Populist parties enjoy stable support in various European countries. The literature on the rise of populism argues that this support especially increases in times of crises. Surprisingly, the German right-wing populist Alternative für Deutschland (AfD) did not increase its support in the wake of the COVID-19 pandemic. Moreover, the party even lost 2.3 percentage points in the 2021 federal election. We address this puzzle and ask why the AfD has not been able to use the crisis to its advantage. Our main argument in answering this question is that, although the AfD pursued the classic populist strategy of fundamental opposition, the support base of the AfD is strongly divided on the preference towards measures containing the spread of COVID-19. This division is reinforced by individual affectedness by the pandemic. Introducing a novel weekly dataset on voter preferences, we show that the AfD support base is strongly divided on the issue with approval of the government measures being a significant and substantial contributor to vote switching away from the AfD. Using regional-level data and a difference-in-differences approach, we further show that western German regions hit especially hard by the pandemic display a lower AfD vote share than other regions. Our findings have important implications for the impact of exogenous shocks on electoral competition and also on the future of populist parties. Alternative für Deutschland (AfD) is losing, not gaining, both in polls and in the 2021 federal election. Focusing on the AfD and the 2021 federal election, our paper answers the question: Why is the COVID-19 crisis not having a positive effect on AfD support? We examine in detail the policy preferences of AfD voters, the attitudes exhibited by individuals who previously supported the AfD but now indicate a different voting intention, and how the AfD performed in regions hit especially hard by the pandemic.The successes of populist parties is often attributed to economic, social, or political crises  supporters towards the government’s COVID-19 measures have contributed to the electoral losses of the AfD or whether the federal election outcome is just a result of the parties stagnating vote share in general. We provide evidence for this effect by outlining a theoretical mechanism and running regression analyses on why the AfD has lost supporters over the course of the pandemic as well as by running difference-in-differences regressions to control other sources of the loss of votes.Wordfish tool introduced by Slapin and Proksch  radical right-wing ideology and public health, as this ideology carries the notion of an imagined healthy ‘national body’ in which people with illnesses and disabilities are excluded from the imagined ‘people’ Schäfer . SolidarThe second commonality between AfD support and rejection of policy measures taken by the government against the COVID-19 pandemic is the fundamental opposition towards government policies, which is rooted in the political distrust as well as disaffection of voters, and captured by the anti-elite rhetoric of the AfD  increases the preference for the implementation of measures to counter the pandemic, (2) raises the salience of the COVID-19 pandemic in the run up to the election even further, and (3) favors the development of a rally ’round the flag effect that draws away support from any opposition party towards the governing parties and their policy response. Based on this, we expect more individuals to switch from the AfD to another party in the regions especially affected by the pandemic, i.e., counties that experienced a more severe pandemic. We test this argument with the following hypothesis:The worse a county is hit by the COVID-19 pandemic, the higher the loss of votes for the AfD.In conclusion, we make three arguments with respect to why the AfD has so far failed to use the crisis for its own benefit. First, the voters of the AfD are internally divided on their policy preferences. Second, voter–party incongruence makes previous supporters withdraw their support from the AfD and vote in accordance with their preferences. Third, the likelihood of withdrawing support from the AfD is higher in regions especially affected by the pandemic, leading to a higher vote share decline in these regions.In order to test the first and second hypotheses, we use a novel and comprehensive weekly survey data on voter attitudes toward government measures to combat the COVID-19 pandemic in 2020 and 2021. The third hypothesis is tested with data on regional infection rates and vote share changes of the AfD between 2017 and 2021. Descriptive insights based on these data are provided in the following sections.To assess voter attitudes to COVID-19 measures, we employ the ‘Trendfragen Corona’ survey by the Press and Information Office of the Federal Government of Germany . The survey was launched in week 12 of 2020 as a weekly survey and was later reduced to a bi and quadweekly survey.Figure Figure The figure again strikingly shows the division among the supporters of the AfD with 53% against and 47% in favor of the measures taken by the government. Even more interesting, the respondents who indicated a switch to another party display a clear pattern of favoring the measures by the government. Only around 22% of the switchers state that they are against the measures, while the vast majority of switchers, i.e., 78%, is in favor of the government’s policy response. Not only does this finding further support our first hypothesis on the internal division of the AfD’s voter base, it also sheds some first light on the second hypothesis according to which the support of measures is correlated with withdrawing support from the AfD. Additionally, it draws attention to the fact that a considerable amount of AfD voters is against the COVID-19 measures. Even more importantly, the group of stable supporters also consists of voters that are in favor of the government’s measures. Based on the assumption that voters select parties based on issue proximity, this can only be explained by the fact that these voters seemingly place a greater importance on the AfD’s policy position towards the party’s core issues of migration and nationalism despite these issues being less salient in the COVID-19 pandemic.To test our second hypothesis, we utilize data on the federal election vote share of the AfD on the regional level. For 2017, we retrieve the county-level AfD vote share from the Federal Election Commissioner’s Office. For 2021, the data has yet to be released from this source, so we based our analysis on the pooled county-level data compiled by ZEIT Online and Fusionbase.To compare the performance of the AfD, we calculate the vote share difference between 2021 and 2017 for each county. Motivated by various contributions that underscore the differences in the electoral support between eastern and western Germany  on the individual voter’s attitude towards the COVID-19 measures by the government (Attitude) using a logistic regression. We account for the panel-like structure in our data by including week-fixed effects. As we assume our sample to be rather heterogeneous, we employ robust standard errors. We also control for possible confounding variables by including a respondent’s gender , age (Age), employment type (Worker), education (Low education), and income (Low income). We also control for the differences between eastern and western Germany through region fixed effects. The baseline model is defined by: Switching is the vote switching away from the AfD of an individual Attitude). Additionally, with Moving to the econometric models, we test our second hypothesis by calculating the probability that a voter who voted for the AfD in the 2017 election supports a party other than the AfD when asked about his or her voting intention in the 2021 election. We regress this vote switching variable (VoteShare) on the regional level between 2017 and 2021. The change in vote share is regressed on the average 7‑day incidence in the respective county (Incidence). Since both variables follow a normal distribution, we employ OLS estimators. We again use robust standard errors and state fixed effects. Motivated by the difference found between eastern and western Germany, we interact the incidence variable with an eastern Germany dummy variable. Apart from this, we also run separate regressions for eastern and western German counties. The baseline model is defined by: The second set of regression models analyzes the correlation between region specific severity of the pandemic and voter migration away from the AfD, as formulated in the third hypothesis. The response variable is the change in the AfD’s vote share (The third set of regression models takes this approach one step further and applies a difference-in-differences (DiD) identification strategy Abadie . For theVoteShare is the AfD vote share in a given region (Treated) and time (Time) dummy, as well as their interaction term , the mean (61), and the mean plus one standard deviation (80). We calculate the DiD estimator by running an OLS regression with a treatment and a time dummy variable. The time variable is 1 if the year is 2021 and 0 if the year is 2017. The treatment variable is 1 if a county’s average 7‑day incidence is above 72. The treatment effect is determined by the interaction between the treatment and time dummy variable. For the difference between eastern and western Germany, we include an additional eastern Germany dummy and add an interaction term with this dummy. The baseline model is defined by: The results of the set of regression models concerned with the correlation between attitudes towards the measures and vote switching (Hypothesis 2) are displayed in Table The next model introduces week and region fixed effects. This does not change the statistical significance of the coefficient and increases its size only slightly. The following models gradually introduce the control variables, while keeping the week and region fixed effects. While the dummy variable concerned with male respondents has no effect on the size and significance of the coefficient of the attitude variable, the inclusion of the age variable is accompanied by a slight drop in the size of the coefficient. In the next model, the size of the coefficient is further reduced by including the dummy variable that controls for employment status. Both control variables are significant, indicating that a higher age increases the probability of vote switching, while being employed as a worker decreases the probability. The low education and income control variables included in the last two models are not statistically significant and only slightly decrease the size of the coefficient. In sum, the coefficient of the main variable of interest is negative and statistically significant throughout the different model specifications. The size of the coefficient is also fairly stable. From this, we can conclude that supporting the COVID-19 measures taken by the government is correlated with an increased probability of vote switching away from the AfD. This finding lends credible support for our second hypothesis.ceteris paribus a 34.4% probability of vote switching if he or she feels that the measures should be more far reaching. This predicted probability is visualized in Fig. In order to analyze whether this correlation is not only statistically significant but also substantially relevant, we estimate the predicted probability of an AfD voter in the 2017 election withdrawing support in the 2021 election, depending on the attitude towards the COVID-19 measures. While a respondent who voted for the AfD in 2017 and who indicated that the measures are too far reaching has only a predicted vote switching probability of 16.6%, the same respondent has Moving from the second to the third hypothesis, we now analyze the correlation between the severity of the pandemic and AfD voting on the county level. The results are displayed in Table Surprisingly, the coefficient becomes statistically significant in the following models when employing a sample split between western and eastern Germany. Even more surprising, and replicating the previous descriptive findings, the correlation is negative in the western German sample and positive in the eastern German sample. This means that regions with high average 7‑day incidences witnessed a decline in AfD voting in western Germany, while the opposite is true for eastern Germany. The next two models introduce state fixed effects. Including the fixed effects slightly decreases the size of the coefficient in the western German sample and more than doubles it in the eastern German sample. The statistical significance is not impacted negatively by including the fixed effects. Based on these findings, the last model is run in the full sample but with an additional interaction term between an eastern Germany dummy and the incidence variable. These results replicate the previous finding from the sample split, with a negative correlation in case of the eastern Germany dummy being 0, i.e., the western German sample, and a positive correlation in case of the eastern German dummy being 1, i.e., the eastern German sample. This finding partially supports the third hypothesis, as we find a negative impact of the COVID-19 incidences on the AfD vote share, but this effect is exclusively restricted to western Germany.After having established the statistical significance of the correlation between negative vote share differences and high average incidences in western Germany and a positive vote share difference and high average incidence in eastern Germany, we again want to analyze the substantial relevance of the correlation found. In order to do so, we calculate the predicted change in the AfD’s vote share in eastern and western German counties in the case of a one-standard deviation increase (18.4) of the average 7‑day incidence. In western Germany, a one-standard deviation increase is correlated with a 0.5-standard deviation decrease (0.69 pp.) in the AfD vote share difference. In eastern Germany, a one-standard deviation increase correlates with a 0.2-standard deviation increase (0.67 pp.). The substantial relevance of this correlation is further illustrated in Fig. In the last step, we employ a DiD approach to further analyze the correlation between the average 7‑day incidence and changes in the AfD’s vote share. The response variable is the AfD’s county-level vote share. The explanatory variable of interest is the interaction term between treatment and time dummy. The results of the regression analysis are displayed in Table The following models again employ a sample split with and without state fixed effects. Other than in the previous simple OLS regression, the DiD approach does not report a positive and statistically significant correlation of the treatment in eastern Germany. Contrary to this, the coefficient is again negative and statistically significant in western Germany. In the last model, we turn to the full sample with an eastern German dummy variable interaction term. For western Germany, the interaction term between time and treatment remains negative and statistically significant, replicating the previous finding. In contrast, the interaction term for eastern Germany has a positive coefficient that is statistically significant only at the 0.1 level. From this follows that the treatment of experiencing a particularly high incidence is correlated with a decrease in the AfD’s vote share in western Germany but not in eastern Germany. If anything, the treatment of a high average incidence is correlated with an increase in the AfD’s vote share in eastern Germany.After having established the robust statistical significance of the correlation in western Germany and the less robust correlation in eastern Germany, we again turn towards calculating the marginal effects. The marginal effects for the DiD regression are displayed in Fig. Focusing on the eastern German counties, we find that the AfD vote share in 2021 (gray diamonds) is higher than the vote share in 2017 (black dots). Other than in western Germany, the difference is larger in the treated counties (right-hand side) than in the not treated counties (left-hand side), although this positive difference is not statistically significant at the .05 level. From the regression analysis, we also know that the difference between the observations is only statistically significant at the 0.1 level. In terms of substantial relevance, we can conclude that experiencing a particularly high incidence is correlated with a reduction in the AfD’s vote share by about 3 percentage points in western Germany. In eastern Germany, however, being particularly affected by the pandemic, it is, if anything, correlated with an increase in the AfD’s vote share also by about 3 percentage points.In conclusion, our empirical analysis finds considerable support for the above derived hypotheses with one exception. We found that the AfD consistently took a position outside the political consensus of other parties throughout the course of the pandemic and that the voters of the AfD show a distinct internal division regarding their attitudes towards the measures taken against the COVID-19 pandemic. The logistic regression analysis also supported the hypothesis that being in favor of the government measures significantly increases the likelihood of vote switching away from the AfD. The hypothesized correlation between particularly high incidences and the loss of votes of the AfD, however, could only be found in western Germany. Contrary to this, in eastern Germany, the correlation is reversed and considerably weaker in statistical terms, leading us to partially reject the last hypothesis.In the robustness section of our analysis, we focus on the previous DiD results and establish different treatment cutoffs. In detail, we run three additional regression analyses. The treatment cutoff points for these regressions are 49 (lower 25th percentile), 61 (mean), and 80 (mean plus one standard deviation). The regression with the cutoff of 49 is displayed in Table Moving to the DiD regression with the cutoff of 80 displayed in Table Additionally, we again plot the marginal effects for the DiD robustness check regressions that use the cutoff of 80. The results are plotted in Fig. Lastly, it is important to address other long-term developments that could potentially affect the vote share of the AfD. The AfD was also not able to substantially gain votes in the state legislative elections prior to the federal election of 2021. Our proposed theoretical model and empirical analysis does not propose that the AfD’s stagnating election results can be entirely attributed to its performance in the COVID-19 pandemic. Rather, we provide theoretical arguments as to why voters might  withdraw their support from the AfD due to incongruences between the demand and supply side. Methodologically, this creates an identification problem, as it is initially not clear whether the loss of votes is related to a long-term decline of populist support or are at least partially rooted in the pandemic shock and the populist response to it. We address this issue in two ways. First, we provide evidence on the individual level that people are more likely to turn away from the AfD with increasing support of the COVID-19 measures introduced by the government. At the same time, it is unlikely that people supporting the government measures are systematically more aware and dismissive of the AfD’s scandals, internal division, or public perception. Second, the same is true on the regional level, as a bias in the DiD regression could only arise, if the AfD’s lack of support would be positively correlated with higher infection rates.In sum, the robustness checks concerned with the DiD regression further strengthen our confidence in our previous findings. While we find strong and robust support for a negative correlation between pandemic severity and AfD support in western Germany, we cannot reach the same conclusion for eastern Germany. Although the correlation found for eastern Germany is substantial, the positive correlation has to be treated carefully due to the lack of robust statistical significance.In this paper, we asked why the COVID-19 crisis does not have a positive impact on AfD support. Our main argument in answering this question is that the AfD lost voters because its support base is internally divided on the position towards the government’s COVID-19 measures. In detail, we argued that other than GAL-TAN issues like migration, EU integration, or nationalism, around which the AfD has built its support base, the issue of COVID-19 poses a cross-cutting issue that does not easily align with established dimensions of electoral competition. Based on this, we argued that although some voters of the AfD will keep supporting the party based on their rejection of the COVID-19 measures, preference for other positions supplied by the AfD or general detachment from the political system, others will switch their support and vote in accordance with their COVID-19 policy preference. While we reason that these voters will have a higher likelihood of vote switching, we also argued that this trend should be especially observable in regions hit particularly hard by the pandemic. From this, we derived three hypotheses. First, the voter base of the AfD is internally divided on how to react to the pandemic. Second, voters with preferences diverging from the AfD’s policy position on COVID-19 have a higher likelihood of vote switching. Third, the more strongly a region is hit by the pandemic, the higher will be the AfD vote loss in that region.We tested the hypotheses with novel and comprehensive survey data on voter attitudes, infection rates, and voting behavior and intention on a weekly basis in 2020 and 2021. In detail, we used micro-level survey data on attitudes toward COVID-19, previous vote choice, and future voting intentions. Additionally, we included county-level data on the federal election results of 2017 and 2021, as well as data on the county-level 7‑day COVID-19 incidence rate.The first hypothesis was evaluated with a descriptive analysis of voter attitudes. Our results suggest that the AfD’s support base shows a strong and unparalleled internal division on the preferences towards COVID-19 policy measures. Taking this finding one step further, running logistic regression analysis supported our proposition that voters with policy preferences diverging from the AfD’s position were significantly more likely to withdraw their support from the AfD. Finally, we used OLS regression analyses and a DiD approach to examine whether the severity of the pandemic had an impact on the AfD’s election results in the 2021 election. As discussed, we were able to show that this is the case for western Germany. However, we did not find a robust, statistically significant correlation in eastern Germany. Thus, our third hypothesis has be partially rejected.Despite employing robustness checks and implementing a fairly robust identification strategy, our analysis has some weaknesses and blind spots, which increases the need for further research of the relationship between the loss of votes for the AfD and the COVID-19 pandemic. An important issue that we have to address is reverse causality. Recent research has found that AfD strongholds in eastern and western Germany in the 2017 election are also the counties that have been hit especially hard by the pandemic (Richter et al. However, while we can arguably rule out the possibility of reverse causality in western Germany due to the direction of the hypothesized and found correlation, this is not the case for eastern Germany. In eastern Germany, reverse causality could be an issue but, of course, only against the backdrop of the nevertheless not statistically robust correlation. Based on the DiD regression, this would still mean that additional voters switched to the AfD in the wake of rising 7‑day incidence rates, which were, in turn, also carried by AfD voters. Hence, reverse causality is not as big of an issue for our analysis as it might seem at first glance. Nonetheless, upcoming state elections should be utilized in future research for a more robust empirical identification strategy that also controls for reverse causality, especially in eastern Germany.Regarding the generalizability of our results, we can only make suggestions on how the mechanism might work in other countries. This goes especially for populists in government, who have seemingly mishandled the pandemic but are to a large extent still in office (Bayerlein et al. Lastly, the uncovered difference between eastern and western Germany demand more research on why the AfD has not lost support in eastern German counties. This question further underscores that different mechanisms of political support are at work when comparing eastern Germany to western Germany. This is, of course, in line with existing literature that already provides diverging findings for eastern Germany and western Germany concerning political attitudes and AfD support (see, e.g., Betz and Habersack Irrespective of the remaining questions, we can conclude that a considerable number of AfD supporters jumped ship in the wake of the COVID-19 pandemic. This finding has important implications for the impact of exogenous shocks on electoral competition and also on the future of populist parties. In our paper, we have provided evidence that exogenous shocks like the COVID-19 pandemic have the potential to shake up the party system, as they are usually cross-cutting issues that increase the likelihood of voter realignment. Additionally, we were also able to show that populist party support is not based on blind allegiance, but is, at least for some voters, rooted in their policy preferences. If these preferences are not in line with the populist party’s position and also become salient, voters seemingly do not hesitate to defect from populist parties. The downside of this is that it took an existential crisis for some voters to realize that sailing on with a populist party could not lead to the promised land but maybe to certain doom.


Current coastal spatial planning in Sweden uses simple methods to account for how flood risks increase owing to sea level rise. Those methods, however, fail to account for several important aspects of sea level rise, such as: projection uncertainty, emission scenario uncertainty and time dependence. Here, enhanced methods that account for these uncertainties are applied at several locations along the coast. The relative importance of mean sea level rise and extreme events for flood risk is explored for different timeframes. A general conclusion for all locations is that, extreme events dominate the flood risk for planning periods lasting a few decades. For longer planning periods, lasting toward the end of the century, the flood risk is instead dominated by the risk of high sea level rise. It is argued that these findings are important for assessments of future flood risk, and that they should be reflected in coastal spatial planning. The global mean sea level rose faster during the last century than during any preceding century in the last 3000 years, and even higher rates of sea level rise are projected for this century  at our six tide-gauge stations is considerably lower than the high tide at many locations in the neighbouring North Sea  affect the probability that a given area will be flooded. This paper extends the results of  distributions fitted to the tide-gauge data from each station. A schematic of the flood risk simulator framework is given in Fig. eronymus . Uncertaeronymus . All simThis joint distribution can be used to determine the probability that a given height above the current mean sea levels will be flooded at least once during the planning period as a consequence of the joint effect of mean sea level change and extreme sea levels. Our analysis focuses on levels above the current mean sea level that has a 1 in 10 000 probability of being reached during the planning period. The usage of this probability level is inspired by the recommendations of the Swedish National Board of Housing, Building and Planning Boverket . HoweverThe flood risk simulator calculates the height of 1 in 10 000 joint sea level events. Joint meaning here that they are caused by both mean sea level change and sea level extremes. The individual heights of the mean sea levels and sea level extremes that contribute to these events are also calculated. Moreover, the simulator calculates the probabilities individually for seeing, at least once within a planning period, a mean sea level and a sea level extreme at least as high as those that contribute to the joint 1 in 10 000 sea level events.Figure p, q] combination. However, as the planning period gets longer the situation reverses. Sometimes almost completely so that a joint 1 in 10 000 sea level event is composed of a 1 in 10 000 mean sea level co-occurring with an extreme sea level that has a high probability of occurrence. In fact, for the longest planning period, 1 in 10 000 joint sea level events are composed of a very low probability mean sea levels and high probability extreme sea levels at almost all stations and  combinations. The only exception is the All stations show a qualitatively similar pattern with respect to length of planning period. For the shortest planning period, 1 in 10 000 joint sea level events are essentially composed of a 1 in 10 000 extreme sea level that co-occurs with a mean sea level that is close to its expected value for the chosen [nd Kalén  showed tThe height above the current mean sea level of 1 in 10 000 joint sea level events and the height of their mean sea level and extreme sea level components are shown in Fig. eronymus  it was sThe heights of the mean and extreme sea level components of 1 in 10 000 joint sea level events change in similar ways at all stations, as functions of the length of the planning period. For short planning periods, 1 in 10 000 joint sea level events are composed of very high extreme sea levels and moderate mean sea levels. Conversely, for long planning periods such events are composed of very high mean sea levels and more moderate extremes. For four of these six stations the mean sea level component is even larger than the extreme sea level component of the joint sea level events when the planning period ends in the year 2100. The dominance of the mean sea level component in these long planning periods owes not only to its monotonic increase with the length of the planning period, but also to a decrease in the extreme component. That is, the extreme components of these events become lower by some tens of centimeters as the planning period goes from 2030 to 2100. Moreover, the yearly probability of the extreme component of these joint sea level events increases from around The reason for this behaviour is that the distribution of plausible future mean sea levels expands its range rapidly as the length of the planning period increases, in particular for high emission scenarios  to a tide-gauge-based return level, which usually corresponds to a return period between 100 and 10 000 years. Sometimes an extra safety margin of arbitrary amount is also added . We argue that this general pattern should be considered in future coastal spatial planning. Optimally, this should be carried out using sophisticated tools such as the flood risk simulator used in this study. However, simple method like those in use today could be enhanced to account for the essence of our findings by basing short term planning on high extreme levels and moderate sea level rise and long term planning on moderate extremes and high sea level rise.In conclusion, we set-up the flood risk simulator (Hieronymus


Cancer evolution consists of a stepwise acquisition of genetic and epigenetic changes, which alter the gene expression profiles of cells in a particular tissue and result in phenotypic alterations acted upon by natural selection. The recurrent appearance of specific genetic lesions across individual cancers and cancer types suggests the existence of certain “driver mutations,” which likely make up the major contribution to tumors’ selective advantages over surrounding normal tissue and as such are responsible for the most consequential aspects of the cancer cells’ gene expression patterns and phenotypes. We hypothesize that such mutations are likely to cluster with specific dichotomous shifts in the expression of the genes they most closely control, and propose GMMchi, a Python package that leverages Gaussian Mixture Modeling to detect and characterize bimodal gene expression patterns across cancer samples, as a tool to analyze such correlations using 2 × 2 contingency table statistics.Using well-defined simulated data, we were able to confirm the robust performance of GMMchi, reaching 85% accuracy with a sample size of n = 90. We were also able to demonstrate a few examples of the application of GMMchi with respect to its capacity to characterize background florescent signals in microarray data, filter out uninformative background probe sets, as well as uncover novel genetic interrelationships and tumor characteristics. Our approach to analysing gene expression analysis in cancers provides an additional lens to supplement traditional continuous-valued statistical analysis by maximizing the information that can be gathered from bulk gene expression data.We confirm that GMMchi robustly and reliably extracts bimodal patterns from both colorectal cancer (CRC) cell line-derived microarray and tumor-derived RNA-Seq data and verify previously reported gene expression correlates of some well-characterized CRC phenotypes.https://github.com/jeffliu6068/GMMchi.The Python package GMMchi and our cell line microarray data used in this paper is available for downloading on GitHub at  Gene expression data are one of the most widely used tools in multiple aspects of understanding cell biology. There is a range of tools developed to evaluate expression levels including the microarray and RNA-seq technologies. Gene expression data produced across all platforms inherently results in a continuous-valued dataset representing the levels of expression of individual genes. The field of computational gene expression analysis is thus an analysis of continuous distributions and the correlative understanding of between genes or sets of genes activities within a biological context .In cancer biology, tumorigenesis is driven by stable or meta-stable alterations at the genetic or epigenetic levels. Many of these changes probably lead to minor changes in the phenotype of a cell and minimal, if any, reproductive advantage. Some genetic mutations or meta-stable epigenetic expression changes that are common across different types of cancers do however lead to phenotypes that confer considerable reproductive advantages characteristic of a tumor, such as uncontrolled proliferation countering the control of normal differentiation, anti-apoptosis, and immune evasion . These mA single selectively advantageous mutation or epigenetic change may often lead to changes in the expression level of a whole set of other genes. This will give rise to a cluster of gene expression changes associated with such a particular mutation or meta-stable epigenetic change. The expectation is then that each expression distribution of a gene in such a cluster, over a range of cancers, will be bimodal with the mutation or epigenetic change associated with one mode and the corresponding normal version with the other mode of the bimodal distribution .The more closely a change in a gene expression level is associated with a given mutation or epigenetic change, the more likely it is to give rise to a bimodal distribution differentiating wild-type and mutation. More generally, on the other hand, the more striking is the distinction between two particular phenotypes over a range of cancers, the more likely it is to have a measurable effect on the biology of the cell, and the more likely it is to be due to a single mutation or stable epigenetic change. This is simply an extension of the principle by which Mendel established his laws of inheritance. Thus, the more distinctive is the bimodality of the expression of a gene over a range of cancers, the more likely it is to be due to, or strongly associated with, a genetic mutation or meta-stable epigenetic change that is giving an advantage to the outgrowth of the tumour, namely a cancer driver. The interest is then to identify striking bimodal gene expression distributions, over a given range of cancers, that lead back to novel mutations or epigenetic drivers. Here this is approached by fitting mixtures of two normal distributions, called Gaussian Mixture Models or GMM, to the gene expression data , 18.2 from micro-array data on the x axis. The distribution for the gene GJC2 (coding for a gap junction protein) is clearly unimodal and well fitted by a single normal distribution, while that for gene CDX1 (coding for a homeobox transcription factor) is clearly well fitted by a mixture of two normal distributions. One of the major challenges for GMM is to test the validity of the assumption that the data either fit a unimodal normal distribution or a mixture of two normal distributions. In Fig. CDH1, encoding E-cadherin, only fits a normal distribution at its upper end, while having a 'tail' of lower values that do not fit a normal distribution. We coined the phrase the ‘tail problem’ to describe a non-normally distributed tail that is inadequately dealt with by traditional GMM which assumes input data are all expected to be mixtures of normal distributions. In this paper, we describe a systematic approach to a novel GMM method capable of identifying normal and non-normal components across data on samples of any given form of expression data and give examples of the method's application to expression data on a panel of CRC derived cell lines.As shown in Fig. Develop an unsupervised method for determining whether a gene’s expression pattern across a set of cell or tissue samples is unimodal or bimodal. If the expression data are a mixture of normal and non-normal components, separate the samples into normal and non-normal components before making the appropriate cutoff separating the samples into individual subgroups.For genes with bimodal expression patterns, analyze their patterns of association into sub-group clusters using 2 × 2 contingency table analysis and explore the associations of these clusters with gene mutations and biological properties, such as patterns of differentiation.The primary objectives of this analytical approach are twofold:This analytical pipeline takes standard normalized and batch-removed gene expression datasets as input obtained either from microarrays or by RNA-Seq.Gaussian Mixture Modeling (GMM) is performed using the sklearn.mixture package in Python. The expression pattern of each gene or probe set is examined as a univariate distribution across the set of samples in the dataset. One or two Gaussian distributions are fitted to each distribution, in the latter case allowing for unequal sub-distribution fractions and variances, using the expectation–maximization algorithm built into sklearn.mixture.GMM. For bimodal gene expression distributions, each sample is given a value 1 or 2 according to the sub-distribution it more likely belongs to, based on the posterior probability outputted by GMM.M with G components, with maximizing parameters Model selection, in particular the choice of either one- or two-Gaussian fitted distributions, is based on maximization of the Bayesian Information Criterion (BIC) defined in Eq.  or, its rameters . The BICrameters , 16. BICanalysis .2 transformation [The challenges faced with directly applying GMM to expression data analysis and the identification of bimodal distributions lie within the nature of gene expression datasets. Although expression data like many other datasets mostly behave as Gaussian distributions after logormation , a large0 diagonal lines when plotted against the theoretical normal distributions, indicating normality for both components of the sample distribution. On the other hand, in Fig. An example of the tail problem in the returned output of GMM is illustrated in Fig. To deal with the tail problem we need to distinguish between a good GMM fit of two normal distributions with different variances and a poor over fit due to a mixture of one or two normal distributions with non-normal tail distributions. To make this distinction we add iterative Chi-square fitting to GMM and call this GMMchi. There are four steps to this iterative optimization process: dynamic binning, tail identification, iterative tail trimming, and a final condition criterion. We now address each step of the pipeline in turn.The most important factors that determine how a series of individual data on a single continuous measurement are presented in a histogram are the number of bins and their widths. This is a particular problem when the number of individual observations is not very large. Both the bin width and the number of bins create problems for the iterative testing of the fit of the observed data to estimated normal distributions.To create a fully automated pipeline, we have incorporated the Mann and Wald bin criterion  defined b is the number of bins while n is the number of samples. This method provides a consistent approach to deciding on the number of bins that best represents the data.Here b. Using b and the range of our data we can then propose that initially the width of our bins be Use the Mann and Wald bin criterion to determine the number of bins Quantify the number of observations per bin.Starting from the upper and lower ends of the distribution, combine adjacent bins that have < 5 observations until all bins have at least 5 observations.Next, since the iterative process within GMMchi is based on the chi-square goodness of fit test as the main criterion for measuring the fit of the mixed normal distribution model to the observed data, it is important for the validity of the Chi-square test to have at least 5 measurements in each bin. This is achieved by an algorithm called dynamic binning, which involves automatically combining bins while applying the least manipulation to the histogram for ensuring optimal results of the underlying chi-square test within GMMchi. The three-step process is as follows:The results of dynamic binning are illustrated in Fig. 2 value for the remaining points as well as the BIC will be calculated to determine the best fitting outcome. The χ2 value is defined in Eq.  or a mixture of non-normal tails and normal distribution(s). If the data fit the non-normal tail criterion, GMMchi will iteratively remove one data point at a time from the extreme end of the tail while fitting the remaining points with GMM. At each tail point removed, the χd in Eq. :4\docume2). To determine at any given point in the pipeline whether there is overfitting of the model with a non-normal tail problem, we have devised a criterion based on the goodness of fit testing.To determine whether there is a non-normally distributed tail the overall input data will first be fitted by GMM after applying dynamic binning. The returned output of the GMM consists of the number of normal components fitted to the data based on BIC, the posterior probability of each data point (x), and the characteristics of each component distribution, namely the mean (µ) and the variance , so that the computed chi-square values are comparable across different iterations of the pipeline. This is an important attribute that allows the comparison of the model fit across sequential iterations when selecting the best fit.2 threshold value for tail detection, we plotted the distribution of the χ2 values of all our GMM bimodally fitted genes in the 78-cell line panel, as shown in Fig. 2 values for the estimated distribution are given in Eq.  with a non-normal tail that should be subjected to iterative tail trimming.At each iteration, we begin by removing one data point from the tail, perform dynamic binning to adjust the appropriate number of observations per bin, and fit the resulting distribution with GMM.2) using parameters from the output to determine how well the revised model fits the input data.Then, we determine the chi-square goodness of fit (χ2n and χ2(n−1), the χ2 values of the current iteration (n) and the previous iteration (n − 1), as well as the BIC score to determine whether the model is fitting better than the previous fit. We use the BIC score to determine whether a lower χ2 value is also the least complex model with the best fit.We next compare χ2 and BIC.The algorithm iterates (1), (2), and (3) until no other conditions produce a lower χIterative tail trimming (ITT) is an optimization process where the algorithm searches for the best fitting datapoint where the cutoff can be made to separate the tail from the rest of the normally distributed data. The process is as follows:2 values representing the fit of the model at each iterative step. The lower the χ2 value the better the model fits. Even though BIC is a criterion that is equal to the χ2 value with a penalizing term, as shown in Eq. , there is another important factor to consider before applying ITT. There are four basic possible distribution patterns a tail problem can give rise to in a GMMchi, as illustrated in Fig. 2 value is > 2 5.52 and fitting the remaining data with one or two mixtures of Gaussian distributions, covering all four possible scenarios mentioned above. The cutoff that determines where each tail starts is based on the intersection of the two overlapping normal distributions. This ensures all scenarios are well-considered within our model and the output represents the optimal tail trimming of the best fitting model. Possible scenarios including unimodal or bimodal distribution with both upper and lower tails are not considered in GMMchi due to their rare occurrence in gene expression data.The algorithm takes all possible scenarios listed above into account in performing ITT on the data. This process runs ITT removing data from both low or high tails if the χThe final criterion is determined by identifying the lowest chi-square value, which indicates the best fit, and the lowest BIC score, which suggests the least complex model, across all 4 possible scenarios. This results in the definition of the tail data points and so the definition of the boundaries between the estimated normal distributions and the tails. The last step, when there is a well fitted bimodal distribution after omitting the tail(s), is to show the boundary between the high and low values of these two distributions by their point of intersection.The resulting best fit can be either the original fit, indicating that a mixture, usually of two normal distributions, best represented the data, or a new fit consisting of a mixture of normal distributions and non-normally distributed tails.The output of GMMchi includes the categorization of input data across all samples and a set of the means and variances that describe the fitted distributions.2 value reached after removing 9 datapoints from the tail.The visual output consists of 4 graphs as in Fig. To test the robustness and sensitivity of the GMMchi pipeline, we performed two separate simulations.Unimodal case: Mixture of 1 normal distribution + 1 non-normal tailBimodal case: Mixture of 2 normal distributions + 1 non-normal tailThe purpose of this simulation is to assess how effective GMMchi is at identifying a simulated mixture of normal and non-normal tails with an increasing spread of the tail. The spread or size of the tail is defined as its range. There are two components to this simulation:m and std are respectively the mean and standard deviation of the normal component. This ensures that the tail starts at the upper end of the distribution when there is just one normal distribution. For 2 normal distributions, we use the m and std of the upper distribution and start the tail at the upper end of this second distribution. The upper bound is determined by n is increased incrementally from 1 to 12 to give a larger spread of the tail with each set of simulations as n increases. We then generate 5000 mixtures of normal components and a non-normal tail for each of the 12 simulations as n increases from 1 to 12. GMMchi analysis is carried out on each of the 5000 simulated mixtures for each of the 12 sets of simulations to compare the GMMchi assessment with the known distribution based on the chosen parameters m, std, the number of data points and the proportion of data points in the tail. The accuracy for each n is then the percentage of correct categorizations.We randomly generate normal distributions choosing means at random between 5 and 10 and standard deviations at random between 0.5 and 5. The total number of data points is chosen at random to be between 50 and 150. The number of data points in the tail is randomly chosen to be between 5 and 25% of the total number of data points. To test how well GMMchi performs with an increasing spread of the tail, we choose the lower bound of the tail to be The results of these simulations for a single normal distribution and a non normal tail are illustrated in Fig. For simulated bimodal distributions with a tail Fig. Unimodal distributionBimodal distributionUnimodal distribution + tailBimodal distribution + tailThe purpose of this simulation is to assess how accurate GMMchi is at predicting the correct categorization with different sample sizes. The four categorizations we included in this simulation are:We generate a total of 500 distributions, 125 for each of the above four categorizations, using the same basic approach as above for simulation test 1, and do this for 12 different sample sizes. The mean of the normal component(s) is randomly chosen between 3 and 10 and the standard deviation similarly between 0.5 and 5. The number of data points in the tail is randomly chosen to be within 5–25% of the total number of data points in the normal component(s). The upper range of the tail is anywhere below 7 standard deviations away from the mean of the upper normal component. After randomly generating 500 simulated mixtures for each of 12 sample sizes from 10 to 2000 we ran GMMchi on each of the mixtures and compared GMMchi's prediction with the known distribution. The accuracy is, as before, the percentage of correct categorizations. The results of all the simulations are illustrated in Fig. In the current data landscape, with many datasets having sample sizes well above 100 and often approaching 1000, such as the TCGA CRC data on 637 samples, GMMchi should provide good data models for the objective analysis of observed mRNA expression level distributions.6.65) and 128 (27). We now use GMMchi analysis of our overall expression data on the 78 CRC derived cell lines to provide an objective statistically based estimate of the background noise level.The background noise threshold is an essential measurement needed when exploring gene mRNA expression data, especially if the data come from a fluorescence-based platform. The background arises necessarily from, for example, the within-array fluorescence noise. An estimate of its level is needed to distinguish convincingly the background from genuine low-level expression. Previous extensive analyses of our microarray data suggested a background noise level between 100  away from the mean (µ = 5.14) of the background distribution to ensure we remove 95% of the background expression level. This ensures more confidence in the identification of genuine positives for further GMMchi analysis.The intersection between the two normal distributions is used to define the background noise threshold, which on this basis was 2Therefore, in further analysis of the CRC cell line data we include the background noise threshold to signify lack of expression, namely any expression value less than the background threshold is counted as no expression. This variable is a built-in function in the current GMMchi pipeline. GMMchi analysis thus categorizes cell lines into three groups: background, low, and high, which we aim to associate with different biological functions.The probe filter removes genes that are considered as non-expressing, following the procedure shown in Fig. The function GMM.probe_filter first removes genes where all cell lines express levels below the background threshold. The pipeline then searches for genes that are on the borderline of being non-expressing, defined as genes that are expressed above the threshold level in fewer than 5% of the cell lines and whose distribution variance is less than 2.5% of the total range of variances across all genes. These values are arbitrarily chosen as the default values to achieve a stringent criterion for complete lack of expression. They can be easily changed in the algorithm.7.16. The criterion ensures we maximize capturing categorical unimodal distributions while discounting any unimodal distributions that are completely background signal or entirely above background threshold.We noticed that nearly half of the unimodal distributions had parts of their expression distribution at less than the background threshold, suggesting no expression, and the rest expressed, but with no clear separation between expression and lack of expression see Fig. . InherenIn microarray-based expression analysis, the expression level of a gene can be linked to multiple probe sets owing to microarray’s probe-based method of detecting the level of a specific DNA or RNA fragment through groups of probe sets that identify a unique region of a gene . The proTo address and facilitate analysis on a gene-by-gene basis, we developed a probe combining pipeline that aims to combine expression levels of multiple probe sets representing a gene into a single gene with a weighted value that encompasses all the combined probe sets. However, sometimes not all probe sets associated with a given gene can be combined to represent that gene. When we identify probe sets that identify the same gene but do not correlate well with most of the other probe sets, we will retain the probe set without combining it with the rest and assume it is a splice-variant of the gene since we expect splice-variants to behave differently from each other.As seen in Fig. The association between different probe sets from the same gene is determined by Pearson’s r correlation. A group of associated probe sets is identified as all those with pairwise correlations above 0.69, the threshold that indicates a significant association determined by the cutoff of the bimodal distribution identified by plotting out all the r-values using GMMchi, as shown in Fig. i; i for a given cell line j, theni expression value because the lower the goodness of fit value the better the fit. If all the weights are the same, then the weighted mean is just the arithmetic mean of the k expression values. The weighted mean is then put through GMMchi to convert those values into groups of 1 (low expression) or 2 (high expression) for further analysis.CDH1–2). The result of this pipeline is then the basis for further downstream analysis.Lastly, probe sets that do not behave similarly with other probe sets  are treated as separate splice variants and kept as a separate probe set annotated with the gene plus the probe number  is the primary transcription factor responsible for enterocyte differentiation with, however, a strong association with the closely related CDX2 [CDX1 and CDX2, and three other differentiation markers, carcinoembryonic antigen-related cell adhesion molecule 5 , villin 1 (VIL1), and glycoprotein A33 (GPA33). All these 5 gene expressions have clear bimodal distributions making them good candidates for 2 × 2 association analysis using the function GMM.run_hits. The 2 × 2 Table CDX1 and the 4 other genes. Note that the association between the expressions of CDX2 and CDX1 is an inclusion, namely with all CDX1 high being CDX2 high, but not the reverse.It was previously shown that caudal-related protein 1  was significantly associated with lumen formation as an inclusion. It seems likely NRCAM, a differentiation marker for neural development, is a nonessential gene for lumen formation that can nevertheless be expressed in lumen-forming cells as a sort of bystander effect [In total there are 22 out of 42 cell lines with the phenotype lumen formation as assessed from previous experiments by lumen formation assays . Table 1r effect .These analyses show how associations between continuous data, such as the gene expression data, can be related to categorical data such as, for example, mutations, distinctive phenotypes, and methylation, by transforming the continuous data into categorical data using GMMchi and so revealing interesting patterns of gene expressions associated with patterns of differentiation.MUC2 (the gene for MUC2) and three other genes  known to be involved in normal goblet cell differentiation are shown in Fig. TFF3, REG4, and FCGBP in Fig. Goblet cells are one of the three differentiated cell types in the colon and are responsible for the production of mucin 2, MUC2, the major mucin that lines the intestine. The GMMchi analysis of The GMMchi pipeline allows the possibility of identifying given cell lines or subsets of the cell lines in the histogram output. Two in-depth examples of this are given in Fig. MUC2 expression are obtained using the function GMM.crosstab_table. Several genes whose function has been shown or suggested to be associated with goblet cell differentiation, and so MUC2 production, occur in this list They include, for example, REG4, a secreted protein found to associate closely with secretory cells as well as playing an important role in intestinal morphogenesis [FCGBP, Fc-γ binding protein suggested to crosslink with MUC2 and have a chaperone like activity on the maturation of MUC2, ST6GALNAC1 and UGT1A8 /// UGT1A9, involved in glycosylation of MUC2 [AGR3, whose product is involved in catalyzing disulfide bond formation in protein folding and so contributes to maintaining the structure of MUC2, CEACAM5, associated with both goblet cell and enterocyte differentiation, and the transcription factor ETS2. A trefoil factor TFF1 is also in the list, and we have included the trefoil factor TFF3, though it comes just below the cutoff point for the top 50 associations, because of its well known almost total association with MUC2 expression in normal colonic tissue. The trefoil factors are also known to play an important role in MUC2 maturation, possibly after its secretion [In Table ogenesis , FCGBP,  of MUC2 , AGR3, wGMM.GMM_modeling This is used to perform GMMchi on an input matrix of samples x features. The input is a continuous mRNA measurement using either RNA-Seq or microarray (as discussed in the main text).GMM.probe_filter This is used to perform probe filtering to remove genes that are considered not to be expressed (as discussed in the main text).GMM.find_hits This is used to calculate 2 × 2 contingency tables from the categorized data returned from the GMM.GMM modeling output.GMM.run_hits This is used to output the full 2 × 2 contingency table from a pre-defined set of genes of interest (as discussed in the main text).GMM.crosstab_table This is used to visualize the full 2 × 2 contingency table from a pre-defined set of genes of interest (as discussed in the main text).A python library was built for easy utilization of the GMMchi pipeline. Built-in methods (functions) within the pipeline include:https://github.com/jeffliu6068/GMM.The whole program is implemented in Python and can be imported as a library; the full code, the library and installation link are online at: The somatic mutations or stable epigenetic changes whose selection drives cancer progression are discrete genetic variations analogous to the germ line genetic variants that are inherited following Mendelian segregation. The consequences of these somatic genetic changes can be seen in changes in the levels of mRNA expression of a wide range of genes. These levels are continuous variables. At the simplest level there could, for example, be a significant change in the message level of a mutated or epigenetically altered gene. A collection of cancers, some of which carry this genetic change and some not, would then be expected to have a bimodal distribution of the message levels for this gene, giving rise to a dichotomised phenotype of high versus low levels of expression. The more closely associated the function of the altered gene, for example having a role in the control of a particular cellular differentiation process, is with other genes also involved in the same function, the more likely it is that these other genes will have similar correlated bimodal distributions to that observed for the altered gene.The aim of this paper has been to create a widely applicable and flexible tool to analyse the relationship between discrete genetic changes and continuously variable gene expression levels by transforming the continuous expression levels into dichotomised phenotypes. Intrinsically dichotomous phenotypes, such as presence or absence of a particular cell type or ability to form a particular differentiated structure or not, can then be included with the derived dichotomised gene expression phenotypes for further analyses of their interrelationships.2 transformed, these usually have a distribution that is well described by a normal distribution. When, however, as is likely for cancers a relatively small subset of the cancers has either a higher or lower expression level than their normal counterparts, these distributions are less likely to be normal. This is the 'tail' problem, and so a primary goal of the analysis has been to find an objective procedure for identifying non-normally distributed tails. This has been done by a combination of iterative tail pruning and GMM, using a χ2 goodness of fit approach to enable the selection of the best fitting mixture of a GMM derived distribution and a non-normal tail. It is important to note that χ2-based methods are sensitive to the total number of bins. As GMMchi determines the numbers of bins based on the Mann and Wald bin criterion, this renders the bin numbers dynamic as data are trimmed away during tail-trimming. Nonetheless, in the case of expression data, almost all tails represent less than 10% of the total data thus will minimally interfere with the χ2 calculation yielding a reliable result. The success of this approach, GMMchi, has been demonstrated using genome wide microarray derived gene expression data from a collection of 78 CRC  cell lines. The complete pipeline starts by filtering out genes that are not expressed in any cell line, or those that are well described by a single unimodal normal distribution and allows for detection of splice variants when multiple oligonucleotide sets are used for a single gene. GMMchi can be applied to obtain dichotomised phenotypes from any similar source of data, such as single cell mRNA data or TCGA RNAseq data, whose analysis will be the subject of a further paper.The obvious first step is to apply GMM (Gaussian Mixture Modelling) to observed gene expression levels for a range of cancers, assuming, as has been widely observed that, when logSimulated mixtures of normal distributions and tails confirm the overall consistency of our pipeline, indicating it reaches an accuracy of 85% once the number of samples is sufficiently large at around a minimum of 90 samples, which is close to the number of cell lines in our demonstration data set. The accuracy of identifying a bimodal distribution with a tail spread beyond 7 standard deviations decreases quite rapidly with increasing tail size. It is, however, rare in practice to find tails that span beyond 7 standard deviations from the main component in gene expression data.The GMMchi pipeline iterative tail pruning process so far allows for only a single tail at either the upper or lower end of the overall distribution. This could readily be extended to allow tails at both the upper and lower ends, though we did not see evidence for the need for this in our analysis of the CRC cell line data.The 2 × 2 analyses associated with the analysis of the lumen formation and goblet cell differentiation phenotypes can be extended to searches for clusters of gene expressions not so far clearly associated with any defined phenotype. If there are different types of continuous measurements on the same set of cell lines, for example mRNA by microarray analysis and RNAseq, then each data set can be separately dichotomised by GMMchi giving rise to discrete phenotypes that can also be jointly analysed. Missing data are simply accounted for in a 2 × 2 analysis by only including pairs of cell lines that have data for both pairs of dichotomised phenotypes. Thus, for all those genes with at least two defined levels of expression, which if there are more than two levels can be condensed into just two levels, it is possible to look for associated clusters using 2 × 2 analyses with a defined minimum high-level r or low-level p value cut off, using the r values as a distance measure for clustering. A cluster identified in this way can be turned into a further dichotomised phenotype by assigning a cell line to a high or low category depending on the consistency of its expression of the members of the cluster at either a high or a low level. Such phenotypes could themselves then be used for further 2 × 2 analysis. This is an approach used for the early identification of HLA types using antisera from multiparous women known, mostly, to have antibodies to two or more HLA specificities .In conclusion, we propose that GMMchi is a general tool for the analysis of the interrelationships between a matrix of measurements of a set of continuous variables on a panel of objects, based on dichotomisation of the continuous variables and the analysis of their 2 × 2 associations.


The effects of post-treatment temperature-based methods for accelerating the ageing of PLA were studied on 1D single-PLA filaments after fused filament fabrication (FFF). The goal was to answer the questions whether the PLA can be safely aged—i.e., without degrading—at higher temperatures; at which temperatures, if any; how long it takes for the PLA to fully age at the chosen temperature; and which are the main differences between the material aged at room temperature and the material aged at higher temperatures. We also share other helpful information found. The use of 1D filaments allows for decoupling the variables related to the 3D structure  from the variables solely related to the material . 1D PLA filaments were aged at 20, 39, 42, 51, 65, 75, and 80 °C in a water-bath-inspired process in which the hydrolytic degradation of the PLA was minimised for the ageing temperatures of interest. Those temperatures were selected based on a differential scanning calorimetry (DSC) scan of the PLA right after it was printed in order to study the most effective ageing temperature, 39 °C, and highlight possible degradation mechanisms during ageing. The evolution of the thermal and mechanical properties of the PLA filaments at different temperatures was recorded and compared with those of the material aged at room temperature. A DSC scan was used to evaluate the thermal and physical properties, in which the glass transition, enthalpic relaxation, crystallisation, and melting reactions were analysed. A double glass transition was found, and its potential implications for the scientific community are discussed. Tensile tests were performed to evaluate the tensile strength and elastic modulus. The flow-induced molecular orientation, the degradation, the logistic fitting, and the so-called summer effect—the stabilisation of properties at higher values when aged at higher temperatures—are discussed to assess the safety of accelerating the ageing rate and the differences between the materials aged at different temperatures. It was found that the PLA aged at 39 °C (1) reached almost stable properties with just one day of ageing, i.e., the ageing rate accelerated by 875% for the elastic modulus and by 1635% for the yield strength; (2) the stable properties were higher than those from the PLA aged at room temperature; and (3) no signs of degradation were identified for the ageing temperature of interest. Additive manufacturing is a growing field that can produce 3D-printed components with few geometrical restrictions. These manufacturing processes are suitable for polymers, metals, ceramics, and composites ,3,4,5,6.g) [Altogether, the facts from our previous work , in whicg) .g. However, if temperatures above Tg are used, the macromolecules have so much kinetic energy that they can no longer be in ordered and stable positions—it is as though the amorphous part were to melt—and the ageing of the PLA is no longer possible.As the macromolecules gain more and more kinetic energy due to an increase in the temperature, they have more energy to reach more stable locations on the microstructure of the polymer, reducing its free space and ageing the material. For effective ageing (in which the properties increase until they stabilise), it would be ideal to use temperatures as high as possible but consistently below the Tg is a second-order transition in which the polymer chains significantly increase their mobility. It can be measured using differential scanning calorimetry (DSC) or fast scanning calorimetry (FSC) as an endothermal drop related to the change in heat capacity [g also depends on a molecular weight distribution which, by its essence, makes the Tg into a distribution of temperatures. Even though the Tg is defined at some point between a changing line in the DSC graph (a transition), there is an onset and offset temperature. There are different theories for defining the specific Tg value from a DSC, but they all provide a Tg value higher than the onset temperature. Thus, lower values than the usually defined Tg should be used to ensure that all the PLA is ageing. More precisely, the ageing temperatures should be lower than the onset temperature measured at a slow heating rate and with as much surface of the PLA in direct contact with the DSC crucible as possible. This will ensure that all the amorphous part of the PLA  remains restricted to low degrees of mobility, leading to an effective ageing process [g value measured at 10 °C, ageing performed at 1 degree below that measured Tg might not age the polymer or, at least, not all of its structure.The Tcapacity , or throcapacity ,32. In t process . Otherwig to improve the mechanical properties of PLA by increasing the crystallinity [Regarding the literature about ageing PLA, many researchers have studied the effect of heating the PLA after it was processed via FFF ,33,34. Fallinity ,34. Alteallinity . Howeverallinity ,14,27,29Is it safe to effectively accelerate the ageing of the PLA by heating it? To answer this question, the degradation of the PLA was analysed with regard to the shifting of peaks in the DSC scans  and the decrease in both the elastic modulus and tensile strength. As the primary degradation mechanism is the hydrolytic degradation with the air humidity, the change in colour of a desiccant inside the ageing device was used to determine the low-humidity conditions during the ageing qualitatively.At which temperature can the PLA be effectively and safely accelerated? How long does the PLA need to be entirely aged at that temperature? Different temperatures were studied. The temperatures were chosen based on a DSC scan on the PLA right after it was printed after finding an optimal ageing temperature and forcing the degradation mechanism. The evolution of the thermal and mechanical properties was measured at different times to ensure that stable properties were reached and the minimum ageing time was set for the central ageing temperature of interest.What are the main differences between the PLA that was slowly aged at room temperature and the one aged at higher temperatures? The PLA aged at room temperature is compared with the material effectively aged at higher temperatures. This makes possible a better understanding of how future 3D-printed samples of PLA will be affected—at least for the thermal and mechanical properties of the 3D structures that are no longer slowly aged during 3 months at 20 °C but accelerated at a higher temperature.Has any other phenomenon of interest been detected? As different properties were studied, the results obtained during this research were analysed to determine other phenomena related to the PLA and, if something interesting is found, to share it with the research community.This paper analyses and discusses the effects solely affecting the PLA when it is accelerated by raising the ageing temperature in a low-humidity atmosphere by studying those 1D filaments. Our research questions to answer are:g) are analysed. This paper complements our previous study on controlling the ageing of the PLA [To answer these questions and solely address the variables related to the material, 1D filaments were studied, following the methodology developed in previous research ,17,23 ba5.Our main aim is to find and understand a reliable, safe, and effective method to accelerate the ageing of PLA 4043D-based materials (decoupled from the effects of the structure) to be sure of how the ageing method is affecting the PLA and reduce the research schedule from months (time required until the PLA is entirely aged at room temperature) to just a few days (as demonstrated in this study).6.An emphasis on the importance of studying aged materials is essential for comparing results from different research with confidence.7.It would allow us to understand and report how the thermal treatments affect the material decoupled from the variables related to the structure. Nonetheless, this study will have some limitations for extrapolation to 3D structures, as the thermal history of the PLA during the printing of 3D models is different depending on printed geometry, printer parameters, and ambient conditions . HoweverAnswering our research questions will provide the following benefits:This study analysed the first three months of the natural ageing of 1D filaments of PLA aged at room temperature and higher temperatures. Our previous study showed that a similar high-molecular-weight PLA required three months of natural ageing at 20 °C in order to stabilise its properties; this is the reason for conducting the ageing studies for up to 91 days. As the primary objective was to compare the samples at room temperature (20 °C) with the samples aged at higher temperatures , no longer times were considered. For a better understanding of the content of this work, a summary of the material production, storage, ageing, and experiments is provided in The material studied was the 4043D (a commonly studied PLA suitable for 3D printing applications) from Nature Works and supplied by Prusa in the form of filaments with a diameter of 1.75 ± 0.02 mm. The material was printed in a Prusa i3MK3S+ with a 0.4mm nozzle at 215 ± 1 °C  and at an extrusion rate of 20 mm/s, a relatively slow printing speed found in accurate printing processes and studied by other authors . FilamenPrinted filaments were stored inside zip bags with a desiccant inside to ensure a low-humidity-controlled atmosphere, essential for avoiding any degradation through the hydrolysis of the PLA. Ageing at room temperature was performed inside these bags, as indicated in The differential scanning calorimetry (DSC) was performed on a Mettler Toledo 822e device at a heating rate of 10 °C/min from 30 to 180 °C d. The deMechanical tests were performed on an Instron 5866 universal tensile test machine with a load cell of 1 kN d. BeforeA, the ageing potential . It describes the gap between the property at zero days  and infinite days of ageing (stabilized).B, is the ageing rate in units of (1/time), which describes the speed at which the material evolves towards the steady state.g (51 °C and 65 °C). When the logistic fitting was not possible because the material did not evolve with the ageing temperature (51 °C in this study), the linear fitting y = mx + c was used. The 65 °C was not fitted to any curve as there were not enough points.This fitting was proven to properly define the evolution of the tensile strength, yield strength and elastic modulus with the natural ageing at room temperature by using just those 3 parameters. The objective was to increase the value of the B parameter, the ageing rate, by increasing the ageing temperature. B was used to quantify the extent to which the ageing rate varied at different ageing temperatures. The logistic fitting is impossible to apply if the material is not ageing, i.e., for the temperatures higher than TThis is the expression used for the logistic fitting:All the results for the thermal properties measured on the DSC are summarised in g. For this reason, Tg was measured on PLA samples right after printing (0.5 and 2 h) to set a maximum ageing temperature. However, how this maximum ageing temperature should be obtained is not further explained. Our goal was to use the highest temperature possible. Two DSC tests were performed to obtain a picture of the Tg reaction on the PLA right after it was printed. It is generally said that the ageing rate of a polymer can be accelerated only at temperatures below Tg, called Tg2, was identified after 0.5 h of the printed material. A slight endothermal process was noticed before the Tg2 on the samples aged for 2 h. The experiment was conducted at heating rates of 10 °C/min during the following days  that the slight endothermal reaction was becoming larger and forming the typical shape of the glass transition; we called it Tg1 as it was the first glass transition to appear on the DSC heating scan. For the Tg1 to show up consistently, at least 2 h of natural ageing at 20 °C was required after the FFF process  any DSC in the literature on PLA samples right after printing at ageing times lower than 2 h, (b) for the typically used procedure “first heating – cooling – second heating” [g2 but not about Tg1. Notice that the PLA 4043D-based materials are commonly studied in the literature [g1 phenomenon is very subtle during the first hours. A large mass of PLA-to-surface contact between the PLA and the aluminium crucible made it even more challenging to detect it. This result shows the importance of carefully placing the material inside the DSC crucible in the most controlled and reproducible manner and with a similar mass (not in the range of 2 to 10 mg). Both variables were responsible for shifts of up to +1 °C in the temperature peaks and a widening of the peaks. The evolution of Tg1 was found to be extremely fast  was 42.9 °C , and 15 degrees lower than the Tg2 measured at 0.5 h after printing. The 51 °C ageing temperature was chosen as it was close to but below the second glass transition (Tg2) found on the PLA right after it was printed in order to demonstrate that the Tg2 measured cannot be directly used as an indicator of the maximum temperature for accelerating the ageing of the PLA, especially if it was measured on a sample right after printing. Temperatures of 65, 75, and 80 °C were used to provide further information on how ageing at higher temperatures could affect the PLA (inside our ageing device) and to show any potential degradation evidence together with the mechanical tests to confirm the safeness of accelerating the ageing of the PLA. The DSC results are discussed together with the mechanical properties.The evolution of TAll the results from the tensile tests on the PLA 4043D are summarised in Thermal and mechanical properties were measured, and they are highly interrelated. For an appropriate discussion, all the results must be considered together. The samples aged at 20 °C were used for setting a reference. In this way, it was possible to compare the evolution of the samples aged at higher temperatures with these controls. As expected, the ageing of these PLA 4043D samples at room temperature was consistent with those in our previous study on a similar PLA, the 2003D. The trends in the evolution of the properties remained similar: (1) the enthalpic relaxation  increased until stabilisation; (2) the enthalpic relaxation enthalpy continued to increase even after the stabilisation of the glass transition; (3) the crystallisation and melting reactions remained unaltered; (4) the elastic modulus and yield strength evolved until they stabilised, requiring times similar to the stabilisation of the glass transition. This similar evolution trend was further proven by applying our proposed logistic fitting  to the mThe logistic fittings in The samples aged at 39 °C showed almost stable properties after just one day of ageing, indicating a significant improvement in the ageing rate, the B parameter on the logistic fitting. The logistic fittings were calculated, and Equations (4) and (5) were obtained:In comparing the parameters obtained in the logistic fittings on samples aged at 20 and 39 °C , we obsequations ).g increases with the ageing of samples after printing. Thus, the maximum ageing temperature could be increased proportionally to the increase in Tg, but fully optimising the accelerated ageing process was not the objective of this work.It is known, and was here observed, that the Tg and enthalpic relaxation. This effect was referred to as the summer effect, as samples which might be stable at ambient conditions, if not stored properly, could evolve to higher values due to summer temperatures. The summer effect also applies for explaining the outstanding results obtained on samples aged at 42 °C, in which even higher values were obtained on the DSC scans after just four days for the thermal properties, reaching the Tg1 63.3 °C and its enthalpic relaxation 65.6 °C and 5.7 J/g without any signal of degradation.The evolution of the properties to higher g1, proving the importance of studying aged materials. The samples, however, had a relatively high Tg, as if they were aged. However, the enthalpic relaxation value was as it would be if the material had been just printed, raising a contradiction for any direct relation between the glass transition and the enthalpic relaxation: samples which are not aged regarding the enthalpic relaxation might show Tg values similar to those of an aged material. This shows that it is difficult, if not impossible, to correlate some of the thermal properties measured on the PLA. For example, a Tg1 of 60 °C, an average value of Tg found on aged PLA materials, is not necessarily a sign of a high enthalpic relaxation enthalpy . However, some trends can be found and modelled [Samples aged at 51 °C did not evolve, as the temperature was over the Tmodelled . The desextruded a, due toextruded b,c and mA decrease in the molecular weight—degradation—could also explain the slight decrease in the elastic modulus due to the degradation of the material. This degradation is especially important when the temperature increases over 50 degrees in a humid ambient environment . AlthougConsidering both the loss of flow-induced molecular orientation and the degradation as the two possibilities for explaining the decreases in the elastic modulus, the PLA samples were aged and tested at even higher temperatures for one day, forcing the mechanisms of both the degradation and the loss of flow-induced molecular orientation. Note that the influence of the crystallinity was discarded entirely, as the measured values were below 3% for all the samples. Three higher temperatures were analysed: 65, 75, and 80 °C. Both the 75 and the 80 °C temperatures were discarded from the mechanical test, as we observed on the DSC that they crystallised after one day of ageing , introducing new variables to consider in the system but which are out of the scope of this study. Nonetheless, the desiccant remained bright yellow, as the positive in The results from the PLA aged at 65 °C were insufficient to provide information to support any of the mechanisms: (1) it had similar mechanical values to the 42-day-aged PLA at 51 °C, and (2) the error bars of the measurements were within the error bars of the reference samples. The DSC results showed that the material was slightly degraded as both a decrease in the crystallisation temperature and an increase in the crystallisation enthalpies -related to higher mobility of the polymeric chains were detected. After all, the same analysis considering the error bars applies to the samples aged at 51 °C. The elastic modulus decreased by 5–6 % considering the average values, but it is also true that the error bars fell within the error bars of the reference sample. This makes it impossible to conclude why the elastic modulus of the PLA aged at 51 °C decreased. However, the results provide useful information: the degradation of the PLA and the loss of flow-induced mechanical orientation are almost negligible for a sample aged for 42 days at 51 °C, thus making those phenomena completely negligible for samples aged at 39 °C—potentially at 42 °C—for some days.With all these data, we can ensure that accelerating the ageing rate of the PLA 4043D at 39 °C for a few days in an oil-bath device is safe, that it effectively ages the PLA, and that it will produce slightly higher properties than if it is slowly aged at room temperature. Furthermore, more critically, no variables related to the 3D structure were mixed in the discussion, making this a potentially essential reference for discussing the effect of heat treatments on 3D-printed parts produced via FFF.Questions 1 and 2:Summarising the results obtained, our answers to our initial research questions are:It is safe to accelerate the ageing rate of the PLA at 39 °C for several weeks in our proposed oil-bath ageing device, which aged the material properties close to the stable ones after just one day, but some extra days are needed to fully reach stable propertiesAny degradation of the PLA during the ageing for some days at 39 °C can be discarded as it is hardly observable in a sample aged for 42 days at 51 °C.The mechanical tests, combined with the DSC, were not suitable for decoupling the potential degradation from the loss of flow-induced molecular orientation.g1 during the ageing process to obtain faster ageing rates and higher properties due to the so-called summer effect.Potentially, higher temperatures could be used with the increase in TQuestion 3:The main difference between the samples aged at 39 °C and those aged at 20 °C was the higher stability of the material aged at 39 °C. This has been defined as the summer effect, in which ageing at higher temperatures produces higher stabilisation properties  due to the higher kinetic energy of the PLA macromolecules that can access more stable locations.g1 and Tg2) were found and characterised which had not been previously described in the literature. For these two glass transitions to be noticed, the printed PLA needed at least 2 h of ageing at 20 °C. This could induce possible mistakes in selecting the ageing temperature, as the 51 °C (below Tg2) was found to be too high for ageing the material.Two glass transitions . This might affect the crystallinity contents of the material. This variable was not considered in this study, as the crystallinity percentages were very low and materials with higher contents (samples aged at 75 and 80 °C) were discarded.