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[EQUATION] Equation ( 15 ) has important consequences (cf. Fig. 4 for illustration). It shows that for long channels, where [MATH] , the fraction of the occupied sites at the jamming transition tends to one half: [MATH] when [MATH] . This means that long channels can be filled up almost to half of their maximal capacit... |
3.1 Jamming and saturation of the flux through the channel Although the transport efficiency [MATH] decreases with the increasing flux [MATH] , the total transmitted flux [MATH] saturates at large fluxes ( [MATH] ) to the limiting value |
[EQUATION] (for [MATH] ). This saturation of the transmitted flux at large incoming flux [MATH] is another manifestation of the jamming of the channel entrance by the particles inside. Indeed, equation ( 13 ) shows that the density at the entrance [MATH] tends to [MATH] , as [MATH] . In other words, the flux saturates ... |
By contrast, the exit site [MATH] is not completely blocked even at high [MATH] and [MATH] ) as [MATH] . Thus, even at very large fluxes, when the entrance site is completely blocked, the channel is not fully occupied. From equation ( 15 ), the number of particles in the channel is |
[EQUATION] In particular, for long channels ( [MATH] ), the channel occupancy in the saturated limit is [MATH] . Also note that the saturated flux is proportional to [MATH] , and that it decreases with [MATH] |
The results of this section closely parallel Michaelis-Maenten kinetics of multi-step enzymatic reactions 46 and are important for the estimation of binding affinities from the channel transport experiments 53 54 56 , as well as for comparison with experiments on flux through artificial nano-channels ( - Sec 4). |
Comparison with experiments In experimental systems, the exit rates and the rates of transport through the channel are determined by a potentially complicated kinetics of binding and unbinding inside the channel. Can the theory adequately describe these experiments? Facilitated diffusion theories produced results consi... |
Briefly, in the experiments of Ref. 16 that we chose for comparison with theoretical predictions, transport of short DNA segments through artificial nano-channels was studied. The flux of the DNA segments through these channels was measured in two cases: 1) empty channels and 2) channels were lined with single-stranded... |
The radius of an empty channel is [MATH] nm and the channel length is [MATH] 16 . The grafted ssDNA hairpins reduce the passageway radius, which, for the purposes of comparison with the theory, we roughly estimate as [MATH] nm for the channels with DNA grafted inside. Using the value for the gyration radius of the tran... |
41 , where [MATH] is the diffusion coefficient of the transported DNA coils outside the channel 60 [MATH] is the viscosity of the solvent, [MATH] is the hydrodynamic radius of the coils 60 and [MATH] is the outside concentration of the transported DNA 42 41 . To model the finite capacity of the channel, we estimate the... |
51 48 49 (cf. also Appendix). Finally, [MATH] , where [MATH] is the reduction in the exit rate due to the trapping inside the channel 27 (cf. also Appendix). We return to the question of how [MATH] is related to the actual binding energy below. The ratio [MATH] and [MATH] are the two independent fitting parameters of t... |
We first tested the model for the case without DNA segments attached inside the channel. The data (black dots) and the fit (black line) with [MATH] and [MATH] are shown in black dots in Fig. 5 Analogously, for the channels with the DNA hairpins inside, the fit of the equation ( ) to the data (red dots) is shown in the ... |
As expected, the transient binding of the transported DNA segments to the DNA hairpins inside reduces the exit rate [MATH] by a factor [MATH] . Because this energy is influenced by many factors that are poorly understood 16 , one can not easily connect the value of Z to the actual binding energy between the transported... |
42 27 35 48 . The authors of 16 measured fluxes through the channel for DNA segments possessing different numbers of mismatches to the DNA grafted inside and found that the flux decreases with the number of mismatches. Thus, assuming as a first approximation that the binding energy [MATH] decreases linearly with the nu... |
That the simplified theory developed in this paper can correctly reproduce the trends in the observed fluxes, and even gives semi-quantitative fit of the data for reasonable values of the parameters, is encouraging. This demonstrates that a theory that is built upon only two essential assumptions, 1) facilitation of di... |
Discussion of other effects observed in 16 , which are attributable to a conformational transition of the hairpin layer during transport, is outside the scope of the present work. |
Summary and discussion Proper functioning of living cells requires constant transport of different molecular signals into and out of the cell, as well as between different cell compartments. To carry out this task, the living cells have evolved various mechanisms for efficient and selective transport. |
One class of transport devices comprises narrow channels whose diameter is comparable to the size of the molecules transported through it. Examples include selective transport through the nuclear pore complex, bacterial porins, 12 11 10 and other non-biological transport systems such as zeolites 40 49 25 A crucial feat... |
Driven by the notion that natural evolution has optimized the function of such devices, large effort is being currently invested into the creation of artificial nano-molecular sorting devices that mimic the function of biological channels 19 13 14 16 15 55 17 18 . The design of such devices requires detailed understand... |
The precise conditions for the optimal transport selectivity through narrow channels still elude our understanding. A large body of experimental work indicates that the selectivity is often based on the differential interactions of the transported molecules with their corresponding transport channels. Moreover, interac... |
Recent theoretical works have shown that selective transport through narrow channels can arise from a balance between efficiency and speed; transient trapping inside the channel increases the probability of a molecule to pass through the channel, but leads to jamming at too high trapping times 29 26 30 27 35 31 34 48 3... |
Extending previous work, in this paper we have analyzed transport through narrow channels in the framework of generalized kinetic theory. We represent a transport channel as a sequence of positions (sites) and the transport through the channel is determined by the hopping rates from one position to another inside the c... |
We briefly summarize our major findings below. In qualitative agreement with previous work, we find that the transient trapping of the particles in the channel increases the transport probability; particles that have high exit rates do not stay in the channel long enough to reach the exit into the destination compartme... |
When the exit rate is too slow or the incoming flux is too high, the rate of particles’ entrance to the channel becomes higher than the rate of exit and the particles start to accumulate inside the channel, because the space inside is limited. This leads to two distinct effects. First, the particles inside the channel ... |
For symmetric channels, this balance between the transport probability and the obstruction of the particle entrance to the channel, determines the optimal exit rate [MATH] , (cf. Fig. 4) which maximizes the transport. This provides a basis for selectivity, whereby different molecules can be selected by the kinetics of ... |
The fact that the transport efficiency has a maximum at a certain value of the exit rate [MATH] provides a natural definition for the ’jamming transition’. Particles with the exit rates faster than [MATH] pass through the channel essentially unhindered by the interactions with other particles because they do not stay i... |
Although many particles can be crowded inside the channel, and the entrance to the channel is blocked, transmitted flux does not disappear even at high fluxes and densities, but rather saturates to the limiting value determined by the trapping time and the channel length (cf. Fig. 3 and Fig. 5.) This closely parallels ... |
In order to determine whether the theory developed in this paper can provide an adequate description of experiments, we compared predictions of the theory to the experiments reported in 16 . That work found that at low concentrations of the transported particles the flux through artificial nano-channels increases if th... |
Thus, we find that the theory based on only two main ingredients: 1) transient trapping of the molecules inside the channel and 2) crowding of the molecules in the limited space inside the channel, captures the essential features of the selective transport through nano-channels. Moreover, the theory provides verifiable... |
We expect that the effects described in this paper should play a role in selective transport through any narrow channel. For instance, the effects described in this paper might be relevant in determining the selectivity of the ion channels, although other factors might be dominant 21 22 23 24 In each particular system ... |
Finally, we note that the theory developed in this paper can also be applied to other signal-transducing schemes, such as signalling cascades and multi-step enzymatic reactions 61 62 63 46 |
The author is thankful to C. Connaughton, B. Chait, I. Nemenman, J. Pearson, A. Perelson, Y. Rabin, K. Rasmussen, M. Rout, N. Sinitsyn, T. Talisman, Z. Schuss for stimulating discussions, P. Welch for comments on the manuscript, and anonymous reviewers for helpful suggestions. This research was performed under the ausp... |
Appendix Single particle occupancy: connection to previous work In this section we show that the model of this paper can be reduced to previous models, in a proper limit. Let us assume, following 27 33 30 35 34 that already when the channel is occupied only by one particle, it prevents the entrance of others. The chann... |
In this case, the problem reduces to a ’single-site’ channel of Sec. 2.1 but with forward exit rate [MATH] , backward exit rate [MATH] that are not independent, but are determined by the internal kinetics of the channel, and are related through the single-particle dwelling time [MATH] and transport probability [MATH] .... |
[EQUATION] From equation ( ), the probability of a single particle to traverse the channel of length [MATH] (for [MATH] ) is [MATH] , and the residence time is [MATH] |
52 . Thus, we get for the transmitted flux: [EQUATION] which is identical to expressions obtained in Ref. 27 , if one bears in mind that the flux is [MATH] , where [MATH] is the concentration of the particles outside the channel. |
In is important to note that the optimal exit rate in this case is [MATH] , that is almost independent of [MATH] for long channels. This is in contrast to the model of Sec. 2.4, which takes into account multiple occupancy of the channel by many particles - where the optimal exit rate decreases with [MATH] . The optimal... |
Connection between continuum and discrete models. Discrete model of equation ( ) reduces to a continuum description of transport inside the channel, if one defines the one-dimensional particle density [MATH] where [MATH] is the distance between the ’sites’, so that [MATH] , with a diffusion coefficient [MATH] |
27 42 41 . For comparison with real systems, one-dimensional diffusion inside the channel must be matched to the three-dimensional diffusion outside the channel, through the choice of [MATH] (see e.g. 41 55 27 28 ). For clarity, we re-derive this connection here without the inter-particle interactions inside the channe... |
We denote the three-dimensional concentration of particles at the left side far away from the channel as [MATH] ; we assume that concentration on the right side far away from the channel is zero. At steady state, a density profile will be established such that the flux through the pore is [MATH] , the (three-dimensiona... |
At steady state, the flux that enters the channel from the left is 41 [EQUATION] where [MATH] is a geometrical pre-factor that depends on the shape of the channel opening; [MATH] for a circular opening 41 . Note that if all the impinging particles would go through the channel, the entering flux would be [MATH] - the fl... |
The flux that exits the channel to the right is 41 27 [EQUATION] The flux inside the channel, for a flat potential profile, is: 29 28 42 26 |
[EQUATION] where [MATH] is the average inverse Boltzmann factor of the attractive energy inside the channel, [MATH] Solving the above equations, we get: |
[EQUATION] And thus the fraction of the transmitted flux is [EQUATION] On the other hand, equation ( ) gives without jamming ( [MATH] |
[EQUATION] Finally, choosing [MATH] , the discrete and continuous formulations become equivalent as long as [MATH] 28 27 42 45 The distance between the ’sites’ models the excluded volume interactions between the particles. In this paper we make the most parsimonious choice: the distance between sites is equal to the si... |
Expressions for [MATH] For completeness, we present here the expressions for the general case [MATH] [MATH] The optimal exit rate (for the values of [MATH] and [MATH] when the optimum exists): |
[EQUATION] The channel occupancy [EQUATION] and the saturation current in the [MATH] limit: [EQUATION] Figure Legends Figure 1. Schematic diagram of transport through a channel |
A. Schematic illustration of the transport through a narrow channel. B. Kinetic diagram of a one-site channel. C. Kinetic diagram of a two-site channel. |
Figure 2. Kinetic diagrams of transport through a channel of an arbitrary length A. Symmetric channel consisting of [MATH] positions (sites). The particles enter the channel at a site [MATH] with an average rate [MATH] B. Equivalent energetic diagram in the case when the exit rates are determined by the interaction (bi... |
C. Equivalent geometry of the channel in the case when the exit rates are due to spatial bottlenecks at the channel ends. Figure 3. |
Efficiency of transport through a channel of an arbitrary length A. Transport efficiency as a function of the exit rate for [MATH] [MATH] for different entrance sites [MATH] . Black line: [MATH] , gray line: [MATH] ; corresponding dashed lines show the probability of a particle to traverse the channel; it is identical ... |
C. Transmitted flux [MATH] - cf. equations (8) and (16), as a function of the normalized incoming flux [MATH] ; black line: [MATH] , dashed line [MATH] [MATH] [MATH] . Note that the transmitted flux saturates to a constant value [MATH] in the jammed regime. D. Optimal exit rate as a function of the channel length [MATH... |
Figure 4. Occupancy of the channel at the jamming transition A. Occupied fraction of the channel at the jamming transition, [MATH] , as a function of the channel length [MATH] , for different values of the incoming flux [MATH] . It shows that the channel can be occupied to a considerable degree - up to half of the avai... |
Figure 5. Flux through nano-channels: comparison with experiment A. Flux through the nano-channel as a function of the outside concentration of the transported ssDNA. Black dots - experimental data from Ref. 16 for a nano-channel without trapping inside. Corresponding black line - theoretical fit from eq. ( ) with [MAT... |
B. Reduction of the flux through the channel as a function of the number of mismatches between transported ssDNA and the ssDNA hairpins grafted inside, relative to the flux of the perfect complement ssDNA measured at the feed ssDNA concentration [MATH] M. Dots - experimental data from Ref. 16 for a single mismatch at t... |
Figure 6. Three-dimensional diffusion outside the channel Schematic illustration of the three-dimensional diffusion outside the channel and one-dimensional diffusion inside. See text in Appendix. |
# Source: arxiv 0811.3389 # Title: Identity and divergence of protein domain architectures after the Yeast Whole Genome Duplication event # Sections: all # Downloaded: 2026-03-02T08:42:23.044207+00:00 |
Identity and divergence of protein domain architectures after the Yeast Whole Genome Duplication event D. Fusco 1,‡ , L. Grassi [MATH] A. L. Sellerio 1,‡ , D. Corà [MATH] B. Bassetti 1,4 , M. Caselle [MATH] , M. Cosentino Lagomarsino 1,4∗ |
Università degli Studi di Milano, Dip. Fisica. Via Celoria 16, 20133 Milano, Italy. Università degli Studi di Torino, Dip. Fisica Teorica, Via Giuria 1, 10125 Torino, Italy. |
I.N.F.N. Torino, Via Giuria 1, 10125 Torino, Italy I.N.F.N. Milano, Via Celoria 16, 20133 Milano, Italy [MATH] corresponding author, email: Marco.Cosentino-Lagomarsino@unimi.it, |
Tel. +39 02 50317477 ; [MATH] equal contribution. Abstract Analyzing the properties of duplicate genes during evolution is useful to understand the development of new cell functions. The yeast |
S. cerevisiae is a useful testing ground for this problem, because its duplicated genes with different evolutionary birth and destiny are well distinguishable. In particular, there is a clear detection for the occurrence of a Whole Genome Duplication (WGD) event in S. cerevisiae , and the genes derived from this event ... |
Introduction Genomes possess a high degree of redundancy in the information they encode for . Considering protein-coding genes, there is strong evidence |
that this redundancy has arisen from gene duplication events. Such duplications can involve individual genes, genomic segments or whole genomes. The yeast S. cerevisiae has arisen from an ancient whole-genome duplication |
The study of gene duplications is useful for understanding the evolution of proteins. Proteins descending from a common ancestor homologs ) are usually identified by sequence alignment methods. However, such methods typically have two main hindrances: ) not taking into account directly the protein folding, which persis... |
, compact structure , function and evolution . Several authors proved the usefulness of structural domain assignments in identifying homology. This implies that the duplicates tend to maintain their structures. This observation raises two interesting questions. The first one is how reliable the structural homology assi... |
We addressed the first question by implementing an algorithm for detecting homology via structural domain assignments and comparing the results with the ones obtained by sequence alignment methods. More specifically, the description of genes at the protein domain level requires: ( ) the construction of a protein domain |
architecture database, containing a description of each protein, in term of the domains that form it; ( ii ) the implementation of homology criteria between the entries of the database. This method is limited by our partial knowledge of protein domains, so that the architecture data suffer from incomplete coverage. Fur... |
The second question arises from the fact that gene duplications drive evolutionary innovation, by providing raw material to develop new functions. In particular it is interesting to understand how the whole-genome duplication event reshape the genome in a distinct way from local duplications and how this is reflected b... |
Results Homology assignment by domain characterization The superfamily domain coverage spans one third of the genomes we examined. According to the SUPERFAMILY database, v. 1.69 |
, for S. cerevisiae there is a total of 6702 sequences, 3346 (50%) of which with at least one assignment. The coverage is approximately 34% of total sequence, and 85% of domains are produced by duplication. The figures for |
K. waltii are similar: 2932 (56%) sequences on 5214 were given at least one assignment, representing the 36% of total sequence covered; 84% of domains are produced by duplication. |
In order to study homology from the structural domain viewpoint, we implemented three homology criteria based on domain architectures |
. Criterion defines two proteins as homologs if their domains architectures coincide (i.e. they contain the same domains in the same order). Criterion |
allows for multiple repetitions of the same domains. The biological hypothesis behind this criterion is that, after duplication, changes may occur to the architecture of the proteins, by mechanisms such as internal duplication (e.g. by unequal crossing over), generating architectures containing multiple repetitions of ... |
We compared homology classes those defined by sequence alignment methods. This test was divided in two different steps. First, we evaluated the fraction of homology relationships identified for the WGD (by Kellis et al |
) and by general sequence alignment methods (Ensembl-Compara ) that are also identified by criteria , and . The results of this analysis are shown in Table . These results confirm the efficiency of domain-based classifications in detecting evolutionary relatedness among proteins (as observed in |
). Specifically, they indicate that even the most stringent homology criterion , is able to find the majority of triplets (72%), pairs (67%) and Ensembl Compara homology classes (64%). The other criteria perform better; in particular, criterion retrieves more than 90% of the information in blocks of conserved syntheny.... |
Secondly, we quantified the fraction of paralogs not recovered by Ensembl, for each paralogy class defined by the three homology criteria (figure ). All three criteria define a significant fraction of classes that are not recovered by Ensembl. Notice that criteria and follow qualitatively similar trends and produce a s... |
paralogy relations. Figure shows the limitations of both criterion and . The former, being more restrictive, builds small homology classes and consequently the probability that a whole class is not recognized by Ensembl is higher. The latter builds wide homology classes associating far away homologs. The consequence is... |
will almost certainly contain some Ensembl homologs, as shown by the small number of classes that are not recovered. On the other hand, the same classes rarely contain Ensembl homologs |
only and consequently are rarely completely covered. Domain architecture evolution in WGD and non-WGD duplicates Duplicate gene pairs must undergo an altered selective regime that leads to an asymmetry emerging at different levels, for example as an increase in the rate of protein sequence evolution. Furthermore, genes... |
. Among the possibilities, there is a process by which one copy maintains the original function, and thus is constrained by selection, leaving the other one free to evolve, as originally hypothesized by Ohno |
and supported by evidence in yeast . However, theoretical and experimental work has argued that both duplicates can evolve independently at the same rate |
. We considered the question of testing the consequences of these processes at the domain level. We followed the evolution of WGD duplicates through their domain architectures, i.e. the ordered sequence of domains forming the proteins. The length of an architecture is the total number of domains and gaps that form it. ... |
duplication, ( ii ) sequence divergence leading to structural changes in domains and ( iii ) domain insertions. In order to quantify globally the changes in protein architectures, we introduced two scoring methods that define a quantitative notion of relatedness between architectures. The first, called “domain score” i... |
To test for asymmetry, we compared for each WGD triplet the two S. cerevisiae WGD paralogs with the respective K. waltii ortholog, detecting the best- and worst- matching paralog. This was done using the domain score and the architecture score between both paralogs and their K. waltii ortholog. Table shows the fraction... |
K. waltii . Furthermore we called F1 the fraction of triplets in which only one of the two S. cerevisiae paralogs has domain (or architecture) score one with the corresponding K. waltii |
ortholog. Comparing proteins with the architecture score we detect 65% of F2 triplets and 16% of F1 triplets, while by using the less restrictive domain score we detect 80% of F2 triplets and 11% of F1 triplets . This indicates that some duplicate proteins tend to evolve without changing their domain composition but ra... |
K. waltii is larger in randomized instances. Consequently, the domain architectures of WGD duplicates are typically more balanced than expected from the null model, rather than more asymmetric. |
We extended the analysis of paralog divergence to non-WGD paralogs, taking into account the duplication date reported by Wapinski et al. |
. Measuring the average domain and architecture scores as a function of duplication age, and their standard deviations on the age sets, we find that the domain score is roughly constant and very close to one (figure ), indicating that even ancient paralogs maintain similar domain composition. The more stringent archite... |
should not be seen at the domain level. In order to test this directly, we have considered the distribution of domain and architecture score for single-copy S. cerevisiae genes versus their K. waltii |
orthologs and we have compared this result with domain and architecture score of double-copy S. cerevisiae genes (WGD duplicates) versus their K. waltii orthologs. The two histograms perfectly overlap for both architecture and domain score (figure S1 ). |
Functional divergence and duplication age In order to gain more insight into the divergence of duplicates at the domain level, we evaluated how the same duplicate proteins tend to diverge in their function. Specifically, we calculated the Gene Ontology (GO) term similarity between paralogs for each of the GO branches (... |
. The results, shown in figure , indicate that for all the three GO branches, recent duplicates tend to be more similar than older ones. Indeed, average GO term similarity values tend to decrease as the duplication time increases. On the other hand, the mean GO term similarity of duplicates in all duplication date grou... |
The same trends are also visible from the histograms of GOsim and domain-based similarity scores of all duplicate pairs (figure ). The pairs of duplicates having high domain-based similarity is consistently higher in number than those with high GO similarity, but this trend is weaker for the “molecular function” taxono... |
(figure ). The latter distribution has the lowest peak at one and the highest value at low scores, confirming that strong migration in protein sequence accompanies stability of domains and functions. |
In order to exclude biases of computational nature that could influence the results, we repeated the analyses with different conditions. Firstly, not all proteins S. cerevisiae are covered entirely by domains, but some have gaps. Excluding from the analysis proteins with gaps should confirm that the functional migratio... |
shows that this is indeed the case. Secondly, Gene Ontology annotations inferred from computational evidence could generate false positives in GO similarity, especially in the case of recent duplicates with significant sequence similarity. To circumvent this possibility, we restricted the analysis to manually curated g... |
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