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Institute
- Mathematisch-Naturwissenschaftliche Fakultät (478) (remove)
Assimilation of pseudo-tree-ring-width observations into an atmospheric general circulation model
(2017)
Paleoclimate data assimilation (DA) is a promising technique to systematically combine the information from climate model simulations and proxy records. Here, we investigate the assimilation of tree-ring-width (TRW) chronologies into an atmospheric global climate model using ensemble Kalman filter (EnKF) techniques and a process-based tree-growth forward model as an observation operator. Our results, within a perfect-model experiment setting, indicate that the "online DA" approach did not outperform the "off-line" one, despite its considerable additional implementation complexity. On the other hand, it was observed that the nonlinear response of tree growth to surface temperature and soil moisture does deteriorate the operation of the time-averaged EnKF methodology. Moreover, for the first time we show that this skill loss appears significantly sensitive to the structure of the growth rate function, used to represent the principle of limiting factors (PLF) within the forward model. In general, our experiments showed that the error reduction achieved by assimilating pseudo-TRW chronologies is modulated by the magnitude of the yearly internal variability in themodel. This result might help the dendrochronology community to optimize their sampling efforts.
Towards the assimilation of tree-ring-width records using ensemble Kalman filtering techniques
(2015)
This paper investigates the applicability of the Vaganov–Shashkin–Lite (VSL) forward model for tree-ring-width chronologies as observation operator within a proxy data assimilation (DA) setting. Based on the principle of limiting factors, VSL combines temperature and moisture time series in a nonlinear fashion to obtain simulated TRW chronologies. When used as observation operator, this modelling approach implies three compounding, challenging features: (1) time averaging, (2) “switching recording” of 2 variables and (3) bounded response windows leading to “thresholded response”. We generate pseudo-TRW observations from a chaotic 2-scale dynamical system, used as a cartoon of the atmosphere-land system, and attempt to assimilate them via ensemble Kalman filtering techniques. Results within our simplified setting reveal that VSL’s nonlinearities may lead to considerable loss of assimilation skill, as compared to the utilization of a time-averaged (TA) linear observation operator. In order to understand this undesired effect, we embed VSL’s formulation into the framework of fuzzy logic (FL) theory, which thereby exposes multiple representations of the principle of limiting factors. DA experiments employing three alternative growth rate functions disclose a strong link between the lack of smoothness of the growth rate function and the loss of optimality in the estimate of the TA state. Accordingly, VSL’s performance as observation operator can be enhanced by resorting to smoother FL representations of the principle of limiting factors. This finding fosters new interpretations of tree-ring-growth limitation processes.
Subsurface microbial communities undertake many terminal electron-accepting processes, often simultaneously. Using a tritium-based assay, we measured the potential hydrogen oxidation catalyzed by hydrogenase enzymes in several subsurface sedimentary environments (Lake Van, Barents Sea, Equatorial Pacific, and Gulf of Mexico) with different predominant electron-acceptors. Hydrogenases constitute a diverse family of enzymes expressed by microorganisms that utilize molecular hydrogen as a metabolic substrate, product, or intermediate. The assay reveals the potential for utilizing molecular hydrogen and allows qualitative detection of microbial activity irrespective of the predominant electron-accepting process. Because the method only requires samples frozen immediately after recovery, the assay can be used for identifying microbial activity in subsurface ecosystems without the need to preserve live material. We measured potential hydrogen oxidation rates in all samples from multiple depths at several sites that collectively span a wide range of environmental conditions and biogeochemical zones. Potential activity normalized to total cell abundance ranges over five orders of magnitude and varies, dependent upon the predominant terminal electron acceptor. Lowest per-cell potential rates characterize the zone of nitrate reduction and highest per-cell potential rates occur in the methanogenic zone. Possible reasons for this relationship to predominant electron acceptor include (i) increasing importance of fermentation in successively deeper biogeochemical zones and (ii) adaptation of H(2)ases to successively higher concentrations of H-2 in successively deeper zones.
The temporal dynamics of climate processes are spread across different timescales and, as such, the study of these processes at only one selected timescale might not reveal the complete mechanisms and interactions within and between the (sub-) processes. To capture the non-linear interactions between climatic events, the method of event synchronization has found increasing attention recently. The main drawback with the present estimation of event synchronization is its restriction to analysing the time series at one reference timescale only. The study of event synchronization at multiple scales would be of great interest to comprehend the dynamics of the investigated climate processes. In this paper, the wavelet-based multi-scale event synchronization (MSES) method is proposed by combining the wavelet transform and event synchronization. Wavelets are used extensively to comprehend multi-scale processes and the dynamics of processes across various timescales. The proposed method allows the study of spatio-temporal patterns across different timescales. The method is tested on synthetic and real-world time series in order to check its replicability and applicability. The results indicate that MSES is able to capture relationships that exist between processes at different timescales.
Central Asia is located at the confluence of large-scale atmospheric circulation systems. It is thus likely to be highly susceptible to changes in the dynamics of those systems; however, little is still known about the regional paleoclimate history. Here we present carbon and hydrogen isotopic compositions of n-alkanoic acids from a late Holocene sediment core from Lake Karakuli (eastern Pamir, Xinjiang Province, China). Instrumental evidence and isotopeenabled climate model experiments with the Laboratoire de Meteorologie Dynamique Zoom model version 4 (LMDZ4) demonstrate that delta D values of precipitation in the region are influenced by both temperature and precipitation amount. We find that these parameters are inversely correlated on an annual scale, i.e., the climate has varied between relatively cool and wet and more warm and dry over the last 50 years. Since the isotopic signals of these changes are in the same direction and therefore additive, isotopes in precipitation are sensitive recorders of climatic changes in the region. Additionally, we infer that plants use year-round precipitation (including snowmelt), and thus leaf wax delta D values must also respond to shifts in the proportion of moisture derived from westerly storms during late winter and early spring. Downcore results give evidence for a gradual shift to cooler and wetter climates between 3.5 and 2.5 cal kyr BP, interrupted by a warm and dry episode between 3.0 and 2.7 kyr BP. Further cool and wet episodes occur between 1.9 and 1.5 and between 0.6 and 0.1 kyr BP, the latter coeval with the Little Ice Age. Warm and dry episodes from 2.5 to 1.9 and 1.5 to 0.6 kyr BP coincide with the Roman Warm Period and Medieval Climate Anomaly, respectively. Finally, we find a drying tend in recent decades. Regional comparisons lead us to infer that the strength and position of the westerlies, and wider northern hemispheric climate dynamics, control climatic shifts in arid Central Asia, leading to complex local responses. Our new archive from Lake Karakuli provides a detailed record of the local signatures of these climate transitions in the eastern Pamir.
Mechanical and/or chemical removal of material from the subsurface may generate large subsurface cavities, the destabilisation of which can lead to ground collapse and the formation of sinkholes. Numerical simulation of the interaction of cavity growth, host material deformation and overburden collapse is desirable to better understand the sinkhole hazard but is a challenging task due to the involved high strains and material discontinuities. Here, we present 2-D distinct element method numerical simulations of cavity growth and sinkhole development. Firstly, we simulate cavity formation by quasi-static, stepwise removal of material in a single growing zone of an arbitrary geometry and depth. We benchmark this approach against analytical and boundary element method models of a deep void space in a linear elastic material. Secondly, we explore the effects of properties of different uniform materials on cavity stability and sinkhole development. We perform simulated biaxial tests to calibrate macroscopic geotechnical parameters of three model materials representative of those in which sinkholes develop at the Dead Sea shoreline: mud, alluvium and salt. We show that weak materials do not support large cavities, leading to gradual sagging or suffusion-style subsidence. Strong materials support quasi-stable to stable cavities, the overburdens of which may fail suddenly in a caprock or bedrock collapse style. Thirdly, we examine the consequences of layered arrangements of weak and strong materials. We find that these are more susceptible to sinkhole collapse than uniform materials not only due to a lower integrated strength of the overburden but also due to an inhibition of stabilising stress arching. Finally, we compare our model sinkhole geometries to observations at the Ghor Al-Haditha sinkhole site in Jordan. Sinkhole depth ∕ diameter ratios of 0.15 in mud, 0.37 in alluvium and 0.33 in salt are reproduced successfully in the calibrated model materials. The model results suggest that the observed distribution of sinkhole depth ∕ diameter values in each material type may partly reflect sinkhole growth trends.
Background
Riociguat is the first of a new class of drugs, the soluble guanylate cyclase (sGC) stimulators. Riociguat has a dual mode of action: it sensitizes sGC to the body’s own NO and can also increase sGC activity in the absence of NO. The NO-sGC-pathway is impaired in many cardiovascular diseases such as heart failure, pulmonary hypertension and diabetic nephropathy (DN). DN leads to high cardiovascular morbidity and mortality. There is still a high unmet medical need. The urinary albumin excretion rate is a predictive biomarker for these clinical events. Therefore, we investigated the effect of riociguat, alone and in combination with the angiotensin II receptor antagonist (ARB) telmisartan on the progression of DN in diabetic eNOS knock out mice, a new model closely resembling human pathology.
Methods
Seventy-six male eNOS knockout C57BL/6J mice were divided into 4 groups after receiving intraperitoneal high-dose streptozotocin: telmisartan (1 mg/kg), riociguat (3 mg/kg), riociguat+telmisartan (3 and 1 mg/kg), and vehicle. Fourteen mice were used as non-diabetic controls. After 12 weeks, urine and blood were obtained and blood pressure measured. Glucose concentrations were highly increased and similar in all diabetic groups.
Results
Riociguat, alone (105.2 ± 2.5 mmHg; mean±SEM; n = 14) and in combination with telmisartan (105.0 ± 3.2 mmHg; n = 12), significantly reduced blood pressure versus diabetic controls (117.1 ± 2.2 mmHg; n = 14; p = 0.002 and p = 0.004, respectively), whereas telmisartan alone (111.2 ± 2.6 mmHg) showed a modest blood pressure lowering trend (p = 0.071; n = 14). The effects of single treatment with either riociguat (97.1 ± 15.7 µg/d; n = 13) or telmisartan (97.8 ± 26.4 µg/d; n = 14) did not significantly lower albumin excretion on its own (p = 0.067 and p = 0.101, respectively). However, the combined treatment led to significantly lower urinary albumin excretion (47.3 ± 9.6 µg/d; n = 12) compared to diabetic controls (170.8 ± 34.2 µg/d; n = 13; p = 0.004), and reached levels similar to non-diabetic controls (31.4 ± 10.1 µg/d, n = 12).
Conclusion
Riociguat significantly reduced urinary albumin excretion in diabetic eNOS knock out mice that were refractory to treatment with ARB’s alone. Patients with diabetic nephropathy refractory to treatment with ARB’s have the worst prognosis among all patients with diabetic nephropathy. Our data indicate that additional stimulation of sGC on top of standard treatment with ARB`s may offer a new therapeutic approach for patients with diabetic nephropathy resistant to ARB treatment.
Climate impacts on transocean dispersal and habitat in gray whales from the Pleistocene to 2100
(2015)
Arctic animals face dramatic habitat alteration due to ongoing climate change. Understanding how such species have responded to past glacial cycles can help us forecast their response to today's changing climate. Gray whales are among those marine species likely to be strongly affected by Arctic climate change, but a thorough analysis of past climate impacts on this species has been complicated by lack of information about an extinct population in the Atlantic. While little is known about the history of Atlantic gray whales or their relationship to the extant Pacific population, the extirpation of the Atlantic population during historical times has been attributed to whaling. We used a combination of ancient and modern DNA, radiocarbon dating and predictive habitat modelling to better understand the distribution of gray whales during the Pleistocene and Holocene. Our results reveal that dispersal between the Pacific and Atlantic was climate dependent and occurred both during the Pleistocene prior to the last glacial period and the early Holocene immediately following the opening of the Bering Strait. Genetic diversity in the Atlantic declined over an extended interval that predates the period of intensive commercial whaling, indicating this decline may have been precipitated by Holocene climate or other ecological causes. These first genetic data for Atlantic gray whales, particularly when combined with predictive habitat models for the year 2100, suggest that two recent sightings of gray whales in the Atlantic may represent the beginning of the expansion of this species' habitat beyond its currently realized range.
Genetic studies of the Eurasian brown bear (Ursus arctos) have so far focused on populations from Europe and North America, although the largest distribution area of brown bears is in Asia. In this study, we reveal population genetic parameters for the brown bear population inhabiting the Grand Kaçkar Mountains (GKM) in the north east of Turkey, western Lesser Caucasus. Using both hair (N = 147) and tissue samples (N = 7) collected between 2008 and 2014, we found substantial levels of genetic variation (10 microsatellite loci). Bear samples (hair) taken from rubbing trees worked better for genotyping than those from power poles, regardless of the year collected. Genotyping also revealed that bears moved between habitat patches, despite ongoing massive habitat alterations and the creation of large water reservoirs. This population has the potential to serve as a genetic reserve for future reintroductions in the Middle East. Due to the importance of the GKM population for on-going and future conservation actions, the impacts of habitat alterations in the region ought to be minimized; e.g., by establishing green bridges or corridors over reservoirs and major roads to maintain habitat connectivity and gene flow among populations in the Lesser Caucasus.
With accelerating climate cooling in the late Cenozoic, glacial and periglacial erosion became more widespread on the surface of the Earth. The resultant shift in erosion patterns significantly changed the large-scale morphology of many mountain ranges worldwide. Whereas the glacial fingerprint is easily distinguished by its characteristic fjords and U-shaped valleys, the periglacial fingerprint is more subtle but potentially prevails in some mid- to high-latitude landscapes. Previous models have advocated a frost-driven control on debris production at steep headwalls and glacial valley sides. Here we investigate the important role that periglacial processes also play in less steep parts of mountain landscapes. Understanding the influences of frost-driven processes in low-relief areas requires a focus on the consequences of an accreting soil mantle, which characterises such surfaces. We present a new model that quantifies two key physical processes: frost cracking and frost creep, as a function of both temperature and sediment thickness. Our results yield new insights into how climate and sediment transport properties combine to scale the intensity of periglacial processes. The thickness of the soil mantle strongly modulates the relation between climate and the intensity of mechanical weathering and sediment flux. Our results also point to an offset between the conditions that promote frost cracking and those that promote frost creep, indicating that a stable climate can provide optimal conditions for only one of those processes at a time. Finally, quantifying these relations also opens up the possibility of including periglacial processes in large-scale, long-term landscape evolution models, as demonstrated in a companion paper.
We propose a network structure-based model for heterosis, and investigate it relying on metabolite profiles from Arabidopsis. A simple feed-forward two-layer network model (the Steinbuch matrix) is used in our conceptual approach. It allows for directly relating structural network properties with biological function. Interpreting heterosis as increased adaptability, our model predicts that the biological networks involved show increasing connectivity of regulatory interactions. A detailed analysis of metabolite profile data reveals that the increasing-connectivity prediction is true for graphical Gaussian models in our data from early development. This mirrors properties of observed heterotic Arabidopsis phenotypes. Furthermore, the model predicts a limit for increasing hybrid vigor with increasing heterozygosity—a known phenomenon in the literature.
The phrase form and function was established in architecture and biology and refers to the idea that form and functionality are closely correlated, influence each other, and co-evolve. We suggest transferring this idea to hydrological systems to separate and analyze their two main characteristics: their form, which is equivalent to the spatial structure and static properties, and their function, equivalent to internal responses and hydrological behavior. While this approach is not particularly new to hydrological field research, we want to employ this concept to explicitly pursue the question of what information is most advantageous to understand a hydrological system. We applied this concept to subsurface flow within a hillslope, with a methodological focus on function: we conducted observations during a natural storm event and followed this with a hillslope-scale irrigation experiment. The results are used to infer hydrological processes of the monitored system. Based on these findings, the explanatory power and conclusiveness of the data are discussed. The measurements included basic hydrological monitoring methods, like piezometers, soil moisture, and discharge measurements. These were accompanied by isotope sampling and a novel application of 2-D time-lapse GPR (ground-penetrating radar). The main finding regarding the processes in the hillslope was that preferential flow paths were established quickly, despite unsaturated conditions. These flow paths also caused a detectable signal in the catchment response following a natural rainfall event, showing that these processes are relevant also at the catchment scale. Thus, we conclude that response observations (dynamics and patterns, i.e., indicators of function) were well suited to describing processes at the observational scale. Especially the use of 2-D time-lapse GPR measurements, providing detailed subsurface response patterns, as well as the combination of stream-centered and hillslope-centered approaches, allowed us to link processes and put them in a larger context. Transfer to other scales beyond observational scale and generalizations, however, rely on the knowledge of structures (form) and remain speculative. The complementary approach with a methodological focus on form (i.e., structure exploration) is presented and discussed in the companion paper by Jackisch et al. (2017).
QuantPrime
(2008)
Background
Medium- to large-scale expression profiling using quantitative polymerase chain reaction (qPCR) assays are becoming increasingly important in genomics research. A major bottleneck in experiment preparation is the design of specific primer pairs, where researchers have to make several informed choices, often outside their area of expertise. Using currently available primer design tools, several interactive decisions have to be made, resulting in lengthy design processes with varying qualities of the assays.
Results
Here we present QuantPrime, an intuitive and user-friendly, fully automated tool for primer pair design in small- to large-scale qPCR analyses. QuantPrime can be used online through the internet http://www.quantprime.de/ or on a local computer after download; it offers design and specificity checking with highly customizable parameters and is ready to use with many publicly available transcriptomes of important higher eukaryotic model organisms and plant crops (currently 295 species in total), while benefiting from exon-intron border and alternative splice variant information in available genome annotations. Experimental results with the model plant Arabidopsis thaliana, the crop Hordeum vulgare and the model green alga Chlamydomonas reinhardtii show success rates of designed primer pairs exceeding 96%.
Conclusion
QuantPrime constitutes a flexible, fully automated web application for reliable primer design for use in larger qPCR experiments, as proven by experimental data. The flexible framework is also open for simple use in other quantification applications, such as hydrolyzation probe design for qPCR and oligonucleotide probe design for quantitative in situ hybridization. Future suggestions made by users can be easily implemented, thus allowing QuantPrime to be developed into a broad-range platform for the design of RNA expression assays.
The novel space-borne Global Navigation Satellite System Reflectometry (GNSS-R) technique has recently shown promise in monitoring the ocean state and surface wind speed with high spatial coverage and unprecedented sampling rate. The L-band signals of GNSS are structurally able to provide a higher quality of observations from areas covered by dense clouds and under intense precipitation, compared to those signals at higher frequencies from conventional ocean scatterometers. As a result, studying the inner core of cyclones and improvement of severe weather forecasting and cyclone tracking have turned into the main objectives of GNSS-R satellite missions such as Cyclone Global Navigation Satellite System (CYGNSS). Nevertheless, the rain attenuation impact on GNSS-R wind speed products is not yet well documented. Evaluating the rain attenuation effects on this technique is significant since a small change in the GNSS-R can potentially cause a considerable bias in the resultant wind products at intense wind speeds. Based on both empirical evidence and theory, wind speed is inversely proportional to derived bistatic radar cross section with a natural logarithmic relation, which introduces high condition numbers (similar to ill-posed conditions) at the inversions to high wind speeds. This paper presents an evaluation of the rain signal attenuation impact on the bistatic radar cross section and the derived wind speed. This study is conducted simulating GNSS-R delay-Doppler maps at different rain rates and reflection geometries, considering that an empirical data analysis at extreme wind intensities and rain rates is impossible due to the insufficient number of observations from these severe conditions. Finally, the study demonstrates that at a wind speed of 30 m/s and incidence angle of 30 degrees, rain at rates of 10, 15, and 20 mm/h might cause overestimation as large as approximate to 0.65 m/s (2%), 1.00 m/s (3%), and 1.3 m/s (4%), respectively, which are still smaller than the CYGNSS required uncertainty threshold. The simulations are conducted in a pessimistic condition (severe continuous rainfall below the freezing height and over the entire glistening zone) and the bias is expected to be smaller in size in real environments.
Background:
All living cells display a rapid molecular response to adverse environmental conditions, and
the heat shock protein family reflects one such example. Hence, failing to activate heat shock proteins can impair
the cellular response. In the present study, we evaluated whether the loss of different isoforms of heat shock
protein (
hsp
) genes in
Caenorhabditis elegans
would affect their vulnerability to Manganese (Mn) toxicity.
Methods:
We exposed wild type and selected
hsp
mutant worms to Mn (30 min) and next evaluated
further the most susceptible strains. We analyzed survi
val, protein carbonylation (as a marker of oxidative
stress) and Parkinson
’
s disease related gene expression immediately after Mn exposure. Lastly, we observed
dopaminergic neurons in wild type worms and in
hsp-70
mutants following Mn treatment. Analysis of the
data was performed by one-way or two way ANOVA, depending on the case, followed by post-hoc
Bonferroni test if the overall
p
value was less than 0.05.
Results:
We verified that the loss of
hsp-70, hsp-3 and chn-1
increased the vulnerability to Mn, as
exposed mutant worms showed lower survival rate and increased protein oxidation. The importance of
hsp-70
against Mn toxicity was then corroborated in dopaminergic neurons, where Mn neurotoxicity was
aggravated. The lack of
hsp-70
also blocked the transcriptional upregulation of
pink1
, a gene that has been
linked to Parkinson
’
sdisease.
Conclusions:
Taken together, our data suggest that Mn exposu
re modulates heat shock protein expression,
particularly HSP-70, in
C. elegans
.Furthermore,lossof
hsp-70
increases protein oxidation and dopaminergic
neuronal degeneration following manganese exposure, which is associated with the inhibition of
pink1
increased expression, thus pot
entially exacerbating the v
ulnerability to this metal.
During the last few decades, the rapid separation of the Small Aral Sea from the isolated basin has changed its hydrological and ecological conditions tremendously. In the present study, we developed and validated the hybrid model for the Syr Darya River basin based on a combination of state-of-the-art hydrological and machine learning models. Climate change impact on freshwater inflow into the Small Aral Sea for the projection period 2007–2099 has been quantified based on the developed hybrid model and bias corrected and downscaled meteorological projections simulated by four General Circulation Models (GCM) for each of three Representative Concentration Pathway scenarios (RCP). The developed hybrid model reliably simulates freshwater inflow for the historical period with a Nash–Sutcliffe efficiency of 0.72 and a Kling–Gupta efficiency of 0.77. Results of the climate change impact assessment showed that the freshwater inflow projections produced by different GCMs are misleading by providing contradictory results for the projection period. However, we identified that the relative runoff changes are expected to be more pronounced in the case of more aggressive RCP scenarios. The simulated projections of freshwater inflow provide a basis for further assessment of climate change impacts on hydrological and ecological conditions of the Small Aral Sea in the 21st Century.
The Aral Sea desiccation and related changes in hydroclimatic conditions on a regional level is a hot topic for past decades. The key problem of scientific research projects devoted to an investigation of modern Aral Sea basin hydrological regime is its discontinuous nature – the only limited amount of papers takes into account the complex runoff formation system entirely. Addressing this challenge we have developed a continuous prediction system for assessing freshwater inflow into the Small Aral Sea based on coupling stack of hydrological and data-driven models. Results show a good prediction skill and approve the possibility to develop a valuable water assessment tool which utilizes the power of classical physically based and modern machine learning models both for territories with complex water management system and strong water-related data scarcity. The source code and data of the proposed system is available on a Github page (https://github.com/SMASHIproject/IWRM2018).
The course timetabling problem can be generally defined as the task of assigning a number of lectures to a limited set of timeslots and rooms, subject to a given set of hard and soft constraints. The modeling language for course timetabling is required to be expressive enough to specify a wide variety of soft constraints and objective functions. Furthermore, the resulting encoding is required to be extensible for capturing new constraints and for switching them between hard and soft, and to be flexible enough to deal with different formulations. In this paper, we propose to make effective use of ASP as a modeling language for course timetabling. We show that our ASP-based approach can naturally satisfy the above requirements, through an ASP encoding of the curriculum-based course timetabling problem proposed in the third track of the second international timetabling competition (ITC-2007). Our encoding is compact and human-readable, since each constraint is individually expressed by either one or two rules. Each hard constraint is expressed by using integrity constraints and aggregates of ASP. Each soft constraint S is expressed by rules in which the head is the form of penalty (S, V, C), and a violation V and its penalty cost C are detected and calculated respectively in the body. We carried out experiments on four different benchmark sets with five different formulations. We succeeded either in improving the bounds or producing the same bounds for many combinations of problem instances and formulations, compared with the previous best known bounds.
On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator in depends Riesz continuously on perturbations of local boundary conditions The Lipschitz bound for the map depends on Lipschitz smoothness and ellipticity of and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.