have exactly the same shape as the aberration profile of the analyser slits.In the original Parrish-Hart diffractometer on Station 2.3 at the Daresbury synchrotron, the analyser slits were 360 mm with a spacing of 0.2 mm between adjacent foils giving an angular 2er foils . Weak saer foils . The parθ scan of the analyser/detector using a very fine incident beam as shown in The inclusion of an analyser crystal in the diffracted beam of a parallel beam diffractometer gives high resolution diffraction patterns with a low background, but the intensity is invariably less than the Parrish-Hart configuration. The aberration profile introduced by the analyser crystal is generally very narrow and can be determined by measuring the rocking curve of the crystal. For a perfect analyser crystal the aberration profile will be determined by the Darwin profile of the analyser crystal. In practice, however, the aberration profile will be broadened by the mosaic structure of the crystal, any stresses in the crystal and any waviness or curvature of the crystal surface . As a coθ on an ideal polycrystalline specimen with a surface radius of curvature ρ. For diffraction from a particular hkl plane the common property of all the diffracted rays from the specimen is that they all deviate through the same angle 2θ. By simple geometry it can be shown that all the diffracted rays converge to a focus on a circle which has the same curvature as the specimen surface. The focus of the diffracted rays defines the position of the receiving slit. In commercial diffractometers the specimen is invariably flat and the diffracted beam no longer focusses perfectly. Good focussing characteristics, however, can be maintained with appropriately chosen slits to limit the equatorial divergence and a diffractometer radius R sufficiently large to reduce defocussing errors to an acceptable level without losing too much diffracted intensity.The basic optics of the focussing powder diffractometer set up for symmetric diffraction is illustrated in α centred on the diffractometer axis, the aberration profile JFS(2θ) is asymmetric and exists only for the region ε = 0. X rays diffracted from the centre of the specimen are detected at 2ϕ = 2θ where as x rays diffracted off-centre are detected at 2ϕ < 2θ as shown in For an incident beam, with an equatorial divergence ε = 2ϕ − 2θ and the distance q from the diffractometer axis at which the ray is diffracted is,α < 2°). Thus, when a specimen is illuminated over a length Q then the x rays diffracted from either extremity of the beam  defines the limiting value of εM of the aberration function JFS(ε) which exists within the range 0 ≥ ε ≥ εM whereα starting from the relation,θ > 10°, εM becomesJFS(ε) for flat specimen error is,α or a fixed illumination length Q. In the fixed α mode the aberration function is broad at low 2θ and εM has a cot θ dependence whereas in fixed Q mode the breadth rises from zero at 2θ = 0 up to a maximum at 2θ = 90°. The extent of the changes in the aberration function JSF(ε) for each mode of operation using typical operating values for α and Q are shown in The relationship between the difference α = 1°, the effects of flat specimen error in commercial diffractometers are discernable as an increase in both the asymmetry and breadth below 2θ ≈ 40° [α the beam size increases with decreasing 2θ and eventually the beam will cover the whole specimen. The angle 2θlim at which this occurs is given by,Lsp is the length of the specimen. Under these circumstances the value of εM is given by Q = Lsp. Below 2θlim the aberration function remains the same as the beam extends beyond the specimen.With a fixed angle of divergence 2θ ≈ 40° . Under cθ values and increase with increasing 2θ. With an illuminated length of 20 mm on the specimen, flat specimen error is clearly discernable at 2θ > 20°. It is not always possible to collect data from fixed Q mode diffractometer over a large 2θ range  for an infinitely thick specimen is given by,µR) sin2θ rad and µ is the linear attenuation coefficient. In low absorbing specimens which cannot be considered to be infinitely thick, the angular variable ε has a lower limit εmin so that in T is the specimen thickness. As a consequence the unit area normalising constant in (Jµ(ε) becomes,Specimen transparency produces asymmetry and broadening of the instrumental profile function. For perfect focussing all the diffraction should occur on the focussing circle, but when the beam penetrates the surface, diffraction will occur over a range of depths within the specimen. The aberration function (ε/δ)δε≤0where δ =µ < 50 cm−1. The contribution of specimen transparency is greatest at 2θ ≈ 90° and at this angle the aberration profile has a FWHM ≈ 0.03° 2θ when µ ≈ 50 cm−1, but this drops to ~0.005° 2θ when µ ≈ 200 cm−1. µ and 2θ. Specimen transparency effects are quite strong in polymeric materials with very low attenuation coefficients (µ ≈ 30 cm−1 or less). They can also show up in loosely bound powders of low atomic number materials  where the porosity ≈50 % or less so that a material with a µ = 100 cm−1 is reduced to a powder with a µ = 50 cm−1.The asymmetry and broadening from specimen transparency is greatest for low absorption materials and is clearly evident when the linear attenuation coefficient ω so that it is no longer at the symmetric condition ω = θ as in asymmetric diffraction or,mis-setting the incident beam angle R of the diffractometer.wrongly positioning the receiving slit so that the distance of the slit from the sample is larger or smaller than the nominal radius For both of these conditions the focus of the diffracted beam will be either in front of or behind the receiving slit as illustrated in Defocussing results in broadened diffraction lines and occurs when the receiving slit is not positioned at the focus of the diffracted beam. The most common causes of defocussing are,JDF(ε) for this condition is a impulse function of angular width ΔDF = D/R (rad) where D is the width of the defocussed beam at the receiving slit. Assuming the equatorial divergence α is small  the angular width ΔDF of the aberration profile is given byR2 and R are related by the equation,ΔDF of the aberration function is,ω − θ|, increases. Defocussing is larger at low 2θ angles and varies more rapidly with ω − θ at low 2θ values. Conversely, at high 2θ values, diffractometers will tolerate quite large errors in ω − θ  without the effects of defocussing being detectable in the line breadth. Also, by reducing the angle of divergence α the defocussing ΔDF can also be reduced, but at the expense of the diffracted intensity. The plot of ΔDF vs (ω − θ) for a range of 2θ angles from 30° to 150° is shown in The aberration profile ϕ other than the true diffraction angle 2θ. The only rays for which 2ϕ = 2θ will be those propagating parallel to the equatorial plane and incident on the diffractometer axis. In practice, axial divergence is most readily recognised by the asymmetry it introduces into low angle diffraction lines (2θ < 30°) where the low angle tails extend further than the high angles tails.In a laboratory diffractometer only a small fraction of the photons that form the incident beam emerge from an x-ray source parallel to the equatorial plane. Likewise most of the x rays that reach the detector slit after diffracting from the sample are angled to the equatorial plane. Under these circumstances, a diffractometer will record x-ray counts over a range of angles 2ϕ for a true diffraction angle 2θ depends on the axial divergence β and γ in the incident and diffracted rays  where the asymmetry is opposite to that at low 2θ. Although axial divergence is as strong in very high angle lines as it is in low angle lines, the positive asymmetry developed tends to be less noticeable as it is overshadowed by the dispersion of the emission profile.For a particular ray path the measured diffraction angle 2rays see . Assuminβ ≈ 0) as in many synchrotron systems or laboratory diffractometers with very narrow incident beam Soller slits, all the axial divergence arises from the diffracted beam and θ = 90°, ε ≤ 0 when 2θ < 90° and ε ≥ 0 when 2θ > 90°. The aberration function JAX(ε) for this condition can be derived analytically [When axial divergence in the incident beam is small  can be derived by determining the intensity profile when the receiving slit is scanned across these cones. The parameters of the diffractometer that define JAX(ε) in this instance are Ls, the axial length of the specimen bathed in x rays and the length Lr of the receiving slit and the function JAX(ε) is given by,θ = 10° and 50° are shown in R = 215 mm, an illuminated specimen length Ls = 12 mm and a receiving slit of length Lr = 16 mm.The emergent diffracted beam from each individual ray in the incident beam in JAX(ε) can no longer be done analytically. Eastabrook [JAX arising from incident rays all with the same axial divergence β. The instrument parameters required to define this function are the axial lengths Lx, Ls, and Lr of the x-ray source, the sample and the receiving slit,calculating the analytical aberration function β and γ in the incident and diffracted beams, respectively. The transmission functions, SI(β) and SD(γ), for the incident and diffracted beam Soller slits, respectively, are each triangle functions with 100 % transmission at β = 0 and γ = 0, and 0 % transmission at β = ± ΔI/2 and γ = ± ΔD/2 where ΔI and ΔD are the angular apertures of the slits (see Δ),incorporating the Soller slits into the calculation as angular intensity filters on the axial divergence JAX(ε) by integrating the aberration functions determined at each β across all the allowed β values,JAX functions. Considerable savings in computing time without loss of accuracy can be achieved by using a smoothing procedure in which the function JAX(ε) is convoluted in ε space with an impulse of width = step size of the data being fitted. The saving in computing time comes from a reduction in the number of summation points at discrete values of β required to accurately reproduce calculate the full aberration profile When allowance is made for axial divergence in both the incident and diffracted beams and for the presence of Soller slits in each of these beams, the calculation of the aberration function stabrook  first destabrook  generalistabrook ,4 have dLx, Ls, and Lr and given by,Δ between 2° and 5°, and in such cases the breadth of the aberration profile is reduced quite dramatically. This is illustrated in ΔI and ΔD ≈ 2°), the dimensions Lx, Lr, and Ls tend to be redundant parameters in the calculation of JAX(ε) as the Soller slits control the maximum axial divergence in the incident and diffracted beams.In the absence of Soller slits the maximum axial divergence in the incident and diffracted beams, θ ≈ 50°. At very high angles, 2θ ≥ 150°, the asymmetry reverses and the extension of the high angle tails increases with increasing 2θ. Broadening from axial divergence is evident at all 2θ values but passes through a minimum in the region 2θ ≈ 110°. However, the shift in peak angle 2θmax at Imax relative to the true 2θB is small at all 2θ values unlike the shift in centre of gravity 2θcg − 2θB which varies quite considerably from very large negative values, near 2θ = 0°, to very large positive values as 2θ approaches180° [The major change brought on by axial divergence is extension of the low angle tails of profiles below 2ches180° .in specimens with strong preferred orientation, such as thin films and rolled or extruded metals, the diffraction cones are no longer of uniform intensity along the arcs of the diffraction cones.α radiation the optical path length in the diffracted beam is increased by ≈100 mm giving an effective radius of the detector arm of 100 + R mm. An example of the extent to which axial divergence is reduced by introducing a graphite diffracted beam monochromator is illustrated in the inclusion of a monochromator in the beam path reduces the axial divergence. When either diffracted beam or an incident beam monochromator is present the optical path length of the beam is extended. With a diffracted beam monochromator the optical path length of the diffracted beam is extended by the optical path between the receiving slit and the detector slit. In a diffractometer with a graphite monochromator tuned to Cu KIn practice it is not always possible to calculate the exact form of the axial divergence aberration function for a particular specimen/diffractometer configuration. The two main reasons for this are,θ −2θ mode with its centre maintained at the focus condition. As the detector is scanned the total diffraction pattern is formed by adding together and averaging the patterns recorded at each step. In this mode a diffraction pattern can be accumulated very rapidly with excellent counting statistics because individual lines are within the detector window for a considerable time. For example a LPSD with an angular window of 10° moving at 5° per min corresponds to a diffraction peak being detected for 120 s. Consequently, in this mode, a 100° 2θ diffraction pattern may only take approximately 20 min to collect. To obtain the same level of counting precision in a conventional single slit diffractometer with 0.02° 2θ steps would take approximately one week. In practice, this gain in time is never fully realised because of the longer dead times of LPSDs and the damage that can be caused on the anode wire by very high localised count rates. Also the diffraction peaks recorded at off-centre positions along the detector window are broadened and asymmetric. Some manufacturers make the angular window a software adjustable parameter to maintain the resolution although this means that only a fraction of the detector is being used.LPS detectors with angular windows up to 10° (at 200 mm) are used in reflection mode on commercial diffractometers to increase the data collection rate particularly for kinetic based studies. In most cases LPS detectors are used in stationary mode and only a fixed angular region of a pattern is recorded. A number of manufacturers offer LPSD systems where the detector can be scanned in flat specimen error including defocussing,parallax error,thermal noise.A full treatment of these aberrations and how they are modified by a scanning LPSD system is given in Cheary and Coelho [In LPSD systems, the receiving slit aberration is no longer relevant and the flat specimen aberration is replaced by an aberration function that embodies three effects which are folded together in the final function;d Coelho . For a sJPSD(ε) depends on the 2θ value of a peak and on the offset angle β of the recorded peak from the centre of the LPSD. The angular variables used to define the LPSD system are given in ζ is limited by the angle of divergence α of the incident beam .In a conventional diffractometer defocussing and flat specimen error can be convoluted together independently, but this is no longer valid in a LPSD system. The effect of both the specimen and the detector being flat is that the defocussing is no longer symmetric about the centre of the LPSD. In a stationary LPSD the aberration function ε and ζ, the difference ε = 2ϕ − 2θ is related parabolically to ζ,JPSD(ε) is obtained by transforming the intensity across the incident beam IB(ζ) from ζ space into ε space, ie.,ζ) is reasonably constant then the un-normalised form of JPSD(ε) = |dζ/dε|. This is readily calculated by differentiating the transformation ε ≤ ε0. When calculating this function allowance has to be made for the fact that ε value and JPSD(ε) possesses a discontinuity at the boundary between two rays and one ray contributing to the aberration function. This is illustrated in θ . When ε0 is outside the range of ε values dictated by the limiting values of ζ in the incident beam  then there are no discontinuities within JPSD(ε), no infinities and JPSD(ε) is finite at all ε as illustrated in β = −2°. It should be noted that the aberration functions for a particular 2θ are not symmetric with respect to β. In β = +2° and β = −2° at 2θ = 20° are distinctly different and have different limits in ε.When the diffractometer is operated symmetrically and the incident beam is centred on the diffractometer axis then assuming small angles for β from the centre of the LPSD is a unit area impulse function with a width ΔPX (°2θ) given by,D is the depth of the detector and R is the radius of the diffractometer. Parallax error can be reduced considerably by increasing the detector gas pressure. In this way the x rays are absorbed by a thin layer of gas beneath the detector window, in which case the diffractometer radius R is defined by the front window of the detector rather than its anode wire.When a diffracted x-ray photon enters a LPSD its path in the detector gas is not perpendicular to the anode wire, except at the centre of the detector, and additional broadening, known as parallax broadening, is introduced into off-centre diffraction lines. This arises because the ionisation caused by incoming photons is likely to occur at any point along its path, but the subsequent avalanche is perpendicular to the anode wire see . For a dµgas > 0. The shape of the aberration profiles for parallax broadening µJPSD(ε) at an angle β to the centre of the LPSD is given by,µg is the linear attenuation coefficient of the detector gas [θ ≈ 40°. For example, when β = ±5° and the detector has a depth D ≈ 8 mm, the breadth of the parallax function Δp can be as large as 0.2°.Most detectors are designed with a quantum efficiency of at least 80 % and have depths somewhere between 5 mm and 10 mm. The aberration profiles for these conditions resemble those shown in ctor gas . ParallaJNPSD(ε) is modelled as a Gaussian function with a FWHM σ,σ is equivalent to a positional uncertainty along the anode wire between Δx ≈ 0.04 mm and 0.20 mm where σ = Δx/R radians. This effect contributes to the broadening much like that from the typical receiving slit width in a conventional diffractometer. Smaller Δx values down to 0.040 mm are possible when the gas in the LPSD is under pressure. Most LPSDs have a positional resolution Δx around 0.1 mm although those that utilise delay lines can be as large 0.2 mm.The ultimate resolution of a LPSD is the spatial uncertainty of the position measurement of an individual x-ray photon incident normal to the anode wire of the detector. This is controlled by the spatial broadening of the discharge produced by incoming photons, by the thermal noise generated in the anode wire and by the accuracy of the position-encoding electronics . In mostW(E) of a single characteristic x-ray emission line well above the threshold energy is Lorentzian and given byE0 is the peak emission energy and Γ is the lifetime broadening given by the sum of the widths of the two relevant atomic levels involved in the transition.The natural shape of the energy distribution θ space it remains Lorentzian, provided dE/d2θ is relatively constant across the profile, with a FWHM Γ2θ in 2θ space, given byΓ and E0 are in eV. The form of the Kα1α2 emission profile from Cu is shown in 1 line nor the Kα2 line has a Lorentzian shape and both the Kα1 and Kα2 lines are asymmetric with extended high angle tails. Moreover, the asymmetries and FWHM values of the Kα1 and Kα2 peaks are different. This is particularly evident with the first series of transition element and is related to the anomaly in the atomic number dependence of the atomic level widths of the LII and LIII levels obtained from x-ray photoelectron spectroscopy data and x-ray emission spectroscopy [Γ, asymmetry indices κ and energies E0 of the four transition elements commonly used for x-ray targets are given in When a Lorentzian energy spectrum is transformed into 2troscopy . The linΓ2θ of the Kα1 line from Cu and other commonly used transition metal anodes is shown in θ = 40° the FWHM Γ2θ is less than or equal to 0.01° (2θ) is considerably smaller than the FWHM values of actual diffraction lines from commercial diffractometers which are typically in the range 0.07° to 0.10° 2θ [θ > 60°, however, the emission profile tends to dominate over the geometrical aberrations. Once 2θ > 100°, the total breadth of the geometrical aberrations is relatively minor and the profile shape conforms closely to the emission profile.In FPPF an accurate model of the emission profile is essential particularly for the analysis of high angle profiles. The FWHM 0.10° 2θ  dependins → 2p, it has been recognised that 3d spectator transitions also contribute up to 30 % of the Kα1α2 emissions [s → 2p, but the second vacancy in the 3d level is not directly involved in the transition. The notation for this transition is 1s3d → 2p3d. A phenomenological representation for accurately describing the asymmetric Cu emission profile was first used by Berger [1 and Kα2 lines were each represented as two Lorentzian profiles as shown in R factor of 1 %, although in some cases it is necessary to use up to seven Lorentzians.The natural asymmetry of the emission lines arises from the multiplet structure of the transitions. In addition to the main transitions involving the change in vacancy state 1missions . In thesy Berger . In this1 emission line in the case of Cu rising uniformly with decreasing atomic number up to ≈1.4 % for Cr [1β3 spectra and appear on both the low energy and high energy tails as the Kβ′ and Kβ″ lines [3α4 non-diagram lines arise from the transition 1s2p → 2p2 in which the actual vacancy transition is 1s → 2p, but in the presence of a 2p spectator hole [Another feature of Kα emission lines which needs to be included in an accurate line profile fitting model is the satellite multiplet structure in the high energy tail as shown in % for Cr . Satelliβ″ lines . The Kα3tor hole ,38. For tor hole ,38. In mW(λ) from wavelength space to 2θ space at 2θ < 130° the Lorentzian shape in λ maps directly into a Lorentzian in 2θ with minimal error. At higher angles the effects of dispersion between λ and 2θ become increasingly evident [θ space for a sum of Lorentzians in λ space isI0i and iΓλ are the relative areas and wavelength FWHM of the ith Lorentzian. The d spacing is defined in relation to the reference wavelength λref of the emission line which in the TOPAS implementation of this procedure is the wavelength of the highest intensity Lorentzian. In the case of the Cu Kα spectrum given in 1a component (λ = 1.540591 Å) so that λref = λKα1a = 2d sinθKα1a. At very high 2θ values  the cosθ terms arising from d2θ/dλ can change quite significantly over a profile and elevate the intensity in the high angle tail. In addition to dispersion, the distortional effects of the Lorentz factor on high profiles should also be incorporated when analysing profile shapes at very high angles [In the transformation of the distribution  evident  and the h angles ,40,41.3), but this is generally a small effect and is not expected to affect the profile shape significantly [IKα2/IKα1 ratio by up to ±10 % depending on the alignment and setting of the crystal. For example, in the monochromated spectrum shown in 1/Kα2 is approximately 0.46 rather than 0.50 as in the unfiltered spectrum.As the natural shape of an emission profile is Lorentzian, the tails can extend a considerable distance from the central peak as shown in ficantly . The inc2 component and 100 % Kα satellites from the CuKα spectrum and, almost completely eradicate the Lorentzian tails of the emission profile.Commercial pyrolytic graphite monochromators are strongly oriented polycrystals bent by a hot pressing operation. The focussing is imperfect and the resolution is relatively poor. Better resolution and, in some cases, high intensities can be obtained from “ground and bent” (Johannson) single crystals. These crystals achieve perfect focussing when correctly aligned and are able to select a very narrow wavelength band. The most common materials used for Johannson monochromators on conventional diffractometers are quartz, germanium and silicon. The example shown in λ space to fit the spectrum entering the detector. In broad terms, monochromators reduce the width of the wavelength distribution and tend to truncate the tails of spectra. A number of approaches can be used to accommodate these changes,represent the components of the wavelength distribution as Voigt or pseudo-Voigt functions rather than Lorentzians, to limit the extension of the profile tails, and modify the relative intensities of the component.T(λ) represented by a simple function, such as a split pseudo-Voigt or split Pearson VII function with up to four refineable parameters, which operates on the tabulated Lorentzian emission profile data. The split functions are used to incorporate asymmetry in the T(λ).represent the effect of the monochromator as a wavelength filter with a transmission function In both cases the parameters of each representation are obtained by analysing and fitting the high angle profiles from either a reference line profile standard, such as LaB6 SRM 660a, or single crystal disc such as a 111 wafer of silicon.In the presence of monochromators, even low resolution graphite monochromators, it is no longer possible to accurately model the wavelength distribution from an x-ray tube using the tabulated unfiltered spectra such as the one in 1 and Kα2 components of the spectrum reflect off the mirror in slightly different directions as shown in 1 and Kα2 peak maxima, Δ2θKα21 = 2θ (Kα2) − 2θ (Kα1), in a line profile is then either larger or smaller than the same profile from a diffractometer with no mirror. When a diffractometer is set up with its mirror in the orientation shown in Δ2θKα21 is smaller than the value expected from the known Kα1 and Kα2 wavelengths by an amount corresponding to the difference Δψ in the directions of the two incident beams on to the specimen. According to Toraya and Hibina [δψ for a conventional long fine focus x-ray with a projected target width of 0.04 mm can rise to 0.0017° for a high resolution mirror, but decreases to a negligible level for low resolution mirrors. In any high accuracy lattice parameter determination using this type of diffractometer it is necessary to incorporate the wavelength dependent zero error into the analysis.The inclusion of a parabolic multilayer mirror in the incident beam of a diffractometer can also introduce a distortion into the wavelength spectrum . This had Hibina  the diffθ values as the measured data points and then a continuous function is formed by interpolating between the calculated points. As the calculated functions are then each a series of linear sections it is possible to calculate the convolution integral analytically [In practice FPPF requires accurate numerical procedures for carrying out multiple convolution integrals. This can be done by representing the calculated profiles as a histogram and reducing the convolution integral to a summation. To avoid systematic errors with this approach the angular step size between calculated intensities needs to be very small but in so doing the operation becomes very time consuming . In the ytically . Some diytically .ΔI and ΔD, of the incident beam and diffracted beam Soller slits are refined independently to a set of profiles, various combinations of refined values ΔI and ΔD can be obtained that give equally good fits to the data. The reason for this is that these two parameters are very strongly correlated and the values can in fact be interchanged without changing either the shape of a synthesised profile significantly or the quality of the fit. As a consequence the best fit with an unconstrained refinement often corresponds to refined parameters that do not make physical sense. When first using the FPPF approach it is advisable to investigate the validity of the refined parameters for a particular diffractometer using a well crystallised reference specimen and establish the parameters of the instrument that allow profiles over the whole 2θ range to be defined with one set of values. It would be unrealistic to expect the refined parameters for a diffractometer to match the directly measured values exactly as there are a number of second order effects in diffractometer profiles that are not incorporated in the fitting model. Moreover, not all of the instrumental aberrations are independent and as such the convolution model is not strictly valid with certain combinations of aberrations [Although FPPF is a powerful method of profile analysis it can fail when used without recognising its limitations. The physical parameters describing a diffractometer are not statistically independent within the least squares refinement procedure and strong correlations exist between many of the refineable parameters. For example, when the angular acceptance angles, rrations . Neverthµ. Absorption effects in a line profile are only evident when µ < 100 cm−1 and under these circumstances there is some justification in making µ a refineable parameter. In specimens with µ ≥ 200 cm−1 refinement of µ normally has little effect on the profile shape because the specimen transparency profile is relatively narrow at all 2θ angles . Consequently, µ is fixed at a representative value and not refined.Once the refined instrument parameters have been established , and these are in reasonable agreement with the actual values, the only instrument parameter that may need refinement from specimen to specimen is the linear attenuation coefficient Tapp and the percent microstrain εrms are frequently included as refineable parameters although their contribution is generally small. Various profile functions can be adopted for crystallite size and microstrain and commonly take the form of a Lorentzian function or a Gaussian function. The most common unit area functions used for crystallite size and microstrain, Bcryst(2θ) and Bµ(2θ), respectively, areHcryst = 180 λ/(πTapp cosθo) °2θ and the FWHM of the Gaussian = θ0 °2θ. In some crushed powders it has been necessary to use a Lorentzian function to represent the microstrain but with a FWHM = ηtanθ where η is a constant related to the microstrain [Even in well crystallised powders with crystallite sizes up to 2 µm crystallite size broadening is detectable in high angle lines. Conversely, in many powders it is not uncommon to have crystallite sizes down to 0.5 µm and crystallite size broadening is evident even in low angle lines. Also, in many crushed powders the action of crushing even in a standard pestle and mortar can induce microstrain broadening. In TOPAS and XFIT therefore, the apparent crystallite size rostrain .θ = 36° to 150° using CuKα radiation. The specimen used in this instance was a 20 mm disc prepared by sintering at 1400 °C for 24 h in air. When a small segment of the disc was examined in a scanning electron microscope the vast majority of crystallites were between 1 µm and 2 µm in diameter. µ, the receiving slit width wr, the angular aperture of the incident beam Soller slits ΔI and the apparent crystallite size Tapp. All the other instrument parameters were fixed at their actual values. Some parameters, such as the equatorial divergence α, the width of the x-ray source and the sample width Ls, have very little effect on the profile shapes and a wide range of values can be accommodated for each of these parameters without affecting the quality of the fit. Although the receiving slit length Lr and x-ray source length Lx can have a significant effect on the profile shape, they were not refined because of their strong correlation with the Soller slit aperture ΔI and the fact that the improvement in fit with their inclusion was minimal. Equally good fits are obtained by fixing Lx and ΔI and allowing Lr to refine. The equatorial divergence angle α and other axial divergence parameters are normally included in a refinement when low angle diffraction lines, preferably below 2θ = 25°, exist in the data set as this region of the diffraction pattern that is sensitive to equatorial and axial divergence. The inclusion of crystallite size broadening in most refinements is very important. In the present MgO refinement its removal from the refinement resulted in the other refined parameters taking on physically unrealistic values and a rise in the Rwp value from 3.5 % to 6.2 %. It was also clearly evident that the calculated profiles no longer fitted the tails of the observed profiles.The quality of the fits obtainable from the FPPF approach and the degree to which the refined instrument parameters agree with the actual values is illustrated below for data collected from a polycrystalline MgO reference specimen over the range 2θ range and slits that will maintain sufficient resolution to monitor line positions in stress analysis. As the profile shape is known, the FPPF technique also provides greater certainty in the identification of weak peaks or impurity lines embedded in the tails of stronger line. In Rietveld analysis or quantitative analysis, FPPF allows the profile shapes across the whole 2θ range to be fitted without any instrument based parameters in the refinement. The focus of the refinement is therefore on the diffraction effects of the specimen and not the instrument. Moreover, line broadening analysis is an integral feature of FPPF for both individual lines or as part of a Rietveld analysis and, correction for instrument broadening is an intrinsic part of the analysis making reference specimens unnecessary in many circumstances. FPPF has its weaknesses. Up until now it has only had limited success as a method of accurate lattice parameter determination and in some diffractometer set-ups the FPPF instrument parameters can differ significantly from the actual values. Although it corrects the 2θ positions of lines for instrument line shift, including zero shift and specimen displacement, it has not been possible to obtain a set of lattice parameters for a particular specimen which are the same, within an uncertainty < ±0.0005 Å, for every hkl line in the pattern [Fundamental parameters profile fitting offers a number of benefits as a method of profile analysis. It is based on a physical model of the diffractometer and its refined parameters should be self consistent with physical dimensions of the diffractometer and the physical properties of the sample. On this basis it can therefore identify whether or not a diffractometer is operating at optimum resolution for the conditions used and provide a means for assessing the performance of a diffractometer in a particular application. For example, a knowledge of the performance of a diffractometer operating under asymmetric conditions can be important in choosing the best 2 pattern . In addi pattern  suggeste


These organisms cause diarrhea, hemorrhagic colitis and the hemolytic uremic syndrome. STEC make two types of Stxs, Stx1 and/or Stx2. Stx2 has one prototype (a) and six subtypes (b–g), but only STEC that make Stx2a, and/or Stx2c, or Stx2d are associated with severe disease. However, Stx2c is about 10-fold less toxic than Stx2d in vivo despite only two amino acid differences in the A subunit at positions 291 and 297. We made mutations at these two sites to create intermediate toxins between Stx2c and Stx2d, and determined the 50% cytotoxic dose on Vero cells before and after heat treatment, and the 50% lethal dose in mice of the toxins. We found that serine 291 was associated with increased toxicity in vivo and that either amino acid change from that in Stx2c to that in Stx2d increased heat stability. We also assessed the secondary structure of Stx2c and Stx2d by circular dichroism (CD) spectroscopy. The CD studies suggest that Stx2c has a less-ordered secondary structure than Stx2d. We conclude that both amino acids at positions 291 and 297 in Stx2c contribute to its decreased stability and in vivo toxicity compared to Stx2d.Shiga toxin (Stx) is an AB Escherichia coli (STEC) have the capacity to produce at least one of the two types of Stxs, Stx type 1 (Stx1) or Stx type 2 (Stx2). Stx1 and Stx2 are highly homologous and have a similar mode of action , was calculated as [θ] = (θ × MR)/(10 × ichroWeb .