Diversity as well as temporal and spatial changes of the proportional abundances of different antibiotics  have been hypothesised to be an effective administrative control strategy in hospitals to reduce the prevalence of antibiotic-resistant pathogens in nosocomial or community-acquired infections. However, a rigorous assessment of the efficacy of these control strategies is lacking. The main purpose here is to present a mathematical framework for the assessment of control stategies from a processual stance. To this end, we adopt diverse measures of heterogeneity and diversity of proportional abundances based on the concept of entropy from other fields and adapt them to the needs in assessing the impact of variations in antibiotic consumption on antibiotic resistance. Thereby, we derive a family of diversity measures whose members exhibit different degrees of complexity. Most important, we extent these measures such that they account for the assessment of temporal changes in heterogeneity including otherwise undetected diversity-invariant permutations of antibiotics consumption and prevalence of resistant pathogens. We apply a correlation analysis for the assessment of associations between changes of heterogeneities on the antibiotics and on the pathogen side. As a showcase, which serves as a proof-of-principle, we apply the derived methods to records of antibiotic consumption and prevalence of antibiotic-resistant germs from University Hospital Dresden (cf. supplement “DiebnerEtAl_Data-Supplement”). Besides the quantification of heterogeneities of antibiotics consumption and antibiotic resistance, we show that a reduction of prevalence of antibiotic-resistant germs correlates with a temporal change of similarity with respect to the first observation of antibiotics consumption, although heterogeneity remains approximately constant. Although an interventional study is pending, our mathematical framework turns out to be a viable concept for the assessment and optimisation of control strategies intended to reduce antibiotic resistance. The drastic increase of antimicrobial resistance worldwide resulting in an alarming increase in morbidity and mortality from clinical infections urges scientists and clinicians to develop counter-strategies . DesigniSince a quick replacement of existing antibiotics is not feasible, antibiotic strategies as “antibiotic mixing”, “antibiotic cycling”, “antibiotic switching”, and “rotation protocols” gained evermore attention in the recent decades –6 . Moreover, some measures of diversity are functions of proportions of “species” within an “ecosystem”, which is why we make use of proportions of consumption. If DDDi(t) denotes the consumption of antibiotics within the antibiotic class i ∈ {1, …, n} at time t, the proportion is given byi may also refer to the active agent.The consumption of an antibiotic is measured in standardised units according to the ATC/WHO definition of defined daily doses, SD(DDD(t)) denotes the corresponding standard deviation, can be used as a rough estimate of “homogeneity” in a properly defined sense. Unfortunately, analogous to the notion of “dispersion,” “homogeneity” is an ambiguous term which deserves clarification. In ecological analyses, the concept of “maximisation of statistical heterogeneity” refers to an approach by means of an entropy or a related diversity measure as discussed in the following section  dispersion. In contrast, physicists prefer to speak of a perfect dispersion, aka a perfect mixture, in such a situation. Thus, an ecosystem or a society close to a monoculture is “homogeneous” whereas a multi-cultural society/ecosystem is called “heterogeneous” which is used synonymously to “diversity.”The coefficient of variation,ion (cf. ). In thiion  heterogeneity, it has several inadequacies including the lack of uniquely capturing temporal changes. In the sequel, we harness methods known in ecological population modelling and other fields of research for an adequate quantification and assessment of antibiotic mixing behaviour and strategies as well as temporal patterns of prevalence of antibiotic resistance.ai and bi with i = 1, …, n be the proportions of species of two n-species populations composed of the same set of species with ai = bi, ∀i = 1, …, n, then SI = 1, i.e., the populations are identical in terms of their species distributions. If, on the contrary, the populations consist of disjoint sets of species, then SI = 0. Similarity index SI scales between 0 and 1. If we now fix say the first population to SI . In terms of richness, a heterogeneous Da is based on an “effective number of species”  giv givDa isRa is a monotonous function of Da, thus, the two measures can be used interchangeably without loss of information since diversity has only a relative meaning, anyway. The same holds for GS and other possible monotonous functions of Da. Entropy Ra, thus Da, independently of a reach their maximum for the fully heterogeneous situation Da = n. From the latter result we conclude that richness might be a sufficient diversity measure for populations close to full heterogeneity. Having said that, heterogeneity HI itself, although it cannot be derived as a special case of a Renyi entropy, shares features of an entropy and is thus a legitimate measure of diversity.The Shannon entropy is a special case of a Renyi entropy defined bySI insofar as it can be interpreted as kind of a self-similarity. Once more, full heterogeneity  G = 1, and maximally unequal proportions  implies G = 0.Finally, the frequently used Gini coefficient deserves to be mentioned:ants see ) is an iHI, as well as measures of diversity, Da, GS, and G, are invariant under permutations of indices. In other words, if the proportions of two species are exchanged, diversity (heterogeneity) does not change. However, similarity index SI is capable to account for such a change after some adaptations, as shown in the Results section. In order to assess both  cycling as well as  mixing behaviour of antibiotic administration or consumption, respectively, the proper similarity index extends tom wards or groups of patients between which antibiotic mixing takes place. In this case, aij and bij refer to the relative abundances before and after the swap of antibiotics administrations between the wards, respectively. In practice, i.e. in the absence of a properly defined study protocol, an allocation of both antibiotic consumption as well as the prevalence of resistant pathogens to precisely defined wards or groups of patients is hampered by the reality of adjusted clinical cycling/mixing. In the following, due to lack of appropriate information on mixing strategies, we present definitions of similarity tailored to our needs without taking strict mixing into account.For what follows, it is important to bring to mind that both heterogeneity, Main purpose of this article is to apply a diversity analysis to records of both clinical consumption of antibiotics as well as prevalence of pathogens exhibiting antimicrobial resistance. We refer to the previous section for a detailed introduction of the applied diversity measures. It remains to mention that the expected impact of diversity of antimicrobial consumption on the prevalence of resistant pathogens is analysed by means of Pearson’s correlations of the corresponding time series.Specifically, slopes along with their significance of differing from zero taken from linear regression quantify the temporal changes of both diversity as well as differential diversity. Of particular interest is the comparison between the time courses of differential diversities of antibiotic consumption and prevalence of resistant pathogens. Such a comparison can be achieved by testing of whether the two corresponding slopes differ significantly or not, or, equivalently, by calculating Pearson’s correlation coefficient along with the corresponding significance test. In the same line, the time series of the relative abundance of resistant pathogens is tested for correlations with the time course of the differential diversity of antibiotic consumption using, as before, Pearson’s product moment correlation analysis.Of note, each of the three time series, i.e. differential diversity of consumption, differential diversity of prevalence, and relative abundance of resistant pathogens, are expected to exhibit autocorrelations. This is a common feature of time series analyses. Nevertheless, Pearson’s correlation is commonly used as a standard technique to quantify the co-variation of two correlated time series and we here regard it to be sufficient for raising hypotheses based on the available preliminary dataset. A more appropriate application of a cross-correlation function with one of two time series being subject to a time lag is not applicable in our case due to the insufficient lengths (and low sampling frequencies) of the time series.Numerical calculations, statistics, and graphics have been performed with R .Before presenting the results of applying the derived diversity measures to observational clinical data, we reemphasise that this work mainly aims in presenting the mathematical framework. The following application to real data has an illustrative purpose and outlines future applications to adequately collected data from a controlled trial. Therefore, in terms of clinically relevant results, the following explanations remain provisional.DDDi(t). The consumptions of 9 classes largely remain constant on a moderate level. One class, that is to say “second-generation cephalosporins”, is characterised by a high consumption at the outset but declines approximately monotonously by more than half towards the end of the observation period. The consumptions of two other classes, in contrast, i.e. “aminopenicillin/beta-lactamase inhibitors” and “narrow-spectrum penicillins” increase approximately monotonously and roughly compensate for the aforementioned decline.DDD are hardly distinguishable from calculations based on the corresponding densities, 100 × DDDi/patient days. Henceforth, due to these minor differences, we skip to report our results with respect to consumption densities since we here primarily deal with an introduction of a methodological concept. The distinction between DDD and the corresponding densities might become important in other contexts, though. The proportions, dddi(t), however, depicted in Calculations based on Mean(DDD(t)), averaged over the 12 observed antibiotic classes (cf. sses cf. . The corV(t) slightly increases again during the final quarters. These results are consistent with the visual impressions from V(t) = 0, means perfect heterogeneity, therefore, V(t) can be interpreted as an inverse measure of heterogeneity.In the present case, variability thus homogeneity in the ecological sense is rather high during the first 4 to 6 quarters compared with the remaining time course. After an approximately monotonous decline until 2016, In the same line, the descriptive analysis can be applied to consumption with respect to active agents. Next step is to account for the Hospital’s functional units. The three panels of pi of antibiotic consumption with respect to antibiotic classes, thus i = 1, …, 12 refers to the 12 antibiotic classes. Ra, for 7 different values of weight parameter a (cf. figure legend), b) the corresponding diversities, a, c) the Gini-Simpson diversity, GS, and d) the Gini coefficient, G. Specifically, a = 0 yields D0 = n = 12  which sHI, defined in HI. Likewise, a mixing strategy, i.e. switching the consumption of two antibiotics between two units would remain unnoticed unless the diversity of the two units are not separately calculated  is more or less constant and is approximately one minus the slope of SI0(t) (SIΔ(t) measures the intensity of the change as a function of time. Thus, we now have a sound basis for the evaluation of changing abundances and possibly related switching strategies.Compared to the Renyi entropy, heterogeneity ated cf. . Since h to 0.69 . However to 0.69 . Due to f SI0(t) . Obviouspi) to information. In this context, the similarity index between two distributions given by ai and bi corresponds to the Kullback-Leibler divergence Reviewers' comments:Reviewer's Responses to QuestionsComments to the Author1. Is the manuscript technically sound, and do the data support the conclusions?The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: PartlyReviewer #2: No**********2. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: YesReviewer #2: No**********3. Have the authors made all data underlying the findings in their manuscript fully available?PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.The Reviewer #1: YesReviewer #2: Yes**********4. Is the manuscript presented in an intelligible fashion and written in standard English?PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.Reviewer #1: YesReviewer #2: Yes**********5. Review Comments to the AuthorPlease use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. Reviewer #1: Authors perform a rigorous mathematical analysis on the observational data, use multiple measures to asses the diversity and the heterogeneity of antibiotic consumption, and connect their output to the resistance prevalence observed in the dataset.Although I think the approach is very valuable, I think there are major points that need to be modified before the publication of this manuscript. My comments are below.Major Points1) I don't understand how measures of diversity or heterogeneity are connected to mixing  or cycling  strategies here. Mixing refers to giving different antibiotics to the different members of the cohort, but that can also end up having a constant DDD over time . Similarly, cycling - the temporal approach - should have way more variation over time, such as excluding certain antibiotics completely from the hospital or the cohort. So I cannot relate the antibiotic consumption observed in this dataset with the common strategies used.2) The authors' most important conclusion is only expressed in one sentence, and not fully clear (lines 362-364). Do they mean almost a no slope by saying that " means of parallel slopes "? Is this really a good way to back their argument up? Given that this is the message of the paper, I really expect more elaboration on this point.3) In general, I understand why authors use so many different measures to asses diversity or heterogeneity, and while reading the manuscript, it was obvious that this is the way they explored the whole research question. But is it necessary to include all these measures in the main text? There is a lengthy section about the explanation and use of so many different measures, but the conclusion and the connection to resistance prevalence is relatively really short, which is supposed to be the main point. I think the manuscript needs a bit of structural revision in this regard.Minor Points1) Figures 4-10 : please also use dots for the data points, like you did for figures 1-3.2) line 14 : these strategies are not really "recent" anymore.3) lines 19-21 : please refer to the key publications that includes the "evidence" you have mentioned.Reviewer #2: Diebner and collaborators present data about antibiotic consumption and pathogen resistance in three administrative units of a hospital over several years. They aim at using these data to test hypotheses related to the efficiency of heterogeneous antibiotic treatments strategies to mitigate the emergence of resistance in hospitals. For this they suggest the use of various measures of heterogeneity  in the context of antibiotic treatment and resistance.I am not convinced of the interest of this framework for the analysis of heterogeneous antibiotic treatment strategies, for 3 reasons:- The measures that go beyond the coefficient of variation are interesting when strictly more than 2 antibiotics are in use. In the "antibiotic mixing" literature, to my knowledge the vast majority of the literature considers the case of 2 antibiotics .- These measures are used in ecology to deal with very heterogeneous data, eg when abundances are very different between different species. This is useful (in ecology) because a species may be several orders of magnitude less abundant than another but still be essential for ecosystem function. But in the case of antibiotic treatments, the point of DDD is that different antibiotics can be compared between them. If the DDD of 2 antibiotics differ by orders of magnitude, it means that the one with the low DDD is virtually absent and has no chance of contributing to the establishment of resistance (or to infection clearance).- In much of the literature (including the one cited by the authors) on heterogeneous antibiotic treatments, a big question is "mixing vs cycling", but I do not see how the measures suggested by the authors can be useful to study cycling. The authors simply do not ever mention cycling after the introduction. It gives me the impression that the papers they cite do not really match the question they want to study.Most importantly, I am also not convinced that the presented data support the claimed result , for several reasons:- Only time series are analysed. In time series, the different time points are not independent from each other, and it is thus not appropriate to decide whether the slope is significantly different from 0 using a linear regression or a correlation test .- Regarding panels 1 and 3 of fig 13 (HI and SIdelta), both the curves for heterogeneity / similarity of antibotic use and pathogen resistances are flat. The authors conclude that the curves for treatment and resistance are similar and thus that there methodology permits to establish that heterogeneous treatments cause heterogeneous resistance. There is absolutely no statistical support (nor proper testing) for this finding. And a more parsimonious explanation for the fact that both curves are flat could be that HI and SIdelta do not capture any property of the data and are always constant. Actually on all figures of the papers, all the temporal curves of HI and SIdelta are indeed flat. I think that questions the interest of these measures.- Regarding pannel 2, the observation (that the slopes of SI0 are the similar for antibiotic use and pathogen resistance) is not uninteresting, but these are only two independent data points  and it is thus impossible to have any statistical support. The authors mention that before pooling all of them into 3 units, they have a larger number of administrative units. Maybe they could test their hypothesis at the level of these smaller units? I do not agree that "a grouping of these small subunits into functional units [is] sufficient" if they want to test this hypothesis.- I think there is an important difference between the question presented in the introduction and in the literature (do mixing startegies reduce antibiotic resistance?) and the question mainly adressed in the results (do heterogeneity of treatment correlate with heterogeneity of resistance?). It is only in the last paragraph of the results that the authors attempt to adress the former.So in the current state, I do not see how the methods and the data presented in this article permit to adress what the authors present as the main question, and can not recommend the publication of this article as long as this is not clarified. But I am sure that there is potential interest in the data presented by the authors .Below are more detailed comments about the manuscript, in three parts .A) I did not find the abstract understandable before reading the rest of the article:"Temporal changes of the proportional abundances of different antibiotics (e.g. mixing or cycling)": unlike cycling, mixing does not implies temporal changes but spatial heterogeneity"Although such a mixing strategy appears to be plausible": at this point mixing is not defined, and the previous sentence seems to rather relate to cycling than mixing ."We adopt diverse measures of heterogeneity and diversity": the authors should at least precise the variable whose they are trying to quantify heterogeneity of diversity"We show that a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption" -> I would suggest to precise the direction of the change in the abstract"we introduce a scheme based on linear regression for the assessment of associations between changes of heterogeneities on the antibiotics and the pathogen side" and "we show that a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption" -> I do not see what is new in the method . All the authors do is plotting two curves next to each other and say they look similar (fig 13), without proper statistical analysis.B) The introduction and methods should be clarified, and need proper references to the literature:L16 "whereas other strategies refer to a scheduled change of the dominantly used class of antibiotics": in the literature (including the one cited by the authors), antibiotic mixing does not involve scheduled change of antibiotic over time, but treatment of different subgroups of patients with different antibiotics.L17 "a fraction of patients": I think the authors should introduce the context better, to explain that these strategies are defined at the scale of a group of patients . This is important because with the definition provided by the authors, cycling could be wrongly understood as applied to individual patients.L19 "Although there is some evidence that": citations needed. Overall there is a huge lack of references to the literature in the article.L23-25 I find the sentence very unclear. What does it mean to "quantify the heterogeneity of [...] time courses of prevalence of antibiotic-resistant pathogens" ?L57 is "DDDi" referring to the DDD or the "consumption density DDD"? If the former, why mentioning "consumption density DDD"?L56-57: At this point, it is not clear what is "the antibiotic group i". The previous paragraph explains that the hospital departments are clustered into 3 administrative units, and that the antibiotics are pooled in 12 antibiotic classes. To which kind of group this sentence refers to ? It only becomes clear much later in the article.L60: Is the coefficient of variation computed for the vector of DDDi(t) for all i? Maybe clarify the formula?The paragraph "Coefficient of variation" must include references. It raises an interesting but rather simple point, and could be shortened.The paragraph "Heterogeneity and entropy" is mostly paraphrasing the literature without proper citations. The authors do not explain how these theoretical ecology measures will be applied to quantify antibiotic resistance and antibiotic uses. I understand that they want to stay general because they will apply the same measures to different data, but everything would be more clear if the "antibiotic groups" had been properly defined (see my previous comment)L90, The definition of the "proportions" are not clear at this point. Do a1+a2+a3+...+an=1, or do ai+bi=1 for all i? Said otherwise, is "ai" the proportion of individuals of population a that belong to species i, or is it the proportion of individuals of species i that belong to environment a ? It only becomes clear much later in the text.C) The data presented in the results  have some potential interest, but at this stage the authors do not show anything convincing.Figure 1 : If I correctly understand, the fact that fig 1a and fig 1b look very similar suggest that the sum of DDD for all the antibiotic groups is almost constant over time. This should be made visible on Fig 1c by making the y-axis start at 0, otherwise a quick look at this panel conveys an inaccurate message.L167-174 I do not see how what it presented in this paragraph is a result. The authors are mostly saying that the coefficient of variation is a good measure of variation.L 192-293 "Please note, the different scales of the y-axes reflect the different total amounts of consumption within each unit due to their different sizes": if this really comes from difference in size of the units (number of patients), Isn't it a strong argument for normalising DDD per patient days? I do not understand why the authors describe this normalisation in the methods but do not use it later in the results while it seems needed and appropriate.L305 "The most important and at the same time most challenging question, in the given context, is whether the mixing behaviour of antibiotic administrations correlates or even causally relates to the prevalence of antibiotic resistances.": the authors do not show that their methods are useful for such a test. Do they really bring something compared to the coefficient of variation?Figure 12 mentions "intensive care" and "normal care" unit, while everywhere else in the article the units are labeled "unit 1", "unit 2" and "unit 3"I do not understand the sentence "We conclude that mixing of antibiotic consumption correlates with the prevalence of antibiotic-resistant bacteria by means of parallel slopes of similarity indexes SI0"Dataset: the authors made the effort of making the full dataset and analysis programs available. Just a minor suggestion: the variable names in R data sheets and the text in the R markdown file are currently in german, I think they would be more useful if translated to english.**********what does this mean?). If published, this will include your full peer review and any attached files.6. PLOS authors have the option to publish the peer review history of their article  digital diagnostic tool,  18 Jun 2020Response to Reviewer #1:Major Points1) I don't understand how measures of diversity or heterogeneity are connected to mixing  or cycling  strategies here. Mixing refers to giving different antibiotics to the different members of the cohort, but that can also end up having a constant DDD over time . Similarly, cycling - the temporal approach - should have way more variation over time, such as excluding certain antibiotics completely from the hospital or the cohort. So I cannot relate the antibiotic consumption observed in this dataset with the common strategies used.Answer: Yes, it is true that cycling or mixing can end up in having the same diversity. This is one of the major aspects of our discussion and lead us to the introduction of a differential measure SI_0. In the revised version, we additionally derived Eq 10 which is appropriate to capture mixing. In fact, the discussion of SI_0 is the most crucial point of our analysis and elaborated on that in the revised version.A general remark with respect to the dataset seems to be due. Main purpose of our work is to derive a sound mathematical framework for the analysis of antibiotic control strategies. The application to the data available for us has to be understood mainly as illustration of how the mathematical methods are applied. We emphasise the fact, that a control in a genuinely defined way either of cycling or mixing, respectively, has obviously not been applied in the Hospital from which we draw data. However, a long-term change of antibiotic administration can be observed which we call „clinical cycling“ following Abel Zur Wiesch et al. 2014. In the light of this rather weak temporal changing behaviour we feel inclined to formulate our (secondary!) finding with respect to the concrete observation in a positive way: Despite the relatively weak differential change of diversity in antibiotic consumption, we observe a positive correlation with the corresponding differential change of prevalent resistant pathogens. We strengthend this point in the revised version. This point is strongly related to the following objection of Reviewer #1 2) The authors' most important conclusion is only expressed in one sentence, and not fully clear (lines 362-364). Do they mean almost a no slope by saying that " means of parallel slopes "? Is this really a good way to back their argument up? Given that this is the message of the paper, I really expect more elaboration on this point.Answer: Throughout the revised manuscript, we clarified this point. In addition, with respect to the paragraph mentioned by Reviewer #1 (lines 362-364 of the first submission), we explicitly added/revised:„To conclude, although a genuine control strategy in the common sense of antimicrobial cycling or mixing was not applied, we observe that a rather long-term temporal change  of consumption of different antibiotics ($SI_0$ applied to antibiotics consumption) correlates with a change of prevalence of antibiotic-resistant bacteria . This correlation is expressed by means of almost equal slopes as well as a corresponding large correlation coefficient, where the slopes are significantly different from zero and approximately equal, of the two similarity indexes $SI_0$ for antibiotics and pathogens, respectively. Whether these temporal changes in antibiotic consumption have a direct causal impact is still speculative but gains additional evidence through the observed reduction of prevalent resistant bacteria. Controlled studies that allow comparisons with more or less static ``control strategies'' and other types of switching behaviours (including mixing) are needed to draw reliable inferences. It is the main intention of this work to supply an appropriate mathematical framework for such studies.“Please also note that we strengthened our arguments in the Results section where we discussed the corresponding findings of how SI_0 behaves.3) In general, I understand why authors use so many different measures to asses diversity or heterogeneity, and while reading the manuscript, it was obvious that this is the way they explored the whole research question. But is it necessary to include all these measures in the main text? There is a lengthy section about the explanation and use of so many different measures, but the conclusion and the connection to resistance prevalence is relatively really short, which is supposed to be the main point. I think the manuscript needs a bit of structural revision in this regard.Answer: It is indeed important for us to introduce and discuss the whole family of diversity measures within the main text and compare it with the antibiotic homogeneity index previously used by other authors. Although all members of the family share common features, the concrete choice nevertheless depends on the concrete structure of studies and related research questions to which the method will be applied. The discussion of features of the proposed measures and their comparisons are indeed crucial points of our manuscript. Minor Points1) Figures 4-10 : please also use dots for the data points, like you did for figures 1-3.Answer: We would like to stick with the rule to only represent actual data points and summary statistics/moments like mean, median, variance etc. by dots but not nonlinear functions like entropies etc. 2) line 14 : these strategies are not really "recent" anymore.Answer: Yes, we agree and changed it accordingly. 3) lines 19-21 : please refer to the key publications that includes the "evidence" you have mentioned.Answer: We added references that show the mentioned evidence including Sandiumenge et al. 2006 Bennett et al. 2007 AbelZurWiesch et al. 2014 Lopez-Lozano et al. 2019 Davey et al. 2013and also one reference for a counter-example: Karam et al. 2016Response to Reviewer #2:Reviewer #2: Diebner and collaborators present data about antibiotic consumption and pathogen resistance in three administrative units of a hospital over several years. They aim at using these data to test hypotheses related to the efficiency of heterogeneous antibiotic treatments strategies to mitigate the emergence of resistance in hospitals. For this they suggest the use of various measures of heterogeneity  in the context of antibiotic treatment and resistance.General answer that applies to several objections raised by Reviewer #2:Main purpose of our work is to derive a sound mathematical framework for the analysis of antibiotic control strategies . The application to data available to us has to be understood mainly as illustration of how the mathematical methods are applied. We emphasise the fact, that a control in a genuinely defined way either of cycling or mixing, respectively, has obviously not been applied in the Hospital from which we draw data. However, a long-term change of antibiotic administration can be observed which we call „clinical cycling“ following Abel Zur Wiesch et al. 2014. In the light of this rather weak temporal changing behaviour we feel inclined to formulate our finding with respect to the concrete observation in a positive way: Despite the relatively weak change of diversity in antibiotic consumption, we observe a positive correlation with the decline of prevalent resistant pathogens. Throughout the revised manuscript, wherever appropriate, we more strictly pointed out our main intention to present a math framework rather than presenting the results of a clinical study. In addition, we strengthend the provisional and somewhat speculative results with respect to data analysis. Even in the virtual absence of „heterogeneous antibiotic treatments“ there is a weak but noticeable impact of the increase of diversity and, more important, of the changes with respect to the initial observation time (captured by SI_0) on the reduction of antibiotic resistance. We clarified and strengthened this point throughout the revised manuscript wherever appropriate. Also confer the following answers.I am not convinced of the interest of this framework for the analysis of heterogeneous antibiotic treatment strategies, for 3 reasons:- The measures that go beyond the coefficient of variation are interesting when strictly more than 2 antibiotics are in use. In the "antibiotic mixing" literature, to my knowledge the vast majority of the literature considers the case of 2 antibiotics .Answer:Cycling and mixing in their extreme forms are rarely met, which is why Abel zur Wiesch et al. 2014 introduced the notion of „clinical cycling“ for the more frequently met adjusted forms. In a real clinical situation, in the absence of a strict mixing plan (with only 2 AB involved), usually more than 2 AB are involved. Surely, in the case of a given strict mixing with only 2 AB and a single given type of infection (pathogen), one would switch to prospective trials , controlled clinical trials (CCT), controlled before-after, cross-over) based on well-defined rotation/switch protocols, thereby, however, excluding continuous real-world processes. Alternatively, one can perform a time series analysis as discussed in Lopez-Lozano et al. 2019. Here the prevalence time series of the resistant pathogen is correlated with the time series of the amount of administered AB. Usually, in such a correlation analysis of two time series, a mutal entropy is used to quantify the correlation. In fact, we see in our approach a generalisation to more than 2 involved AB but this does not exclude the application to only 2 AB.We agree that we used the notions of cycling and mixing in a somewhat vague way. We added clarifying explanations and, most important, we added Eq. 10 in the revised version which explicitly accounts for mixing in the strict sense. Even in the case of only 2 AB these measure results in an interpretable output, although in such a case one would prefer to switch to the time series correlation analysis of 2 time series, as mentioned above. - These measures are used in ecology to deal with very heterogeneous data, eg when abundances are very different between different species. This is useful (in ecology) because a species may be several orders of magnitude less abundant than another but still be essential for ecosystem function. But in the case of antibiotic treatments, the point of DDD is that different antibiotics can be compared between them. If the DDD of 2 antibiotics differ by orders of magnitude, it means that the one with the low DDD is virtually absent and has no chance of contributing to the establishment of resistance (or to infection clearance).We have problems in fully understanding this objection. A typical application in ecology is the impact of crop diversity on the vulnerability of the system to pests. One may hypothesise, that a monoculture (a low diversity) may lead to high vulnerability. This hypothesis is driven by real observations. This vulnerability might be reduced either by implementing a diverse crop culture or by annually switching the type of crop  or a combination of that. We hope that this gives a vivid illustration that there are indeed many analogues between ecosystem and antibiotic stewardship. To quantify the impact of diversity as well as the impact of temporal and spatial changes that might leave overall diversity invariant, an appropriate measure is needed, which we presented (SI_0). We do not see why such measures should not be applicable to situations where a diverse (>2 species) system is compared to a 2-species-system including strategies of cycling and mixing. Furthermore, we do not see why this should not be translated into the area of antibiotic stewardship. By the way, with respect to DDD, even the reverse direction might be of interest, namely to introduce an abundance measure that renders crops comparable with each other by means of their contribution to vulnerability. This is of course much beyond the scope of our approach and we are lucky to already have a working definition of DDD.More specific, given a fixed number of different species, diversity  reaches is maximum if all species are equally abundant, no matter of whether the whole ecosystem consists of only 2 or more different species. Whether or not diversity correlates positive or negative or not at all with a target parameter as, e.g., prevalence of resistant germs, has to be shown. This is the first of our objectives of presenting diversity measures and we are not quite sure what should be wrong with this approach. Relating the prevalence of resistant pathogens to diversity is what other authors approached earlier, e.g. Sandiumenge et al. 2006 or Pluess-Suard et al. 2013, however, using the antibiobtic heterogeneity index (HI in our notation) only. Two things should be mentioned in this context. Firstly, looking at the mere diversity may not suffice, but one should also treat the effect of changes of the amount of AB even if these changes leave diversity invariant. This is what we did when introducing differential measure SI_0. Whether such effects exist or not is still open, but we supply the proper method to test it in the future, and, importantly, we supplied initial evidence through our analysis. Secondly, we used other (better in certain contexts) measures in form of Renyi entropies and compared them to HI. HI fails to capture weak changes of diversity. If low DDD of a given AB really has no impact, than it should be reflected in the impact of a low diversity. - In much of the literature (including the one cited by the authors) on heterogeneous antibiotic treatments, a big question is "mixing vs cycling", but I do not see how the measures suggested by the authors can be useful to study cycling. The authors simply do not ever mention cycling after the introduction. It gives me the impression that the papers they cite do not really match the question they want to study.We agree that it has not always been clearly formulated of whether „only“ diversity is addressed or whether changes in administration  are addressed that might in principle leave overall diversity invariant. We addressed this lack of clear formulation in the revised version of our manuscript. Particularly, we introduced Eq 10 in the revised version that explicitly captures mixing strategies. Most important, SI_0 captures cycling very well and we put more effort into explaining this fact in the revised version. Most importantly, I am also not convinced that the presented data support the claimed result , for several reasons:- Only time series are analysed. In time series, the different time points are not independent from each other, and it is thus not appropriate to decide whether the slope is significantly different from 0 using a linear regression or a correlation test .Answer: Yes, time series are often autocorrelated, but also non-autocorrelated time series exist. More important beyond this general remark, two time series may be mutually correlated. Why is the use of time series analyses formulated as an objection or as a limitation ? It is the very essence and the strength of our processual approach to evaluate whether such a mutual information (correlation) of two time series  exists. For an exquisite example that points in this direction see Lopez-Lozano et al. 2019. We have quarterly data over several years and why should we not use this information on the observed time courses? Moreover, the processual approach is applicable to field-like real-world observational studies. If sequential observations at several observation time points are given, why should one not use this time series? The crucial point is to aggregate the time series of many abundances into a manageable and meaningful proxy time series: diversity and/or differential similarity.Unfortunately, for the data available to us, we cannot attribute time series of specific antibiotic treatment to the emergence of a specific resistant pathogen as in Lopez-Lozano et al. What we have is the time series of diversity (HI or Renyi entropies) and (auto-) similarity measures (e.g. SI_0) and we relate these time series for antibiotics and pathogens with each other. We observe a correlation. Moreover, SI_0 is a measure which captures adjusted administration of AB including cycling even if diversity is unchanged. Unfortunately, for our data, we do not observe a full cycle but rather a long-term change which can be interpreted as the onset of cycling . In our case, the adjusted administration appears in form of a linear change of SI_0 and we think it is legitimate to ask whether this change differs significantly from zero (by means of inference based on confidence).  As long as we do not have experimental (prospective) data but have to rely on an observational study only, we cannot conclude that this correlation is due to causality. The only thing we can do is raising a hypothesis, which is what we did. Perhaps, using a more complex multivariate statistical approach to time series would be the ultimate improvement. However, using diversity and similarity measures, as we did, as surrogate measures to capture essential aspects of the time series appears to be a legitimate approach with medium (i.e. comprehensible) complexity. In the revised version, we elaborated on our processual stance mostly within the introduction but also throughout the manuscript. In addition, we emphasised the relevance of SI_0 and we extended the according analysis.- Regarding panels 1 and 3 of fig 13 (HI and SIdelta), both the curves for heterogeneity / similarity of antibotic use and pathogen resistances are flat. The authors conclude that the curves for treatment and resistance are similar and thus that there methodology permits to establish that heterogeneous treatments cause heterogeneous resistance. There is absolutely no statistical support (nor proper testing) for this finding. And a more parsimonious explanation for the fact that both curves are flat could be that HI and SIdelta do not capture any property of the data and are always constant. Actually on all figures of the papers, all the temporal curves of HI and SIdelta are indeed flat. I think that questions the interest of these measures.Answer: HI (and likewise the Renyi entropy) measures per definition diversity. It remains constant for the observed data. This is, in our opinion, a relevant and most illuminating result. We do not understand why diversity should not capture a relevant property of the data. If the abundances of a low-abundant and a highly-abundant species are exactly swapped, diversity/heterogeneity does not change. This means, diversity/heterogeneity (particularly HI) does not capture the swap of species abundances. The very strength of our approach is, that a measure for the strength of cycling can indeed by derived: SI_0. So far, to the best of our knowledge, only HI has been used in the literature to describe antibiotic diversity which does not capture such a swap. For our data, HI is almost constant over time, although throwing a glance onto the time courses of the consumptions confirms that indeed the amount of essentially three antibiotics switched in the course of time.Thus, we think that it is one of the strengthes of our work having shown that HI or any other pure diversity does not suffice to capture relevant changes in consumption. To capture changes in abundances – INCLUDING those which leave diversity invariant – we introduced SI_0 which detects whether a change took place or not.With respect to the analysed data, there is a more or less constant shift of some of the antibiotics abundances, which is captured by a linear decline of similarity with respect to the first observation. SI_0. The slope is a measure of the intensity of these shift. SI_Delta is an approximation to this slope and the magnitude of 1-SI_Delta is an exquisite measure of the strength of cycling. It is weak and constant in our case, but this is a particular and relevant finding for this case. Thus, HI (i.e. diversity) might in concrete real observations, as is the case for our data, turn out to exhibit no change at all, although there have been changes in consumption. Cycling and mixing might leave heterogeneity/diversity invariant, therefore, we introduced a „differential“ measure to account for such changes. In a sense it is a nice coincidence that our data exhibits an almost constant diversity, although changes in administration happened, because this supplies additional evidence (in form of a proof-of-principle) for the need of our newly introduced „differential diversity measure“ SI_0. SI_Delta is in a certain sense redundant, since it captures the slope of SI_0 which is constant in case of linear SI_0. However, it is arguably a good index that quantifies the strength of cycling in a general sense, including adjusted forms of AB prescriptions.- Regarding pannel 2, the observation (that the slopes of SI0 are the similar for antibiotic use and pathogen resistance) is not uninteresting, but these are only two independent data points  and it is thus impossible to have any statistical support. The authors mention that before pooling all of them into 3 units, they have a larger number of administrative units. Maybe they could test their hypothesis at the level of these smaller units? I do not agree that "a grouping of these small subunits into functional units [is] sufficient" if they want to test this hypothesis.Answer: We completely agree, for a statistical support particularly of the impact of mixing, it is necessary to look at a more fine-grained level of administrative units. „Sufficient“ here refers to the sufficiency for the illustrative purpose with respect to the math framework. We added Eq 10 in the revised version that clarifies this point. However, we would like to keep the focus on the math framework instead of going into depth with the analysis of a dataset that has been recorded in the absence of a well-defined or documented administration strategy. Particularly, information on unit-specific prevalences is missing. We think that making the math framework public even without the analysis of an adequate controlled study has priority. Moreover, our quantification will become most important when being compared to the outcome of other hospitals. Nevertheless, the fact that the „cycling-like“ changes of prevalences of pathogens measured by SI_0 follows the (weak) clinical cycling is, in our opinion, quite striking. In this respect, what we presented is within-unit statistical significance. As mentioned, between-unit variability should be the next step. Our result is, as we mentioned in our manuscript, provisional. Our intention is to supply researchers with our method and see it applied to other records of antibiotic consumptions in order to see how other hospitals perform. - I think there is an important difference between the question presented in the introduction and in the literature (do mixing startegies reduce antibiotic resistance?) and the question mainly adressed in the results (do heterogeneity of treatment correlate with heterogeneity of resistance?). It is only in the last paragraph of the results that the authors attempt to adress the former.Answer: We agree, the notations for different strategies have been used in a somewhat confusing way. Our aim is to provide a method that is capable to deal with diversity and with administrative strategies including cycling and mixing that may leave diversity invariant but are nevertheless relevant. Although mixing refers to a spatial permutation, this permutation takes place at a certain instance of time. This is why we carelessly generalised to refer to temporal changes. In our revised version we addressed the different strategies in a much more careful way. So in the current state, I do not see how the methods and the data presented in this article permit to adress what the authors present as the main question, and can not recommend the publication of this article as long as this is not clarified. But I am sure that there is potential interest in the data presented by the authors .Below are more detailed comments about the manuscript, in three parts .A) I did not find the abstract understandable before reading the rest of the article:"Temporal changes of the proportional abundances of different antibiotics (e.g. mixing or cycling)": unlike cycling, mixing does not implies temporal changes but spatial heterogeneity"Although such a mixing strategy appears to be plausible": at this point mixing is not defined, and the previous sentence seems to rather relate to cycling than mixing ."We adopt diverse measures of heterogeneity and diversity": the authors should at least precise the variable whose they are trying to quantify heterogeneity of diversity"We show that a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption" -> I would suggest to precise the direction of the change in the abstract"we introduce a scheme based on linear regression for the assessment of associations between changes of heterogeneities on the antibiotics and the pathogen side" and "we show that a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption" -> I do not see what is new in the method . All the authors do is plotting two curves next to each other and say they look similar (fig 13), without proper statistical analysis.Answer: We addressed all points and substantially revised the abstract. With respect to objection „All the authors do is plotting two curves next to each other and say they look similar (fig 13), without proper statistical analysis.“ , our answer is that it depends a bit to which school of statisticians you belong. We did not just plot the curves next to each other, but we did a legitimate proper statistical inference based on confidence (overlapping CIs of slopes) rather than trusting too much in p-values. Anyhow, we now added a correlation analysis, but as expected, the inference does not change because the correlation is essentially the same as properly comparing the slopes in this case. B) The introduction and methods should be clarified, and need proper references to the literature:L16 "whereas other strategies refer to a scheduled change of the dominantly used class of antibiotics": in the literature (including the one cited by the authors), antibiotic mixing does not involve scheduled change of antibiotic over time, but treatment of different subgroups of patients with different antibiotics.Answer: We agree, as mentioned above, that the notations have been used in a confusing way. We revised accordingly. See also the above answers.L17 "a fraction of patients": I think the authors should introduce the context better, to explain that these strategies are defined at the scale of a group of patients . This is important because with the definition provided by the authors, cycling could be wrongly understood as applied to individual patients.Answer: We had „groups of patients“ in mind and revised the text accordingly.L19 "Although there is some evidence that": citations needed. Overall there is a huge lack of references to the literature in the article.Answer: We added references. L23-25 I find the sentence very unclear. What does it mean to "quantify the heterogeneity of [...] time courses of prevalence of antibiotic-resistant pathogens" ?Answer: We changed to„… quantify heterogeneity of both, antibiotic consumption as well as prevalence of antibiotic-resistant pathogens as a function of observation time.“L57 is "DDDi" referring to the DDD or the "consumption density DDD"? If the former, why mentioning "consumption density DDD"?Answer: We used DDD and added a corresponding clarification.L56-57: At this point, it is not clear what is "the antibiotic group i". The previous paragraph explains that the hospital departments are clustered into 3 administrative units, and that the antibiotics are pooled in 12 antibiotic classes. To which kind of group this sentence refers to ? It only becomes clear much later in the article.Answer: Antibiotic class was meant and we revised accordingly.L60: Is the coefficient of variation computed for the vector of DDDi(t) for all i? Maybe clarify the formula?Answer: We clarified that point: „… the mean is taken over the antibiotic classes and SD(DDD(t)) denotes the corresponding standard deviation.“The paragraph "Coefficient of variation" must include references. It raises an interesting but rather simple point, and could be shortened.Answer: We considerably reduced redundancy and added references to that paragraph.The paragraph "Heterogeneity and entropy" is mostly paraphrasing the literature without proper citations. The authors do not explain how these theoretical ecology measures will be applied to quantify antibiotic resistance and antibiotic uses. I understand that they want to stay general because they will apply the same measures to different data, but everything would be more clear if the "antibiotic groups" had been properly defined (see my previous comment)Answer: Intention was to address antibiotic classes and we revised accordingly. Please also see the related previous answer. L90, The definition of the "proportions" are not clear at this point. Do a1+a2+a3+...+an=1, or do ai+bi=1 for all i? Said otherwise, is "ai" the proportion of individuals of population a that belong to species i, or is it the proportion of individuals of species i that belong to environment a ? It only becomes clear much later in the text.Answer: a_i is the proportion of individuals of population a that belong to species i . We revised accordingly.C) The data presented in the results  have some potential interest, but at this stage the authors do not show anything convincing.Figure 1 : If I correctly understand, the fact that fig 1a and fig 1b look very similar suggest that the sum of DDD for all the antibiotic groups is almost constant over time. This should be made visible on Fig 1c by making the y-axis start at 0, otherwise a quick look at this panel conveys an inaccurate message.Answer: We changed the scale of the y-axis accordingly.L167-174 I do not see how what it presented in this paragraph is a result. The authors are mostly saying that the coefficient of variation is a good measure of variation.Answer: With respect to the coefficient of variation (CV), the application belongs to a descriptive part. We discussed the application of CV mainly for the sake of completeness, for comparisons with other measures, and because CV is found in the literature as a measure of homogeneity. We occasionally refer to CV in the text and we think that it might be beneficial for some readers to see how the application of CV in such a case behaves. L 192-293 "Please note, the different scales of the y-axes reflect the different total amounts of consumption within each unit due to their different sizes": if this really comes from difference in size of the units (number of patients), Isn't it a strong argument for normalising DDD per patient days? I do not understand why the authors describe this normalisation in the methods but do not use it later in the results while it seems needed and appropriate.Answer: All essential measures we discuss use proportions (relative abundances) as entry argument with the exception of CV. However, we find it illuminating to see the absolute DDDs. With respect to the results based on the diversity measures, it does not matter whether absolute DDDs or normalised DDDs are used. Relative abundances turn out to be almost identical for the two cases. We mentioned this but we did not show data due to redundancy. Moreover, the recorded patient days in the dataset may not be as reliable as the reports on antibiotic consumption. In addition, the surgical unit plays a uniqe role of treating mostly prophylactically. Residency in this unit per patient is short and patient days are not really meaningful. An absolute consumption of AB is arguably more informative.L305 "The most important and at the same time most challenging question, in the given context, is whether the mixing behaviour of antibiotic administrations correlates or even causally relates to the prevalence of antibiotic resistances.": the authors do not show that their methods are useful for such a test. Do they really bring something compared to the coefficient of variation?Answer: I do not see how the coefficient of variation can be modified such that a differential measure turns out comparable to SI_0. CV may reflect to some extent heterogeneity/diversity but it does – very much like HI - neither capture cycling nor mixing in an adequate way. However, HI (entropies) CAN be modified accordingly to yield SI_0. Figure 12 mentions "intensive care" and "normal care" unit, while everywhere else in the article the units are labeled "unit 1", "unit 2" and "unit 3"Answer: Within the text we usually used the clear text in combination with the code to get an intuition of which unit we talk of. We introduced the code for labeling figs and the clear text for the fig legends, but unfortunately not in a consistent way. We revised the figures and text accordingly.I do not understand the sentence "We conclude that mixing of antibiotic consumption correlates with the prevalence of antibiotic-resistant bacteria by means of parallel slopes of similarity indexes SI0"Answer: This objection is perhaps due to our confusing usage of „mixing“ and „cycling“ (see previous answers). What the correlation of SI_0 for AB and SI_0 for pathogens show is, that the magnitude of effective (rather weak) long-term clinical cycling given by the slope of SI_0 is the same as the magnitude of effective cycling-like changes in pathogen prevalences. This might be trival, because reducing a certain AB may reduce the prevalence of pathogens resistant to that AB and vice versa. But the almost perfect correlation is nevertheless quite striking, in our opinion. Moreover, we have shown that the amount of „cycling“ expressed by SI_0 correlates strongly with a reduction of the prevalence of resistant bacteria. Please see previous answers as well as the substantially improved explanations throughout the revised text. Dataset: the authors made the effort of making the full dataset and analysis programs available. Just a minor suggestion: the variable names in R data sheets and the text in the R markdown file are currently in german, I think they would be more useful if translated to english.Answer: We plan to upload the revised R-script to github upon acceptance of the manuscript.AttachmentResponse to Reviewers.docxSubmitted filename: Click here for additional data file. 11 Aug 2020PONE-D-19-25350R1Mathematical basis for the assessment of antibiotic resistance and administrative counter-strategiesPLOS ONEDear Dr. Diebner,Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.journal webpage.I appreciate your efforts in carefully addressing most of the points raised by both Reviewers. However, as you may see in their detailed comments below, although most of the points raised by Reviewer 2 have now been clarified, some concerns still remain about your time series statistical analysis. As a potential reader of your study, I also miss some more details on the specific statistical analysis that you conducted. This point is especially important for PLOS One publication criteria, available in the plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.Please submit your revised manuscript by Sep 25 2020 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at Please include the following items when submitting your revised manuscript:A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). 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For instructions see: We look forward to receiving your revised manuscript.Kind regards,Ricardo Martinez-GarciaAcademic EditorPLOS ONE[Note: HTML markup is below. Please do not edit.]Reviewers' comments:Reviewer's Responses to QuestionsComments to the Author1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.Reviewer #2: (No Response)**********2. Is the manuscript technically sound, and do the data support the conclusions?The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #2: Partly**********3. Has the statistical analysis been performed appropriately and rigorously? Reviewer #2: I Don't Know**********4. Have the authors made all data underlying the findings in their manuscript fully available?PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.The Reviewer #2: Yes**********5. Is the manuscript presented in an intelligible fashion and written in standard English?PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.Reviewer #2: Yes**********6. Review Comments to the AuthorPlease use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. Reviewer #2: Both in their answer and in the updated manuscript, Diebner and collaborators clarified that they see the mathematical framework as the main contribution of their article, and the application to the hospital dataset as a mere example of what can be achieved using this framework.While most of my objections regarding the analysis of this hospital dataset and the interpretation of the findings remain, I guess they should thus not be seen as a blocking point. Besides, I think the authors clarified many points regarding their methodology and the context of their work. Many of my methodological questions are now addressed, in the manuscript or in the answer. As a biologist, it is still hard for me to judge the interest of the measures proposed by Diebner and colleagues without seeing a convincing biological application. So all I can say is that the methods and the context are clearly described, especially with the clarifications made in this revision, and that the figures are easily understandable.Still, I am surprised that the authors do not see the problem with correlating two time-series which are both highly autocorrelated (L400-408). The statistical analysis is not described in details so I may be wrong, but I looked at the code and it seems that the authors directly compute a Pearson correlation between the two vectors (each vector being a time series). If so, I think this is quite inappropriate (because the data points of each vector are not independently sampled from a given distribution). This is what I was trying to point in my initial review by stating that ``only time series are analysed''. I have nothing per se against the analysis of time series, but it should be done correctly with the appropriate statistical tools .**********what does this mean?). If published, this will include your full peer review and any attached files.7. PLOS authors have the option to publish the peer review history of their article  digital diagnostic tool,  14 Aug 2020Response to ReviewersManuscript PONE-D-19-25350R1Reviewer #2:Both in their answer and in the updated manuscript, Diebner and collaborators clarified that they see the mathematical framework as the main contribution of their article, and the application to the hospital dataset as a mere example of what can be achieved using this framework.While most of my objections regarding the analysis of this hospital dataset and the interpretation of the findings remain, I guess they should thus not be seen as a blocking point. Besides, I think the authors clarified many points regarding their methodology and the context of their work. Many of my methodological questions are now addressed, in the manuscript or in the answer. As a biologist, it is still hard for me to judge the interest of the measures proposed by Diebner and colleagues without seeing a convincing biological application. So all I can say is that the methods and the context are clearly described, especially with the clarifications made in this revision, and that the figures are easily understandable.Still, I am surprised that the authors do not see the problem with correlating two time-series which are both highly autocorrelated (L400-408). The statistical analysis is not described in details so I may be wrong, but I looked at the code and it seems that the authors directly compute a Pearson correlation between the two vectors (each vector being a time series). If so, I think this is quite inappropriate (because the data points of each vector are not independently sampled from a given distribution). This is what I was trying to point in my initial review by stating that ``only time series are analysed''. I have nothing per se against the analysis of time series, but it should be done correctly with the appropriate statistical tools .Answer:The reviewer raised an important point regarding the correlation of time series. Yes, strictly speaking, Pearson’s correlation requires independent sampling. Usually, the correlation of two time series is analysed by applying a cross-correlation-function which uses a time-lagged version of one of the time series. Unfortunately, this is not applicable here due to the low sampling frequency and very short time series. Thus, our argument here is that Pearson’s correlation is sufficient to at least raise hypotheses. Arguably, this supports the idea of a proof-of-principle. Based on the proposed analysis using diversity measures it is planned to design a proper study and we think that also other researchers dealing with antibiotic stewardship will benefit from our mathematical framework. We added the subsection “Statistical analysis” at the end of the “Materials and methods” section to elaborate on these points.AttachmentResponse to Reviewers.docxSubmitted filename: Click here for additional data file. 24 Aug 2020Mathematical basis for the assessment of antibiotic resistance and administrative counter-strategiesPONE-D-19-25350R2Dear Dr. Diebner,We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.http://www.editorialmanager.com/pone/, click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. 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For more information, please contact Kind regards,Ricardo Martinez-GarciaAcademic EditorPLOS ONEAdditional Editor Comments :Reviewers' comments: 25 Aug 2020PONE-D-19-25350R2 Mathematical basis for the assessment of antibiotic resistance and administrative counter-strategies Dear Dr. Diebner:I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department. onepress@plos.org.If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact plosone@plos.org. If we can help with anything else, please email us at Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staffon behalf ofDr. Ricardo Martinez-Garcia Academic EditorPLOS ONE


H-Boltzmann theorem. The key point lies in that the definition of majorization allows choosing a wide range of convex functions H-Boltzmann theorem is obtained as a special case for In this work we show how the concept of majorization in continuous distributions can be employed to characterize mixing, diffusive, and quantum dynamics along with the  The concept of majorization has shown a particular interest in the last decades, mainly due to its wide range of applicability in information and quantum theory, among others ,6,7,8,9.Based on the majorization of discrete finite-dimensional vectors, Hardy, Littlewood and Pólya introduced the continuous version for integrable functions ,23, giviThe goal of this paper is to study the relationship between the increasing direction of the majorization ordering and the temporal evolution of a continuous dynamics in order to characterize mixing dynamics, diffusion phenomena and quantum evolutions. Thus, the present contribution also could shed light towards a geometrical definition of the arrow time in the context of continuous majorization.H-theorem in terms of majorization is given. Next, we illustrate some consequences of continuous majorization applied to the Frobenius–Perron operator associated to a model of exponential population dynamics. Finally, in The work is structured as follows. In Here, we give the necessary elements for the development of the present work. We begin by recalling the concept of continuous majorization along with some properties.I if for all I. Given f is majorized by g, denoted by I), the partial order obtained is weak majorization, symbolized by g and h are the infimum and the supreme of ordered chain by majorization if for all We say that a real function  f≺g, if ,23,26(1)n-dimensional vectors. Then, we say that Complementary, Hardy also defined the discrete majorization and showed its relation with the continuous case, as follows. Let Here, Schur-convex if for all quasi-convex if for all Two important concepts that are related to the convexity are the Schur-convexity and the quasi-convexity. A function g result ,23,24,25Theorem 1. Let  1.1. of ):(a) (b) (c) (d) Note that Equation . If neceWhen the functions or vectors represent probability distributions, the majorization adopts an intuitive interpretation. To illustrate this, we consider the discrete version and the same argument can be applied to the continuous case.n-dimensional probability distributions, i.e., n-dimensional probability vectors, the uniform and the delta distributions are the infimum and the supreme elements, respectively.Let Equation , we haven possible outcomes ith outcome occurs) then discrete majorization says that the distribution with the minimal information (the uniform one) is majorized by Physically, if we have an experiment with t where x represents a continuous variable in t, i.e., In this section, we consider a system described by a continuous distribution ble in 0,. We focuLemma 2. The following statements are equivalent:(I) (II) The function In turn, Proof. t.t, then, for all t, its derivative must be negative for all t, thus Finally, let us assume The content of Lemma 2 is that, when we have a dynamics satisfying Lemma 3. Assuming hypothesis andProof. Due to t, thus When the time evolution preserves majorization in the sense of condition Next step is to study what kind of phenomena can be compatible with continuous majorization.X of the variable x is the real interval To study what type of dynamics can be obtained from continuous majorization, in this section, we illustrate the results with some examples. We begin by discussing the H-Boltzmann theorem. In all the examples, the abstract space terval 0,, in ordeH-theorem. In 1872, Ludwig Boltzmann introduced a functional H to describe, in an elegant way [Boltzmann’s H-theorem [H-theorem). H-theorem provides a justification to the equilibrium approach and the increasing of the associated entropy H , and more fundamental, negentropy can be defined by means of Equation Hölder–Minkowski inequality [Riesz representation theorem [Now, let us assume the dynamics satisfies condition equality  and f∘T0 theorem  to the fH is proposed in the form [t, Disordered and thermal molecular motion is macroscopically characterized as diffusion phenomena of a net flux of particles from one region to other. Under Markovian assumptions and making the passing to the continuum, the discretized master equation for the probability transition states becomes the Fokker–Planck equation (FPE) for the probability distribution. Recently, a generalization of the FPE ,33,34 ∂p=ϕ′n the energy index,We analyze how the continuous majorization can characterize quantum dynamics. We consider that the set H may be non-Hermitian, for instance in open quantum systems [n. To verify if the dynamics prescribed by Equation (n. Doing this and using that x of the eigenfunctions Equation  expresse systems . In the Equation  preservewe obtainλϕ′(t)=2γCase I: Hermitian dynamics  From Equation , the transfer operator P associated  is defined as (see Equation (3.2.2) of [A. The Frobenius–Perron operator P belongs to the class of Markov operators, which is a relevant family of operators in dynamical systems theory since they allow to define the evolution of distributions functions on abstract spaces.Given a discrete map .2.2) of )(17)∫APρf is a differentiable and bijective function, given a probability distribution For unidimensional maps and when .2.7) of )(18)Pρ)=Equation  but in tg majorizes g spreads along its evolution, thus showing a diffusive behavior as the time increases. Thereby, the characterization of the Frobenius–Perron operator in Equation  that allow characterizing the dynamics towards the equilibrium for all system whose states are represented by probability distributions, where the temporal order can be expressed univocally as a majorization order, in which, as the distribution evolves along the time, it is majorized by its corresponding earlier times. As a result of this characterization, the equilibrium distribution and the initial one correspond to the minimum and maximal elements, respectively.H-theorem has been obtained as a special case by choosing the convex function After this characterization of a continuous dynamics by means of continuous majorization, we have analyzed some different examples. Specifically, under the assumption of a dynamics satisfying continuous majorization (condition H-theorem by means of Theorem 4 could serve to shed light on the question of the thermodynamical arrow of time. In this sense, we can see that the continuous majorization contains a proper arrow of time governed by its induced order in a continuous dynamics. It is worth noting that the continuous majorization removes some relevance of the thermodynamical entropy as the main functional to be considered; for instance, in the context of dynamical systems, we have seen that the convex function to obtain the mixing property corresponds to the module function. This kind of flexibility by means of the choice of the convex function In summary, our contribution is two-fold. On the one hand, Lemmas 2 and 3 provide a general framework for majorization in a continuous dynamics that can be exported in specific contexts, as illustrated in Finally, an examination of these consequences with more examples is expected in future contexts (for instance, in fractional nonlinear systems , quantif


Clostridium difficile (CD) is the primary cause of health-care-associated infectious diarrhea, afflicting approximately 1% of hospitalized patients. CD may be simply carried or lead to a mild disease, but in a relevant number of patients, it can cause a very severe, potentially fatal, disease. In this narrative review, the present possibilities of CD infection (CDI) prevention will be discussed. Interventions usually recommended for infection control and prevention can be effective in reducing CDI incidence. However, in order to overcome limitations of these measures and reduce the risk of new CDI episodes, novel strategies have been developed. As most of the cases of CDI follow antibiotic use, attempts to rationalize antibiotic prescriptions have been implemented. Moreover, to reconstitute normal gut microbiota composition and suppress CD colonization in patients given antimicrobial drugs, administration of probiotics has been suggested. Finally, active and passive immunization has been studied. Vaccines containing inactivated CD toxins or components of CD spores have been studied. Passive immunization with monoclonal antibodies against CD toxins or the administration of hyperimmune whey derived from colostrum or breast milk from immunized cows has been tried. However, most advanced methods have significant limitations as they cannot prevent colonization and development of primary CDI. Only the availability of vaccines able to face these problems can allow a resolutive approach to the total burden due to this pathogen.For many years, it has been known that  Clostridium difficile (CD) is a toxin-producing, Gram-positive, spore-forming, anaerobic pathogen that can simply colonize the intestinal tract or be associated with gastrointestinal manifestations of various severity from mild to life threatening that can frequently recur. Severe cases, such as those with pseudomembranous colitis, toxic megacolon, perforation of the colon, and septic shock are diagnosed in about 1% of hospitalized patients and death can occur in up to 17% of them, with the highest values during outbreaks [utbreaks . Accordiutbreaks . For manutbreaks . A studyutbreaks . In the utbreaks . Even hiutbreaks  or was cutbreaks . In geneutbreaks . In the utbreaks , and morutbreaks . In Euroutbreaks . Prolonged antibiotic use is the main risk factor for CDI. Other favoring factors are prolonged hospital stay, age >65 years, immunosuppression, the presence of severe underlying illness, and the use of antiulcer and chemotherapeutic medications ,14. TogeTo reduce the total CD burden, infection prevention has been repeatedly recommended, and several clinical practice guidelines have been developed . AntibioAs with other bacteria, the skin and environment of a CD-colonized or infected subject become quickly contaminated and are the basis for CDI diffusion among other patients and healthcare professionals. To reduce this risk, all the guidelines for CDI prevention recommend that general infection control measures be systematically applied where potential patients are located ,18. TheyEarly diagnosis of CD carriage or CDI could prevent the spread of the pathogen and the development of recurrences. However, the optimum method of laboratory diagnosis remains controversial. In some cases, a two-step approach beginning with enzyme immunoassays for the detection of glutamate dehydrogenase (GDH) followed by a toxin test and/or a nucleic acid test is recommended. In other cases, a nucleic acid test alone is considered adequate . It is rStaphylococcus aureus and vancomycin-resistant Enterococcus [Hand washing with alcohol-based hand rubs or soap and water has been demonstrated to be very effective for the prevention of the transmission of several infectious diseases, including those due to rococcus ,27,28,29rococcus . Consequrococcus . Unforturococcus , is poorrococcus  and can rococcus ,34,35,36rococcus . Moreoverococcus . Regardless of the limitations of each of the measures for infection control, the prevention of CDI can be strongly influenced by healthcare worker compliance with contact isolation precautions. Several studies have clearly shown that compliance is generally poor . Adherenp = 0.0029). Different characteristics of the papers not included in the previous study and different methodologies used for analysis may explain this difference. The impact of the association of AS programmes with infection control measures was analyzed considering together the prevention of CD infection and the prevention of infection and colonization with antibiotic-resistant bacteria. This revealed that the association was more effective than AS alone , especially when AS was associated with hand-hygiene interventions .Antibiotic stewardship (AS), i.e., the implementation of programmes for the optimization of antibiotic use, is considered one of the most important measures to reduce the selection of host microbiota and the emergence of antimicrobial-resistant bacteria. Moreover, AS has been found to be associated with a significant reduction in antibiotic consumption, related adverse events, and costs for assistance in both hospital and community settings . Use of As antibiotic-related gut dysbiosis with the emergence of CD is the most important cause of CDI , it was Lactobacillus acidophilus + Lactobacillus casei than with Lactobacillus rhamnosus . Finally, probiotics were safe, well tolerated, and capable of reducing the risk of adverse events associated with antibiotic administration by 17% .A good example of the limited knowledge presently available regarding the prophylactic use of probiotics is given by some recent, well-conducted meta-analyses. The first of them is a Cochrane Review including 31 trials published up until March 2017 and 8672 individuals of different ages . Only 10p < 0.0001). However, compared to single-species probiotics, prophylaxis with multispecies probiotics was only slightly more effective, with a difference that did not reach statistical significance . Moreover, the effect was more evident when prophylaxis was prescribed in settings with a risk of CDI ≥ 5%. As several different preparations were used, differences in protection offered by the various probiotics could not be evaluated. However, comparison between Saccharomyces boulardii, the most frequently administered probiotic, and all the other bacteria did not show any difference (OR 0.86). Finally, serious adverse events were relatively uncommon and more frequent in subjects treated with high-risk antibiotics and in those receiving multiple antibiotic therapy. However, they were never considered attributable to probiotics and were equally distributed between subjects with and without probiotics in the unadjusted analyses  and adjusted analyses . A more recent meta-analysis of randomized controlled trials published up until April 2016 including 6851 participants and most of the studies already considered in the previous meta-analysis was mainly directed to evaluate the impact of probiotics on the risk of CDI after antibiotic administration . Most ofEscherichia coli and Shigatoxin-producing Escherichia coli that have occurred following use of FMT [As development of CDI, particularly recurrent cases, has been associated with significant variations of gut microbiota composition favoring CD colonization, restoring normal microbiota by fecal transplantations (FMT) from a healthy donor has been considered as a logical and practical solution to reduce the risk of new CDI episodes for many years . Howevere of FMT . StartintcdA and tcdB, respectively, and are located within a five-gene locus known as the PaLoc. A restricted number of CD strains produce a third toxin named CD transferase [Clinical manifestations of CDI are due to bacterial toxins. Most of CD strains produce two members of the large clostridial cytotoxin family, toxin A and toxin B, which are encoded by the genes nsferase . As highnsferase ,103,104,nsferase , it was nsferase . HoweverVaccines against CD are mainly intended for geriatric populations and adults at risk of CDI because of receiving frequent antibiotic treatment or with previous history of CDI. Among those based on CD toxins, a formalin-inactivated preparation of the toxins has been found safe and immunogenic in phase II clinical trials ,108. DesThe most advanced toxin-based CD vaccine is a recombinant preparation that contains full-length genetically and chemically inactivated toxins A and B ,110,111.Additionally, worthy of interest is a vaccine based on a recombinant fusion protein comprising relevant epitopes of CD toxins. Even in this case, a phase I study has shown good safety and tolerability and excellent immunogenicity in healthy subjects aged ≥18 years. Moreover, a phase II trial (NCT02316470) in which three doses of the vaccine were given to healthy subjects aged 50 to > 65 showed that seroconversion was achieved in up to 83% of participants for antibodies against both toxins and up to 97% against toxin A alone. Unfortunately, as of now, no phase III trial has been planned, as the manufacturing company is seeking a partner to perform it.Vaccines based on components of CD spores are in the very initial stage of development. Starting from the evidence that the inactivation of collagen-like exosporium protein BclA1 reduces the ability of CD to colonize mice , a vacciAnother CD vaccine candidate based on structural components of the exosporium layer of pathogen spores is the CdeM vaccine. CdeM is a protein that controls the assembly of the exosporium layer , and a vp = 0.06). The difference in recurrence rate reached statistical significance when patients with more than one previous episode of CDI were separately considered  [The administration of monoclonal antibodies against CD toxins A and B has been considered a potential measure to prevent CDI recurrences. Two different preparations, supposed to directly neutralize toxin A (actoxumab) and toxin B (bezlotoxumab), respectively, have been developed. Combined intravenous administration of these vaccines showed significant efficacy in an established hamster model of CDI . Moreove= 0.006) . However= 0.006) . As poor= 0.006) ,104, fur= 0.006)  enrollin= 0.006)  and the = 0.006)  only for= 0.006) . However= 0.006) .A particular aspect of passive immunization is oral administration of hyperimmune whey derived from colostrum or mature milk of cows immunized to obtain secretory immunoglobulin A against the two toxins, TcdA and TcdB, as well as CD. Studies carried out in experimental animals are promising as whey protected animals from death, in contrast to standard antibiotic therapy . HoweverCDI is a significant clinical, social, and economic problem.


The combination of CRISPR/Cas12a-assisted endogenous protein labeling and super-resolution microscopy represents a powerful tool for cell biological research with molecular resolution.Single-molecule localization microscopy (SMLM) reports on protein organization in cells with near-molecular resolution and in combination with stoichiometric labeling enables protein counting. Fluorescent proteins allow stoichiometric labeling of cellular proteins; however, most methods either lead to overexpression or are complex and time demanding. We introduce CRISPR/Cas12a for simple and efficient tagging of endogenous proteins with a photoactivatable protein for quantitative SMLM and single-particle tracking. We constructed a HEK293T cell line with the receptor tyrosine kinase MET tagged with mEos4b and demonstrate full functionality. We determine the oligomeric state of MET with quantitative SMLM and find a reorganization from monomeric to dimeric MET upon ligand stimulation. In addition, we measured the mobility of single MET receptors   •CRISPR/Cas12a enables endogenous protein labeling for super-resolution microscopy•HEK293T cells were generated with MET endogenously labeled with mEos4b•Quantitative PALM microscopy reports efficient dimerization of MET receptor•Single-particle tracking shows increased MET immobilization upon ligand treatment Biochemistry; Molecular Biology; Biophysics; Biological Sciences Research Methodologies Super-resolution microscopy enables the investigation of protein organization and dynamics at the nanoscale and opens the door for a molecular view on protein function in cells . Single-Several options of tagging target proteins with fluorescent proteins are available. Plasmids encoding fusion proteins can be produced synthetically in a short time and introduced into mammalian cells, e.g., by transfection. Transfected cells typically show a range of protein expression levels, which is particularly interesting for the analysis of diseases such as cancer that are associated with overexpression. However, if healthy conditions are to be studied, an endogenous expression level of the protein of interest is required . For thiin vivo (www.pcr-tagging.com). Compared to Cas9, the Cas12a enzyme requires only one crRNA that is expressed from a gene located on the PCR cassette. Generation of a gene-specific PCR cassette (by PCR) is thus quick and it can then directly be transfected into cells together with a Cas12a expression plasmid. The crRNA is expressed from the PCR cassette and can thus together with the Cas12a enzyme produce the site-specific DNA cut. The DNA cut is then repaired by homology directed repair of the PCR cassette, which labels the protein of interest with the desired tag. This makes PCR tagging a simple and time-efficient alternative for gene labeling with which high labeling efficiencies are achieved , which 321-ATTO 647N, which also targets the MET receptor and was characterized in earlier studies by single-molecule methods . In summary, these data demonstrate that we have generated a cell line that expresses fully active MET-mEos4b at an expression level of the wild-type MET receptor.Upon the verification of efficient labeling and functionality of MET, the next step was to investigate its correct localization at the cell membrane. For that, we performed PALM measurements of HEK293T wild-type cells and the stable cell line expressing MET-mEos4b. While wild-type cells showed only unspecific signal, cells expressing MET-mEos4b showed a clear punctate fluorescence signal at the basal cell membrane with total internal reflection illumination D–1F. We Stoichiometric and efficient labeling is particularly important for quantitative imaging of proteins with microscopy. We were explicitly interested to explore the potential of PCR tagging in quantitative super-resolution microscopy. Quantitative SMLM demands for high-efficient stoichiometric labeling and provides information on protein organization within nanoclusters in cells that is otherwise inaccessible . These nq-value of 0.35, which is slightly lower compared to that of CTLA4-mEos3.2 (q = 0.39)  to analyze the oligomerization of MET-mEos4b in its resting state and found 321-ATTO 647N  from the single-molecule “blinking” statistics A and S2B2 (0.32) , thus sh2 (0.32) C. Previo2 (0.32) . We gene2 (0.32) D and S2E2 (0.32) . Next, w2 (0.32) F and S2G = 0.39) . A q-val dimeric A and S2Hc  (321-ATTO 647N approximately doubled the immobile fraction to Receptor mobility was found to respond to ligand binding for various receptor tyrosine kinases . TherefosptPALM)  of MET-msptPALM) A. In consptPALM) B. ConsidsptPALM) . In rest)   measuremrecently . This enreceptor . The MSDIn summary, these results indicate that endogenous tagging by CRISPR/Cas12a in combination with quantitative super-resolution imaging and SPT is a powerful tool to address the molecular organization and dynamics of membrane receptors at an endogenous expression level.The generation of stable cell lines, that express fluorophore-labeled proteins at an endogenous expression level is time intense and often yields poor labeling efficiencies. CRISPR/Cas9 in combination with PALM was used in some studies to label intracellular proteins. The CRISPR/Cas9 method requires (i) a crRNA and tracrRNA expression vector and (ii) cloning of the donor template for homologous insertion of the protein tag. The generation of crRNA expression vectors is usually straightforward, however, requires up to one week for the complete verification of the generated vector. In addition, the synthesis of a donor template by, e.g., Gibson assembly, requires several assemblies to combine all important fragments such as the homologous regions, the tag and an antibiotic resistance . In compWe successfully generated a stable cell line expressing MET-mEos4b A–4E. Thi2) of MET-mEos4b in HEK293T cells. Higher cluster numbers were reported for endogenous MET in HeLa cells in DNA-PAINT measurements (www.proteinatlas.org), we expect a lower receptor cluster density for HEK293T cells, which fits to our PALM data and supports an endogenous expression level.Further, we verified the activity of the MET-mEos4b fusion protein by western blots. To highlight the various benefits of PCR tagging, we analyzed endogenously tagged MET as transient transfection can induce artifacts due to non-native expression levels . We showurements . Howeverin vivo  , where tin vivo  and cutsin vivo .321, we observed a similar increase in dimerization. In previous single-molecule photobleaching experiments, a lower degree of MET dimerization was observed in HeLa cells  imaging mode.heileman@chemie.uni-frankfurt.de.Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Prof. Dr. Mike Heilemann, Additional information including reasonable requests for materials such as plasmids should be directed to and will be conducted by the Lead Contact.The data included in this study are available from the Lead Contact on reasonable request.All methods can be found in the accompanying