2012: Ten Converging Signs
This finding may indicate that effectors collectively target different parts of the overall network rather than a functionally coherent subnetwork. Gene Ontology GO enrichment analysis on TAIR10 annotations of effector targets returned mostly high level categories of regulatory processes Table S2 , including defense signaling, potentially due to our previous data being incorporated into the TAIR annotation.
Both of these phytohormones play important roles in plant pathogen interactions Robert-Seilaniantz et al. The functional categories are consistent with analysis of specific bacterial effectors and their targets Deslandes and Rivas, ; Win et al. The frequency distribution of the randomly observed values obtained in these simulations was used to calculate an experimental P value for OEC interactions with host proteins Figure 2B. Thus, OEC effectors converge onto a small set of host proteins. Applying this novel analysis to Hpa and Psy effectors revealed the same striking and significant intraspecies convergence as observed for OECs Figure 2D,E ; exp.
Thus, effectors of pathogens from diverse kingdoms exhibit intraspecies convergence onto host proteins. We performed simulations for all pairwise, and the three-fold, combinations of the three pathogens. In each case the experimentally observed overlap was significantly higher than expected by chance Figure 2 G—H ; exp.
Thus, we observed significant intra- and interspecies convergence of effectors from three evolutionarily highly diverse hemi- biotrophic pathogens. This strongly suggests that the convergence is the product of natural selection, and that the respective host proteins are functionally relevant to the pathogen. We focused on exon insertions early in genes and tested independent alleles when available Table S3. We did not confirm each line as an mRNA null leaving the formal possibility of phenotypic false negatives.
Emwa1, Emoy2, and Noco2. The Hpa isolates were selected to detect both enhanced disease susceptibility eds and enhanced disease resistance edr phenotypes. The 63 interactors with infection phenotypes will be referred to as effector targets. These host proteins may facilitate pathogen sustenance in the host. Alternatively, they may repress an activator of immune signaling, a function possibly stabilized by the interacting virulence effectors; thus, in the absence of the putative negative regulator, effector action is neutralized and host resistance increases.
Their existence suggests that these host proteins may have disease-specific functions. Heat-map summarizing the outcome of phenotypic analyses of mutants in genes encoding the indicated effector-interactors in infection assays with the noted pathogens and developmental stages. Host proteins are sorted by the number of pathogens interacting with them, then by number of observed phenotypes and performed assays. Mutant lines for 59 proteins interacting with effectors from a single pathogen did not show any disease phenotype and are not shown. Refer to Table S3 for raw data for all phenotyped loci and Figure S3 for complete results for all tested lines.
Fraction of mutant lines for proteins interacting with effectors from the indicated number of pathogens that exhibited an edr , eds or divergent phenotypes across the assays.
We wondered whether the host proteins that interact with effectors from multiple pathogens were also targeted repeatedly by the suite of effectors from any individual pathogen. Before learning of the JOR result, Jason and I had discovered a uniform bound on the number of -fluctuations that works in the more general setting of a uniformly convex Banach space. Our results suggested that effectors from these evolutionarily diverse pathogens converge onto common host proteins, which were characterized by a high interaction degree and a central position in the host protein network. We confirmed these findings via co-immunoprecipitation for single effectors from each pathogen Figure 4D. Archived from the original on 18 September
Pie chart representation of the phenotype density; the number of observed phenotypes relative to individual assays performed for that group. Each pie displays data for proteins that interacted with effectors from the number of pathogens given in the center. Fraction of mutant lines for proteins targeted by the indicated number of Gor effectors for which edr or eds phenotypes were observed. Numbers above bars indicate the number of targets in that class. As in D, but for proteins targeted by the indicated number of Hpa effectors for which edr or eds phenotypes were observed.
Numbers above bars indicate the number of effector-interactors in the class. The nonrandom nature of effector-host protein connectivity suggested that the network topology of the plant immune system is the product of natural selection, and consequently that the convergence we observed is biologically meaningful. We explored whether a relationship exists between intra- and interspecies effector convergence and altered pathogen infection phenotypes. A host protein was considered a point of intraspecies convergence when at least two effectors from the same pathogen interacted with it, and an object of interspecies convergence when effectors from different pathogens interacted with it.
Effector-interactors were binned by whether they interacted with effectors from three, two or one pathogen s Figure 2F. We noted a positive correlation between the degree of interspecies convergence and the probability of observing an infection phenotype in that bin Figure 3B. To exclude that our observation was due to deeper phenotypic interrogation of the most highly targeted proteins, we also calculated the phenotype density for proteins interacting with effectors from three, two, or one pathogen as the fraction of assays individual squares in Figure 3B in which an edr or eds phenotype was observed divided by the total number of assays performed in this group.
This analysis confirmed the correlation between convergence and phenotypic relevance of the targeted host protein Figure 3C. We then evaluated the phenotypic relevance of genes encoding host proteins that are objects of intraspecies convergence Figure 2C—E. We binned host proteins according to the number of effectors from each pathogen interacting with them and evaluated how often an altered immunity phenotype could be observed with the respective pathogen.
We found edr or eds phenotypes for all mutants in genes encoding the two proteins targeted by more than five Gor or Hpa effectors Figure 3D,E. The fraction of phenotypically validated host targets decreased proportional to the degree at which effectors are connected to the respective plant proteins Figure 3D,E. Thus, the extent of intraspecies effector convergence onto host targets is also directly correlated to the functional relevance of the targeted proteins.
We wondered whether the host proteins that interact with effectors from multiple pathogens were also targeted repeatedly by the suite of effectors from any individual pathogen. All nine Arabidopsis proteins targeted by effectors from all three pathogens are also intraspecies convergence points for at least one pathogen Figure S3. Furthermore, 16 of 23 proteins targeted by effectors from two pathogens are also points of intraspecies convergence.
TCP14 was the most targeted host protein, interacting with 23 distinct OECs, 25 Hpa effectors and four Psy effectors, and exhibiting disease phenotypes in all assays Figure 3A. The related family members TCP13, TCP15 and TCP19 were also targeted multiple times by effectors from at least two pathogens and exhibited altered infection phenotypes.
These findings suggest an important and possibly universal role of this class of TFs during infection, consistent with their emerging role as targets of phytoplasma effectors Sugio et al. To independently validate the convergence concept using cell biological methods, we tested TCP14 for co-localization with 11 of the 25 interacting Hpa effectors, focusing on those effectors demonstrated to localize to the nucleus Caillaud et al. We confirmed these findings via co-immunoprecipitation for single effectors from each pathogen Figure 4D.
We thus validated in planta the majority of effector interactions with the most heavily targeted host protein, TCP14, consistent with our claim that the observed convergence onto the plant protein is not an artefact. The same settings were then applied to all assays below. The lower panel exhibits an enlarged view of a representative nucleus boxed in the upper panel. The histogram illustrates the intensity of fluorescent signal across the path indicated by the red arrow. All confocal pictures were taken 40—48 h after infiltration of Agrobacterium strains expressing the different fluorophore—tagged proteins.
See also Figure S4 and Table S4. We sought evidence for the evolutionary relevance of our effector targets from population genomics. We used the complete genomes of 80 accessions sequenced in the context of the Genomes project and mapped on the Col-0 reference genome Cao et al.
These were collected in eight regions distributed over Europe and Asia, where Arabidopsis naturally occurs and thus provide a large spatial and phylogenetic sample of genotypes adapted to different environments http: In addition, we constructed consensus protein sequences from 81 Arabidopsis accessions 80 plus the Col-0 reference by majority voting Altmann et al. We asked whether the direct effector-interactors exhibit evidence for balancing selection, as indicated by positive D T values Figure S5A.
No significant deviation from random expectation could be detected for any of four effector-interactor groups: For three of the four groups the mean of D T for effector-interactors is lower than that of random controls, whereas for the group targeted by three pathogens the mean is slightly higher. The lack of a strong signal can likely be explained by our previous observation that many effector targets are central proteins in the network, which likely cannot tolerate much variation without adverse effects on protein function.
We therefore asked whether instead there might be evidence for the selective pressure imposed by pathogens in the network neighborhood of the effector-interactors. To this end, we explored whether the AI-1 MAIN interaction partners of effector-interactors are subject to balancing selection, but no such evidence could be detected for any of the effector-interactor groups P 0.
It is possible that a majority of interacting proteins mediating non-immune functions may mask any potential signal from the few interacting proteins involved in immune functions. We therefore adopted an inverse approach and investigated whether effector-interactors are preferential interaction partners of proteins encoded by genes under balancing selection. The number of interacting effector-interactors in the real AI-1 MAIN network is always significantly higher than random across a range of cut-offs, demonstrating a preferential interaction of proteins encoded by genes under balancing selection with our effector-interactors.
These findings are supported by similar results obtained with an AAP based ranking, although with slightly different top-ranking proteins. These polymorphic proteins show the greatest signal with effector-interactors targeted by effectors from three pathogens. Schematic illustration of the analysis in B—E: Effectors are shown for illustration only and not included in the analysis. Analysis as described in A. Data from AI-1 MAIN are shown as red dots, the black line shows the median of 1, randomly rewired networks, grey shaded areas show the 25 th and 75 th percentiles of values from rewiring controls.
As in B but counting proteins interacting with effectors from two or three pathogens. As in B but counting proteins interacting with effectors from three pathogens. As in B but counting proteins whose mutation caused altered immune phenotypes. Among the 13 interaction partners of the five most selected proteins are eleven effector-interactors, including the five most targeted proteins. See also Figure S5 and Table S5. The underlying biological reasons of how the increased genetic variation is beneficial in the evolutionary battle remain to be elucidated.
The evidence for preferential interaction of proteins encoded by genes under balancing selection with effector-interactors contrasts with the conservation of effector-interactors themselves, which show signs of purifying selection Figure 5F. Together these data demonstrate that at least a subset of proteins targeted by multiple evolutionary distant pathogens are under purifying selection and in such instances variation at the level of neighbors in the protein interaction network becomes a substrate for balancing selection.
We identified interactions between candidate virulence effector proteins from the obligate biotrophic powdery mildew fungus Golovinomyces orontii and proteins from its host, Arabidopsis. We added these to previously defined interactions between the same set of host proteins and effector suites from two pathogens derived from different kingdoms. Exercise 11 of this previous blog post.
In the abelian case, we used the largest for which was non-trivial as the degree of. This turns out to not be a good choice in the nilpotent case, because the crucial ultratriangle property 3 does not hold for this concept of degree. For instance, if is a two-step nilpotent group, and are non-commuting elements of , then the sequences would ostensibly have degree with this definition, but the product.
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The symmetry property 4 can also be shown to break down. Fortunately, the theory of polynomial sequences in nilpotent groups has been understood since the work of Leibman. The trick is not to view the coefficients appearing above as roaming unrestrictedly in the whole -step nilpotent group , but to restrict some or all of these coefficients to subgroups in the lower central series , defined by setting and for all.
Given natural numbers , we then say that a sequence has filtered degree at most if, when using the Taylor expansion 7 , we have whenever. Thus, for instance, if , and , then the sequence has filtered degree at most. A fundamental result of Leibman proven for instance in this previous post asserts that if the sequence is superadditive in the sense that whenever , then the collection of polynomial sequences of filtered degree at most form a group. A related fact is that if a sequence has filtered degree at most for some superadditive , then any derivative of has filtered degree at most which is still superadditive.
If we let be the set of all superadditive degree sequences, we can order such sequences lexicographically by declaring if there is an with and for all. This makes a well-ordered set, and then we can define the filtered degree of a polynomial sequence to be the minimal in for which has filtered degree at at most. Thus, for instance, in an -step nilpotent group, a sequence with non-trivial would have filtered degree. If we view as being smaller than any element of , we see that the ultrametric axioms are still obeyed, and one can still run the argument more or less exactly as given above; we leave the details to the interested reader.
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- Walsh’s ergodic theorem, metastability, and external Cauchy convergence.
Jason Rute and I recently noticed that in the case of the mean ergodic theorem, one has something even stronger than a uniform metastability result. In the Hilbert space setting, a very elegant variational inequality due to Jones, Ostrovskii, Rosenblatt implies that a sequence of ergodic averages has at most many -fluctuations.
Before learning of the JOR result, Jason and I had discovered a uniform bound on the number of -fluctuations that works in the more general setting of a uniformly convex Banach space.
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Our result is not sharp when specialized to the case of a Hilbert space, however, and we were unable to strengthen it to anything like the JOR inequality. We are curious to know how far this quantitative uniformity extends. Does anything like the JOR square function inequality carry over? In an appendix he also considers nonstandard formulations of such uniformities. So maybe your nonstandard argument can be adapted to yield the stronger result?
But it is certainly a good question…. Christoph Thiele pointed me towards this recent paper of Do, Oberlin, and Palsson which establishes such a variational result for a dyadic version of a double average such as. In practice, these dyadic harmonic analysis arguments can often be adapted to the non-dyadic case, though the details can get messier in the process.
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What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. This theorem is a norm convergence theorem in ergodic theory, and can be viewed as a substantial generalisation of one of the most fundamental theorems of this type, namely the mean ergodic theorem: While this theorem ostensibly involves measure theory, it can be abstracted to the more general setting of unitary operators on a Hilbert space: Given a commutative probability space, we can form an inner product on it by the formula This is a positive semi-definite form, and gives a possibly degenerate inner product structure on.
For future reference we record the inequalities for any , which we will use in the sequel without further comment; see e. Nonstandard analysis and metastability — We will assume some familiarity with nonstandard analysis , as covered for instance in these previous blog posts. For instance, given a sequence of standard functions , one can form their ultralimit from the nonstandard space to the nonstandard space by the formula As usual, we call a nonstandard real bounded if we have for some standard , and infinitesimal if we have for every standard , and in the latter case we also write.
We will need the following fundamental properties about nonstandard analysis: Overspill If an internal set an ultraproduct of standard sets, also known as a nonstandard set of nonstandard numbers contains all unbounded natural numbers, then there exists a standard natural number such that contains all nonstandard numbers larger than.
Loeb measure , hyperfinite case If is a non-empty nonstandard finite set i. Then the following are equivalent: Now we consider more general sequences, in which the above notions of convergence begin to diverge: We say that the sequence is internally Cauchy if for every nonstandard , there exists a nonstandard such that for all nonstandard. We say that the sequence is externally Cauchy or metastable if for every standard , there exists a standard such that for all standard.
We say that the sequence is asymptotically stable if whenever are unbounded. These three notions are now distinct, even for a simple nonstandard metric space such as the ultrapower of the unit interval with the usual metric, as the following examples demonstrate: If is an unbounded natural number, then the nonstandard sequence is internally Cauchy, but not externally Cauchy or asymptotically stable. If is an unbounded natural number, then the nonstandard sequence is internally and externally Cauchy, but not asymptotically stable. If is an unbounded natural number, then the nonstandard sequence is externally Cauchy, but not internally Cauchy or asymptotically stable.
If is an unbounded natural number, then the nonstandard sequence is asymptotically stable and externally Cauchy, but not internally Cauchy. Any monotone bounded nonstandard sequence of nonstandard reals is automatically both externally Cauchy and internally Cauchy, but is not necessarily asymptotically stable, as the example above shows. The property of being externally Cauchy is only dependent on an initial segment of the sequence: The same claim is certainly not true for the notions of internally Cauchy or asymptotically stable, as can be seen by considering examples such as , and.
The property of being externally Cauchy is closed under external uniform limits; if is a nonstandard sequence such that for every standard one can find an externally Cauchy sequence with for all , then is itself externally Cauchy. The same claim holds as well for asymptotically stability, but not for the internal Cauchy property unless one allows to be nonstandard.
One can equate these three nonstandard notions of convergence with standard notions as follows: The nonstandard sequence is internally Cauchy if and only if the standard sequences are Cauchy for all sufficiently close to. The nonstandard sequence is externally Cauchy if and only if for every standard and standard , there exists a standard such that for all and all sufficiently close to. The nonstandard sequence is asymptotically stable if and only if for every standard , there exists a standard such that one has for all standard and all sufficiently close to.
The nonstandard sequence is externally Cauchy if and only if there exists an unbounded such that is asymptotically stable up to , in the sense that for all unbounded. A key property for us is that external Cauchy convergence is also preserved by hyperfinite averages involving a nonstandardly finite number of sequences: For each standard and standard natural number , let denote the subset of given by the formula These sets are not internal subsets of , but are instead -internal i.
We then have for that As can be arbitrarily small, this gives the external Cauchy convergence of as desired. We first observe that if one takes the bounded elements of and forms the Hilbert space completion using the standard norm one obtains a Hilbert space. Thanks to the bound for all bounded in , we see that these ergodic averages can be defined in , and so it suffices to show that the averages are externally Cauchy in for all.
We expand out where all expressions have been extended to nonstandard values of or in the usual fashion, and operations are extended from the bounded elements of to by continuity. With a little bit of rearrangement, this expression can be rewritten as , where the dual function for any is defined by the formula The terminology of dual functions originates from this paper of Ben Green and myself. We expand Fixing , we now restrict to the regime , thus for all standard. If we then return to the range , this creates an error of norm , and so in for all. We consider the averages and our task is to show that the form an external Cauchy sequence in.
We have the easily verified bound Because of this, the linear operator can be uniquely continuously extended to a linear operator from to , where is defined as the Hilbert space completion of the bounded elements of under the norm In particular quotients out all the elements of infinitesimal norm.
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With a little bit of rearrangement, this expression can be rewritten as , where the dual function for any is defined by the formula Thus, if we let be the linear span in of all functions of the form with unbounded and , and is orthogonal to , then vanishes in for any unbounded. Namely, we expand Fixing , we now restrict to the regime , thus for all standard.
If we then return to the range , this creates an error of norm , and so with both sides being interpreted in. It is easy to see that any polynomial nonstandard function can be uniquely expressed in the discrete Taylor expansion form for some finite number of group elements with non-trivial or with if is trivial. We observe the ultratriangle inequality with the inequality being equality if have different degree; we also have the symmetry property Also, we observe the key fact that if is a non-trivial polynomial sequence, then for any , the derivative defined by is a polynomial sequence of strictly smaller degree.
We can now place an ultrametric on , with the distance between two polynomials defined as with the convention that. We define the weight of to be the weight of the augmented tuple relative to the final element of the tuple: Now let be a nonstandard integer, and consider the -reduction where. We first consider the weight of the augmented tuple relative to. Observe that for any , one has thus, relative to , and are in the same equivalence class. Performing an -reduction, we obtain with a weight vector now reduced to. Performing another -reduction, we obtain but now we need to permute to move which has maximal degree and minimal distance to to the end, giving with a weight vector now of.
Performing another -reduction, we obtain which after eliminating duplicates and moving which has maximal degree and minimal distance to to the end, gives with a weight vector of. Another -reduction then gives note the elimination of all quadratic terms which after eliminating duplicates becomes with a weight vector of. Performing yet another -reduction gives with a weight vector of. If is a polynomial nonstandard sequence, then by many applications of discrete analogues of Baker-Campbell-Hausdorff formula, we can as before place uniquely in the Taylor expansion form for some standard finite number of group elements of ; see e.
For instance, if is a two-step nilpotent group, and are non-commuting elements of , then the sequences would ostensibly have degree with this definition, but the product where is the commutator of and , would then have degree , thus contradicting 3. Recent Comments Anonymous on B, Notes 1: The Lagrangian… Anonymous on B, Notes 1: The Lagrangian… peeterjoot on B, Notes 1: The Lagrangian… B, Notes 1: The L… on PCM article: Differential… B, Notes 1: The L… on A, Notes 2: Weak solutions… B, Notes 1: The L… on A, Notes 0: Physical deriva… Anonymous on A, Supplemental: This suggests that the very high rainfall in December over the Appalachians and the northeastern United States has led to greater-than-normal amounts of sediment in the rivers and streams of the Ohio River watershed.
Jialing River on the right stretches kilometers. In the city of Chongqing it falls into Yangtze River. The clean water of Jialing River meets the brownish yellow water of Yangtze River. The Yangtze becomes more powerful after it absorbs the water of Jialing as it continues its path, passing through the Three Gorges and stretching thousands of miles. For 6 km 3. It is one of the main tourist attractions of Manaus, Brazil. This phenomenon is due to the differences in temperature, speed and water density of the two rivers.
It winds its way south into Utah, turning east into Colorado and finally back south down into Utah where it terminates at the confluence of the Colorado River in Canyonlands National Park in San Juan County. The contrast is striking as the clear Thompson River water joins with the muddy Fraser. Devprayag is a town and a nagar panchayat municipality in Tehri Garhwal district in the state of Uttarakhand, India.
The Alaknanda rises at the confluence and feet of the Satopanth and Bhagirath Kharak glaciers in Uttarakhand. The headwaters of the Bhagirathi are formed at Gaumukh, at the foot of the Gangotri glacier and Khatling glaciers in the Garhwal Himalaya.