BACKGROUND
wnt induced signaling pathways play essential roles in development and disease  <cit> .
currently, two initiation models of the canonical wnt/β-catenin signaling pathway have been proposed as illustrated in figure  <dig> <cit> .
one could be referred to as the sequential recruitment/amplification model, in which wnt stimulation is proposed to recruit the scaffold protein axin to approach the membrane through the bridging interactions between frizzled  and dishevelled , as well as between dvl and axin.
gsk <dig>  in association with axin thereafter is able to phosphorylate the lrp5/ <dig> pppp motif near the membrane.
initial phosphorylation creates a docking site for axin and thereby recruits additional axin-gsk <dig> to promote further lrp <dig> phosphorylation  <cit> .
the second model is the signalosome/aggregation model.
recent results showed that a signalosome is formed by aggregated lrp <dig> and axin when wnt is present.
clustering of lrp <dig> leads to the phosphorylation of t <dig> by ck1γ  and subsequent phosphorylation of the pppp motif by gsk <dig>  <cit> .
phosphorylated lrp <dig> recruits axin resulting in the disruption of the “β-catenin destruction complex”, which comprises axin, apc , gsk <dig> and ck1α  <cit> .
this results in the stabilization of a cytoplasmic pool of β-catenin.
free β-catenin enters the nucleus and activates gene transcription by binding to the tcf/lef  family of transcription factors  <cit> .
thereafter, the activation of lrp5/ <dig> is indispensable to initiate the downstream intracellular wnt signaling cascade, in order to stabilize β-catenin.
lrp6/lrp5/arrow belongs to a subfamily of ldl receptors   <cit> .
human lrp <dig> is a type i single-pass transmembrane protein.
its modular extracellular domain has three basic domains; the ywtd -type β-propeller domain, the egf -like domain, and the ldlr type a  domain.
this region has crystal structures present in pdb database  <cit> .
during signaling pathway initiation, wnt binds the cysteine-rich domain of fzd proteins and exhibits a wnt1-dependent association with lrp <dig> extracellular domains in vitro  <cit> .
however, the interaction between wnt and lrp <dig> is weaker compared to the interaction between wnt and frizzled  <cit> .
it is therefore more likely that a lrp6-fzd complex binds to the lrp <dig> extracellular domain.
after deletion of its extracellular domain, lrp <dig> protein could still activate the wnt/β-catenin signaling pathway  <cit> .
the lrp <dig> intracellular domain is rich in prolines and serines.
sequence alignment shows that it includes a s/t cluster and downstream five reiterated ppppx motifs, each of which contains a pppp motif phosphorylated by gsk <dig> and juxtaposed to a ck <dig> phosphorylation site  <cit> .
such dual phosphorylation is essential to stabilize the pool of β-catenin in the cytoplasm  <cit> .
the phosphorylation of the s/t cluster has also been characterized, particularly the phosphorylation of t <dig> by ck1γ  <cit> .
it is believed that the phosphorylated s/t cluster promotes downward ppppx phosphorylation  <cit> .
he’s group previously showed that a lrp <dig> mutant lacking most of the intracellular domain is a loss-of-function  <cit> .
in addition, a truncated lrp <dig> comprising its transmembrane and intracellular domain is constitutively active in wnt signaling transduction  <cit> .
a single pppp motif transferred to a lrp <dig> variant lacking the extracellular domain activates the wnt/β-catenin signaling pathway.
phosphorylated ppppx motifs provide docking sites for axin, which associates with gsk <dig> to promote proximity to lrp <dig>  <cit> .
so far, no stable structure has been obtained from this isolated and contiguous lrp <dig> intracellular domain in current structure databases.
the lrp <dig> intracellular domain is expected to be natively unstructured  since its composition is enriched with proline whose conformation is limited  <cit> .
there has been little study on the conformational behavior of lrp5/ <dig> before activation, when wnt induces signal transduction.
no matter which initiation model applies to the canonical wnt/β-catenin signaling pathway, the conformation of lrp <dig> has to face spatial constraints imposed by  the plasma membrane and  a nearby molecule or molecular assemblies, which could be neighboring lrp <dig> molecules or a wnt-fzd-dvl-axin-gsk <dig> assembly.
we hypothesize that these two spatial constraints would restrict lrp <dig> intracellular domain conformational space and so cause its conformation to adopt a more extended or elongated form before it is activated and docked by axin.
we applied our trades software package  <cit>  to simulate the conformational space of the lrp <dig> intracellular domain, as well as constraints to demonstrate an elongated conformational change of this domain occurs during wnt/β-catenin signaling pathway initiation.
we also tested whether close packing can induce a statistically observable structural change in an ensemble of unfolded states in a sequence independent manner.
RESULTS
lrp <dig> intracellular domain is predicted to be unfolded
no stable structure has been documented for the lrp <dig> intracellular domain in current structure databases.
this region is expected to be natively unstructured because it is enriched with proline and serine.
several protein disorder prediction tools predict that this region is disordered or unfolded.
figure s <dig>  gives the prediction results from disorder predictors; ronn  <cit> , iupred  <cit> , globplot  <cit> , pondr-fit  <cit>  and foldindex  <cit> .
this unfolded intracellular protein region most probably tends to have random coiled conformations, which auto-inhibits the structure itself to avoid interactions with other molecules  <cit> .
like most other disordered protein regions, it exists as an ensemble of structures which can be generated by trades in this simulation study.
radius of gyration distribution
radius of gyration  measures the openness of the whole structure.
a structure with a larger rgyr has more sparse atoms within it.
figure  <dig> displays the rgyr distribution of the initial conformational ensemble  in the lrp <dig> intracellular domain simulation experiment.
the number of generated conformers and average rgyr are provided in the second column in table  <dig>  conformers with different values of rgyr were checked.
it was observed that conformers with smaller rgyr have more compact structures; while, conformers with larger rgyr tend to adopt more open or extended conformations.
in this distribution, the conformation of the generated conformers changes from compact, to more open, to more extended as their rgyr increases.
simulation experiments were carried out both for the lrp <dig> intracellular domain  and the control sequence.
see results in table  <dig> and table  <dig>  a unix script was written to obtain  <dig> conformers that pass constraint  <dig> and constraint  <dig>  out of the initial structural ensemble.
these  <dig> conformers were then filtered by constraint  <dig> with parameter δ set to  <dig> Å and  <dig> Å. this parameter represents the distance from the vertical plane to the plane defined by the transmembrane helix and origin point .
the average rgyr of the structural ensemble gets larger after each constraint is applied, as shown in table  <dig>  table  <dig> and figure  <dig>  in both the lrp <dig> intracellular domain and control sequence simulation experiments, after each filtration, conformers in the structural ensemble surviving from the spatial constraints tend to possess more open or extended conformations.
this was also indicated by the observation that after each constraint more fractions of structural ensemble appear to have rgyr larger than the average .
in addition, when the distance δ gets smaller, the average rgyr in the structure ensemble that survived following constraint  <dig> gets bigger.
the control sequence has lrp <dig> transmembrane region and poly <dig> polypeptide substituting lrp <dig> intracellular domain.
the theoretical maximum rgyr for a completely extended chain  <dig> polypeptide) is calculated from trades to be around 90Å.
the poly <dig> polypeptide is constrained to take a β-strand conformation in its trajectory distribution file .
end-to-end distance distribution
five end-to-end distances with equal length were calculated in the lrp <dig> intracellular domain simulation experiment.
each distance contains at least one conserved ppppx motif.
see table  <dig>  for each end-to-end distance, the difference between the average rgyr of the initial structural ensemble and that of the structural ensemble after constraint  <dig> is provided in the column titled as “∆mean ”.
the difference between the average rgyr of the initial structural ensemble and that of the structure ensemble after constraint  <dig> under different values of distance δ are shown in the columns under the title of “∆mean ”.
for both the lrp <dig> intracellular domain and control sequence, out of the five end-to-end distances, the distribution of d <dig> displays largely increased mean values of the structural ensembles after each constraint.
this was indicated by the positive differences in table  <dig>  it also proves that after each constraint the average value of d <dig> gets larger.
in figure  <dig>  the d <dig> distribution curves of structural ensembles after constraint  <dig> and after constraint  <dig> move to the right of the d <dig> distribution curve of the initial structural ensemble.
meanwhile, more fractions of structural ensemble after each constraint are found at a larger value of d <dig> on the distribution curves.
this indicates that the region corresponding to d <dig>  within the lrp <dig> intracellular domain conformers in the surviving structural ensemble, prefers an elongated or extended conformation.
this region starts right from the beginning of the lrp <dig> intracellular domain and extends to the end of the first ppppx motif.
it is the closest membrane region inside the lrp <dig> intracellular domain.
additionally, as the distance δ gets smaller, the mean of d <dig> of the structural ensemble after constraint  <dig> also gets bigger.
however, for both the lrp <dig> intracellular domain and control sequence, the rest four end-to-end distances’ distributions show no prominent changes compared to the change in d <dig> distribution after applying each constraint and furthermore they overlap with each other as seen in figure s <dig> .
the table shows the length, motif contained, starting and ending positions in lrp <dig> simulation sequence, and the differences between the average rgyr of structural ensembles after each constraint and the initial structural ensemble.
discussion
lrp <dig> intracellular domain structure ensemble favors an elongation form when the wnt/β-canonical pathway initiates
in the lrp <dig> intracellular domain simulation experiment, greater proportions of the structural ensemble after each spatial constraint are observed to have rgyr of a larger value than the average in comparison with the initial structural ensemble .
it shows that the two spatial constraints make the lrp <dig> intracellular domain likely to adopt a more open or elongated conformation.
the plasma membrane and neighboring assemblies formed by wnt-fzd-dvl-axin-gsk <dig> or neighboring lrp <dig> aggregation could limit the lrp <dig> intracellular domain to form fewer numbers of random coiled structures.
instead, the lrp <dig> intracellular domain tends to form more elongated conformations as the spatial constraints exclude its volume near the membrane in the cell.
in vivo, plasma membrane and nearby assemblies or molecules could result in a natural elongation of the lrp <dig> intracellular domain when a wnt signal triggers the pathway.
such elongation behavior might grant kinases ck <dig> and gsk <dig> open access to the phosphorylation sites within the lrp <dig> intracellular domain, and subsequently lrp <dig> could be activated through these phosphorylation events.
we propose that when the wnt pathway initiates, the lrp <dig> intracellular domain is elongated to reduce the auto-inhibition before it is activated.
a conformational change occurs to the lrp <dig> intracellular domain structural ensemble after applying spatial constraints.
it is intriguing to investigate if any conformational change could occur in the subsequences within the lrp <dig> intracellular domain during wnt canonical pathway initiation.
the distributions of the five calculated end-to-end distances could reflect the openness of the subsequences in the lrp <dig> intracellular domain.
the first end-to-end distance d <dig>  which measures the openness of the region that is closest to the membrane on the lrp <dig> intracellular domain.
the distribution curves for d <dig> show that this region gets longer in more conformers out of the structural ensemble after filtration .
this implies that this near-membrane region in the lrp <dig> intracellular domain elongates or extends when its conformational space is limited by the plasma membrane and nearby assemblies or molecules.
such an extended conformation could allow ck <dig> to more easily reach the s/t cluster and initiate phosphorylation.
this may also explain the experimental finding that s/t cluster phosphorylation by ck <dig> promotes the downward activation of the ppppx motif  <cit> .
on the contrary, the end-to-end distances d <dig>  d <dig>  d <dig>  and d <dig> hardly show any changes in their distribution curves between original and filtered structural ensembles .
the means of these distributions of structural ensemble after filtration are in fact smaller than that of the initial structural ensemble.
this suggests the regions corresponding to these distances are on average less extended in the conformers surviving from filtration.
the protein regions corresponding to the five end-to-end distances are gradually further away from the transmembrane helix, which determines the location of the plasma membrane.
the region corresponding to d <dig> is the closest to the plasma membrane followed by d <dig>  the observations on the distribution curves of these distances suggest that the spatial constraints exclude to a great extent the volume of the lrp <dig> intracellular domain at the near-membrane location in the cell.
additionally, the same behaviors are observed in rgyr distributions and end-to-end distributions in the simulation experiment as for the control sequence.
we observed structural changes that can be demonstrated in a hypothetical sequence with as much as 50% proline and 50% serine.
such changes could function as a mechanism by which high rates of mutations could be tolerated whilst conserving function.
hence, it can be concluded that such an elongation process induced by membrane and neighboring assembly/aggregation is sequence independent but maybe compositional dependent.
effects of the two spatial constraints
near-membrane serves as the key point in the simulation study.
the membrane-anchor issue has been discussed in several published papers that claimed the lrp <dig> intracellular domain needs anchoring to the membrane to process signaling  <cit> .
arrow/lrp5/lrp <dig> mutants without the extracellular domain with which to anchor to the membrane constitutively activate the β-catenin pathway in mammalian cells.
the lrp <dig> intracellular domain is unable to activate the signaling pathway unless it is anchored to the membrane  <cit> .
in the simulation, the horizontal plane mimics the constraint imposed by the membrane plane.
the vertical plane mimics the constraint imposed by nearby assemblies or molecules.
experiments show that the components in the assembly, for example, dvl, axin and gsk <dig> accumulate near the membrane when wnt interacts with fzd and initiates the pathway  <cit> .
furthermore, ck <dig> that is responsible for the s/t cluster phosphorylation events is a near-membrane kinase  <cit> .
the second constraint also occurs near the membrane.
if the second constraint is more stringent and the vertical plane gets closer to the conformer , the spatial volume of the conformer is excluded more.
such an excluded volume effect forces the conformer of the lrp <dig> intracellular domain to go through an elongation process.
we propose this elongation might be necessary for the phosphorylation of the lrp <dig> intracellular domain.
liu and colleagues  <cit>  demonstrated that a truncated lrp <dig> comprising of its transmembrane and cytoplasmic domains is expressed as a constitutively active monomer whose signaling ability is inhibited by forced dimerization.
also, wnts are shown to activate canonical signaling through lrp <dig> by inducing an intracellular conformational switch which relieves allosteric inhibition imposed on the intracellular domains.
this paper published in  <dig> is the only one until now on the conformational behaviour of the lrp <dig> intracellular domain through experiments.
there is however no evidence to prove such a conformational switch in terms of indicating the changes in the lrp <dig> intracellular domain structural ensemble.
in the paper published by yasui et al.
<cit> , the authors conclude that the lrp <dig> extracellular domain does not form homodimers in solution and speculate that weak dimerization may occur only at the cell surface where the receptors are confined in the 2d plane.
in our current simulation study, we focus on the conformational change of lrp <dig> intracellular domain under spatial constraints in the initiation of the canonical wnt signalling pathway.
our results show that the spatial constraints cause the structural ensemble of the intracellular domain to adopt an extended or elongated form which relieves the allosteric inhibition.
this provides another explanation for why wild-type lrp <dig> and lrp <dig> mutant without an extracellular domain behave differently with or without the presence of wnt.
the lrp <dig> mutant without an extracellular domain is free from the auto-inhibitory effect imposed by its extracellular domain.
the lrp <dig> intracellular domain anchored to the plasma membrane only faces the spatial constraint caused by the plasma membrane.
it can adopt a more open or elongated conformation to relieve the auto-inhibition caused by this unfolded region itself, to allow ck <dig> and gsk <dig> access.
for wild-type lrp <dig>  without the presence of wnt, membrane constraint is not enough to relieve the auto-inhibition caused by its extracellular and intracellular domains.
it requires another constraint to relieve the auto-inhibitory effect caused by the extracellular domain.
when wnt is present, it forms a complex with fzd and interacts with the lrp <dig> extracellular domain.
though this interaction may be weak, the conformational space of the lrp <dig> intracellular domain is excluded.
the domain is therefore forced to adopt a more open or extended structure for it to reduce the tension.
wnt-fzd hence imposes another spatial constraint to lrp <dig>  in the initiation complex, wnt is not the only component; fzd, dvl, axin and gsk <dig> also participate in the process.
hence, they together could form the second spatial constraint on lrp <dig> to amplify the reduction of auto-inhibitory effect.
such amplification would be helpful to the activation of the lrp <dig> intracellular domain and the stabilization of β-catenin.
our model and results can help explain the results obtained by liang et al who recently discovered that the previously functional unknown protein tmem <dig> is able to promote lrp <dig> phosphorylation in the wnt signalling pathway.
tmem <dig> functions as a membrane scaffold protein, assembling kinases and substrates into a higher-molecular-weight complex prior to phosphorylation, but it promotes lrp <dig> phosphorylation through a mechanism independent of fzd and dvl  <cit> .
like fzd, tmem <dig> could recruit ck <dig> as well as other molecules to form a nearby molecular assembly close to lrp <dig>  any nearby assembly, together with the membrane, can impose the spatial constraints to the conformational space of the lrp <dig> intracellular domain so that this region will be elongated for kinases ck <dig> and gsk <dig> to gain easy access for phosphorylation.
liang et al.
observe tmem <dig> to associate with lrp <dig>  however, unlike fzd, the interaction is likely mediated by the transmembrane domains between lrp <dig> and tmem <dig> which can bring the tmem198-ck <dig> complex more close to lrp <dig>  the findings in liang et al.
also demonstrate that near-membrane is the key point in the simulation model.
the interaction between tmem <dig> and lrp <dig> at transmembrane domains takes place at the membrane plane.
it amplifies the vertical spatial constraint by recruiting ck <dig> which is near-membrane localized.
the spatial constraints can come from any nearby molecules or molecular assemblies.
these include neighboring lrp <dig> molecules in the signalosome/aggregation model, wnt-fzd-dvl-axin-gsk <dig> assembly in the sequential/amplification model or other discovered molecular assemblies such as the tmem198-ck <dig> complex reported in the paper by liang et al.
elongation makes the phosphorylation of unfolded protein regions easier
we proposed that the elongation form may be required for the phosphorylation of the lrp <dig> intracellular domain.
since there is no structure for the lrp <dig> intracellular domain present in the structural databases, we used a structure complex  to demonstrate that in a general case, elongation is required for phosphorylation taking place in the conformation of an unfolded protein region.
contains a camp dependent protein kinase catalytic subunit and its inhibitor, a 22aa long peptide binding to the kinase catalytic site.
the simulation procedure is provided in the additional file  <dig>  we constructed a 100mer sequence that includes the binding peptide in the middle and a repeated proline-serine extending to both terminals.
we used trades to generate a structural ensemble with the constructed sequence.
out of the  <dig> conformers generated,  <dig> passed the aligning, merging and crashes-checking requirements.
these survival conformers are available for docking.
figure 4a shows some examples of the conformers that are available and unavailable for docking.
we calculated the rgyr and end-to-end distances of region 1- <dig>  region 31- <dig>  region 61- <dig> along the sequence and compared the distribution curves between  <dig> conformers that are available for docking and  <dig> conformers that are not available for docking .
we used a t-test to see whether there is a big difference between the means of the datasets .
for rgyr and end-to-end distance of the three regions, the p-values are significantly small .
this indicates that the structures available for the kinase to access and bind on average have larger rgyr and longer end-to-end distances.
we can therefore say that the structural ensemble available for docking becomes more open and elongated for kinases to access through.
this proposition fits as elongation can reduce the auto-inhibition of the unfolded protein’s random coiled structure.
in general, the elongation form induces the phosphorylation sites to become more exposed so that kinases may easily interact.
alternative hypothesis:
the table shows t-test results on the rgyr distributions and end-to-end distance distributions of docking and undocking structural ensembles of the constructed 100mer peptide.
CONCLUSIONS
we compared the rgyr distributions of structure ensembles of the lrp <dig> intracellular domain before and after applying spatial constraints.
we observe that the whole structure becomes open or extended and find that the near-membrane region appears to be elongated with given horizontal and vertical spatial constraints.
during the initiation, the spatial constraints caused by the plasma membrane and nearby assemblies or molecules force an elongation form to dominate the conformational space of the lrp <dig> intracellular domain.
we demonstrated that such an elongation process is required for unfolded protein structures because it can relieve the auto-inhibitory effect and grant kinases easy access.
the near-membrane lrp <dig> intracellular domain extension could expose the s/t cluster phosphorylation site for ck <dig>  which subsequently promotes downward ppppx phosphorylation events.
this study elaborates details on the activation of lrp <dig> through its conformational change in the current wnt/β-catenin pathway initiation models.
trades provides a new way to investigate signal transduction mechanisms through computational simulations and bioinformatics methods.
more importantly, it demonstrates a way to study the conformational behavior of other proline-rich unfolded protein regions including those in signaling pathways and mechanobiological systems.
the wnt/β-catenin signaling pathway plays important roles in cancer and diseases.
the lrp5/ <dig> mutation is responsible for bone density disorders, ocular disorders and disorders of cholesterol and glucose metabolism.
the findings in this study could pave the way to the development of new therapeutics through structure based drug design with the consideration of spatial constraints imposed by cellular components.
experiments proposed to validate the lrp <dig> elongation model are single-molecule fluorescence resonance energy transfer  and time-resolved fluorescence resonance energy transfer .
trades was originally validated with successful comparison to tr-fret distribution  <cit> .
sm-fret and tr-fret have been applied to study the conformations of full-length p <dig>  which has both folded and intrinsically disordered domains  <cit> .
sm-fret can measure the radius of gyration of the lrp <dig> intracellular domain.
tr-fret can measure the end-to-end distance distribution within the lrp <dig> intracellular domain.
collectively these experiments will provide significant validation of the findings presented in this study.

 wnt induced signaling pathways play essential roles in development and disease. currently, two initiation models of the canonical wnt/β-catenin signaling pathway have been proposed as illustrated in figure. one could be referred to as the sequential recruitment/amplification model, in which wnt stimulation is proposed to recruit the scaffold protein axin to approach the membrane through the bridging interactions between frizzled and dishevelled, as well as between dvl and axin. gsk in association with axin thereafter is able to phosphorylate the lrp5/ pppp motif near the membrane. initial phosphorylation creates a docking site for axin and thereby recruits additional axin-gsk to promote further lrp phosphorylation. the second model is the signalosome/aggregation model. recent results showed that a signalosome is formed by aggregated lrp and axin when wnt is present. clustering of lrp leads to the phosphorylation of t by ck1γ and subsequent phosphorylation of the pppp motif by gsk. phosphorylated lrp recruits axin resulting in the disruption of the “β-catenin destruction complex”, which comprises axin, apc, gsk and ck1α. this results in the stabilization of a cytoplasmic pool of β-catenin. free β-catenin enters the nucleus and activates gene transcription by binding to the tcf/lef family of transcription factors. thereafter, the activation of lrp5/ is indispensable to initiate the downstream intracellular wnt signaling cascade, in order to stabilize β-catenin. lrp6/lrp5/arrow belongs to a subfamily of ldl receptors. human lrp is a type i single-pass transmembrane protein. its modular extracellular domain has three basic domains; the ywtd -type β-propeller domain, the egf -like domain, and the ldlr type a domain. this region has crystal structures present in pdb database. during signaling pathway initiation, wnt binds the cysteine-rich domain of fzd proteins and exhibits a wnt1-dependent association with lrp extracellular domains in vitro. however, the interaction between wnt and lrp is weaker compared to the interaction between wnt and frizzled. it is therefore more likely that a lrp6-fzd complex binds to the lrp extracellular domain. after deletion of its extracellular domain, lrp protein could still activate the wnt/β-catenin signaling pathway. the lrp intracellular domain is rich in prolines and serines. sequence alignment shows that it includes a s/t cluster and downstream five reiterated ppppx motifs, each of which contains a pppp motif phosphorylated by gsk and juxtaposed to a ck phosphorylation site. such dual phosphorylation is essential to stabilize the pool of β-catenin in the cytoplasm. the phosphorylation of the s/t cluster has also been characterized, particularly the phosphorylation of t by ck1γ. it is believed that the phosphorylated s/t cluster promotes downward ppppx phosphorylation. he’s group previously showed that a lrp mutant lacking most of the intracellular domain is a loss-of-function. in addition, a truncated lrp comprising its transmembrane and intracellular domain is constitutively active in wnt signaling transduction. a single pppp motif transferred to a lrp variant lacking the extracellular domain activates the wnt/β-catenin signaling pathway. phosphorylated ppppx motifs provide docking sites for axin, which associates with gsk to promote proximity to lrp. so far, no stable structure has been obtained from this isolated and contiguous lrp intracellular domain in current structure databases. the lrp intracellular domain is expected to be natively unstructured since its composition is enriched with proline whose conformation is limited. there has been little study on the conformational behavior of lrp5/ before activation, when wnt induces signal transduction. no matter which initiation model applies to the canonical wnt/β-catenin signaling pathway, the conformation of lrp has to face spatial constraints imposed by the plasma membrane and a nearby molecule or molecular assemblies, which could be neighboring lrp molecules or a wnt-fzd-dvl-axin-gsk assembly. we hypothesize that these two spatial constraints would restrict lrp intracellular domain conformational space and so cause its conformation to adopt a more extended or elongated form before it is activated and docked by axin. we applied our trades software package to simulate the conformational space of the lrp intracellular domain, as well as constraints to demonstrate an elongated conformational change of this domain occurs during wnt/β-catenin signaling pathway initiation. we also tested whether close packing can induce a statistically observable structural change in an ensemble of unfolded states in a sequence independent manner.  lrp intracellular domain is predicted to be unfolded no stable structure has been documented for the lrp intracellular domain in current structure databases. this region is expected to be natively unstructured because it is enriched with proline and serine. several protein disorder prediction tools predict that this region is disordered or unfolded. figure s gives the prediction results from disorder predictors; ronn, iupred, globplot, pondr-fit and foldindex. this unfolded intracellular protein region most probably tends to have random coiled conformations, which auto-inhibits the structure itself to avoid interactions with other molecules. like most other disordered protein regions, it exists as an ensemble of structures which can be generated by trades in this simulation study. radius of gyration distribution radius of gyration measures the openness of the whole structure. a structure with a larger rgyr has more sparse atoms within it. figure displays the rgyr distribution of the initial conformational ensemble in the lrp intracellular domain simulation experiment. the number of generated conformers and average rgyr are provided in the second column in table conformers with different values of rgyr were checked. it was observed that conformers with smaller rgyr have more compact structures; while, conformers with larger rgyr tend to adopt more open or extended conformations. in this distribution, the conformation of the generated conformers changes from compact, to more open, to more extended as their rgyr increases. simulation experiments were carried out both for the lrp intracellular domain and the control sequence. see results in table and table a unix script was written to obtain conformers that pass constraint and constraint out of the initial structural ensemble. these conformers were then filtered by constraint with parameter δ set to Å and Å. this parameter represents the distance from the vertical plane to the plane defined by the transmembrane helix and origin point. the average rgyr of the structural ensemble gets larger after each constraint is applied, as shown in table table and figure in both the lrp intracellular domain and control sequence simulation experiments, after each filtration, conformers in the structural ensemble surviving from the spatial constraints tend to possess more open or extended conformations. this was also indicated by the observation that after each constraint more fractions of structural ensemble appear to have rgyr larger than the average. in addition, when the distance δ gets smaller, the average rgyr in the structure ensemble that survived following constraint gets bigger. the control sequence has lrp transmembrane region and poly polypeptide substituting lrp intracellular domain. the theoretical maximum rgyr for a completely extended chain polypeptide) is calculated from trades to be around 90Å. the poly polypeptide is constrained to take a β-strand conformation in its trajectory distribution file. end-to-end distance distribution five end-to-end distances with equal length were calculated in the lrp intracellular domain simulation experiment. each distance contains at least one conserved ppppx motif. see table for each end-to-end distance, the difference between the average rgyr of the initial structural ensemble and that of the structural ensemble after constraint is provided in the column titled as “∆mean ”. the difference between the average rgyr of the initial structural ensemble and that of the structure ensemble after constraint under different values of distance δ are shown in the columns under the title of “∆mean ”. for both the lrp intracellular domain and control sequence, out of the five end-to-end distances, the distribution of d displays largely increased mean values of the structural ensembles after each constraint. this was indicated by the positive differences in table it also proves that after each constraint the average value of d gets larger. in figure the d distribution curves of structural ensembles after constraint and after constraint move to the right of the d distribution curve of the initial structural ensemble. meanwhile, more fractions of structural ensemble after each constraint are found at a larger value of d on the distribution curves. this indicates that the region corresponding to d within the lrp intracellular domain conformers in the surviving structural ensemble, prefers an elongated or extended conformation. this region starts right from the beginning of the lrp intracellular domain and extends to the end of the first ppppx motif. it is the closest membrane region inside the lrp intracellular domain. additionally, as the distance δ gets smaller, the mean of d of the structural ensemble after constraint also gets bigger. however, for both the lrp intracellular domain and control sequence, the rest four end-to-end distances’ distributions show no prominent changes compared to the change in d distribution after applying each constraint and furthermore they overlap with each other as seen in figure s. the table shows the length, motif contained, starting and ending positions in lrp simulation sequence, and the differences between the average rgyr of structural ensembles after each constraint and the initial structural ensemble. discussion lrp intracellular domain structure ensemble favors an elongation form when the wnt/β-canonical pathway initiates in the lrp intracellular domain simulation experiment, greater proportions of the structural ensemble after each spatial constraint are observed to have rgyr of a larger value than the average in comparison with the initial structural ensemble. it shows that the two spatial constraints make the lrp intracellular domain likely to adopt a more open or elongated conformation. the plasma membrane and neighboring assemblies formed by wnt-fzd-dvl-axin-gsk or neighboring lrp aggregation could limit the lrp intracellular domain to form fewer numbers of random coiled structures. instead, the lrp intracellular domain tends to form more elongated conformations as the spatial constraints exclude its volume near the membrane in the cell. in vivo, plasma membrane and nearby assemblies or molecules could result in a natural elongation of the lrp intracellular domain when a wnt signal triggers the pathway. such elongation behavior might grant kinases ck and gsk open access to the phosphorylation sites within the lrp intracellular domain, and subsequently lrp could be activated through these phosphorylation events. we propose that when the wnt pathway initiates, the lrp intracellular domain is elongated to reduce the auto-inhibition before it is activated. a conformational change occurs to the lrp intracellular domain structural ensemble after applying spatial constraints. it is intriguing to investigate if any conformational change could occur in the subsequences within the lrp intracellular domain during wnt canonical pathway initiation. the distributions of the five calculated end-to-end distances could reflect the openness of the subsequences in the lrp intracellular domain. the first end-to-end distance d which measures the openness of the region that is closest to the membrane on the lrp intracellular domain. the distribution curves for d show that this region gets longer in more conformers out of the structural ensemble after filtration. this implies that this near-membrane region in the lrp intracellular domain elongates or extends when its conformational space is limited by the plasma membrane and nearby assemblies or molecules. such an extended conformation could allow ck to more easily reach the s/t cluster and initiate phosphorylation. this may also explain the experimental finding that s/t cluster phosphorylation by ck promotes the downward activation of the ppppx motif. on the contrary, the end-to-end distances d d d and d hardly show any changes in their distribution curves between original and filtered structural ensembles. the means of these distributions of structural ensemble after filtration are in fact smaller than that of the initial structural ensemble. this suggests the regions corresponding to these distances are on average less extended in the conformers surviving from filtration. the protein regions corresponding to the five end-to-end distances are gradually further away from the transmembrane helix, which determines the location of the plasma membrane. the region corresponding to d is the closest to the plasma membrane followed by d the observations on the distribution curves of these distances suggest that the spatial constraints exclude to a great extent the volume of the lrp intracellular domain at the near-membrane location in the cell. additionally, the same behaviors are observed in rgyr distributions and end-to-end distributions in the simulation experiment as for the control sequence. we observed structural changes that can be demonstrated in a hypothetical sequence with as much as 50% proline and 50% serine. such changes could function as a mechanism by which high rates of mutations could be tolerated whilst conserving function. hence, it can be concluded that such an elongation process induced by membrane and neighboring assembly/aggregation is sequence independent but maybe compositional dependent. effects of the two spatial constraints near-membrane serves as the key point in the simulation study. the membrane-anchor issue has been discussed in several published papers that claimed the lrp intracellular domain needs anchoring to the membrane to process signaling. arrow/lrp5/lrp mutants without the extracellular domain with which to anchor to the membrane constitutively activate the β-catenin pathway in mammalian cells. the lrp intracellular domain is unable to activate the signaling pathway unless it is anchored to the membrane. in the simulation, the horizontal plane mimics the constraint imposed by the membrane plane. the vertical plane mimics the constraint imposed by nearby assemblies or molecules. experiments show that the components in the assembly, for example, dvl, axin and gsk accumulate near the membrane when wnt interacts with fzd and initiates the pathway. furthermore, ck that is responsible for the s/t cluster phosphorylation events is a near-membrane kinase. the second constraint also occurs near the membrane. if the second constraint is more stringent and the vertical plane gets closer to the conformer, the spatial volume of the conformer is excluded more. such an excluded volume effect forces the conformer of the lrp intracellular domain to go through an elongation process. we propose this elongation might be necessary for the phosphorylation of the lrp intracellular domain. liu and colleagues demonstrated that a truncated lrp comprising of its transmembrane and cytoplasmic domains is expressed as a constitutively active monomer whose signaling ability is inhibited by forced dimerization. also, wnts are shown to activate canonical signaling through lrp by inducing an intracellular conformational switch which relieves allosteric inhibition imposed on the intracellular domains. this paper published in is the only one until now on the conformational behaviour of the lrp intracellular domain through experiments. there is however no evidence to prove such a conformational switch in terms of indicating the changes in the lrp intracellular domain structural ensemble. in the paper published by yasui et al., the authors conclude that the lrp extracellular domain does not form homodimers in solution and speculate that weak dimerization may occur only at the cell surface where the receptors are confined in the 2d plane. in our current simulation study, we focus on the conformational change of lrp intracellular domain under spatial constraints in the initiation of the canonical wnt signalling pathway. our results show that the spatial constraints cause the structural ensemble of the intracellular domain to adopt an extended or elongated form which relieves the allosteric inhibition. this provides another explanation for why wild-type lrp and lrp mutant without an extracellular domain behave differently with or without the presence of wnt. the lrp mutant without an extracellular domain is free from the auto-inhibitory effect imposed by its extracellular domain. the lrp intracellular domain anchored to the plasma membrane only faces the spatial constraint caused by the plasma membrane. it can adopt a more open or elongated conformation to relieve the auto-inhibition caused by this unfolded region itself, to allow ck and gsk access. for wild-type lrp without the presence of wnt, membrane constraint is not enough to relieve the auto-inhibition caused by its extracellular and intracellular domains. it requires another constraint to relieve the auto-inhibitory effect caused by the extracellular domain. when wnt is present, it forms a complex with fzd and interacts with the lrp extracellular domain. though this interaction may be weak, the conformational space of the lrp intracellular domain is excluded. the domain is therefore forced to adopt a more open or extended structure for it to reduce the tension. wnt-fzd hence imposes another spatial constraint to lrp in the initiation complex, wnt is not the only component; fzd, dvl, axin and gsk also participate in the process. hence, they together could form the second spatial constraint on lrp to amplify the reduction of auto-inhibitory effect. such amplification would be helpful to the activation of the lrp intracellular domain and the stabilization of β-catenin. our model and results can help explain the results obtained by liang et al who recently discovered that the previously functional unknown protein tmem is able to promote lrp phosphorylation in the wnt signalling pathway. tmem functions as a membrane scaffold protein, assembling kinases and substrates into a higher-molecular-weight complex prior to phosphorylation, but it promotes lrp phosphorylation through a mechanism independent of fzd and dvl. like fzd, tmem could recruit ck as well as other molecules to form a nearby molecular assembly close to lrp any nearby assembly, together with the membrane, can impose the spatial constraints to the conformational space of the lrp intracellular domain so that this region will be elongated for kinases ck and gsk to gain easy access for phosphorylation. liang et al. observe tmem to associate with lrp however, unlike fzd, the interaction is likely mediated by the transmembrane domains between lrp and tmem which can bring the tmem198-ck complex more close to lrp the findings in liang et al. also demonstrate that near-membrane is the key point in the simulation model. the interaction between tmem and lrp at transmembrane domains takes place at the membrane plane. it amplifies the vertical spatial constraint by recruiting ck which is near-membrane localized. the spatial constraints can come from any nearby molecules or molecular assemblies. these include neighboring lrp molecules in the signalosome/aggregation model, wnt-fzd-dvl-axin-gsk assembly in the sequential/amplification model or other discovered molecular assemblies such as the tmem198-ck complex reported in the paper by liang et al. elongation makes the phosphorylation of unfolded protein regions easier we proposed that the elongation form may be required for the phosphorylation of the lrp intracellular domain. since there is no structure for the lrp intracellular domain present in the structural databases, we used a structure complex to demonstrate that in a general case, elongation is required for phosphorylation taking place in the conformation of an unfolded protein region. contains a camp dependent protein kinase catalytic subunit and its inhibitor, a 22aa long peptide binding to the kinase catalytic site. the simulation procedure is provided in the additional file we constructed a 100mer sequence that includes the binding peptide in the middle and a repeated proline-serine extending to both terminals. we used trades to generate a structural ensemble with the constructed sequence. out of the conformers generated, passed the aligning, merging and crashes-checking requirements. these survival conformers are available for docking. figure 4a shows some examples of the conformers that are available and unavailable for docking. we calculated the rgyr and end-to-end distances of region 1- region 31- region 61- along the sequence and compared the distribution curves between conformers that are available for docking and conformers that are not available for docking. we used a t-test to see whether there is a big difference between the means of the datasets. for rgyr and end-to-end distance of the three regions, the p-values are significantly small. this indicates that the structures available for the kinase to access and bind on average have larger rgyr and longer end-to-end distances. we can therefore say that the structural ensemble available for docking becomes more open and elongated for kinases to access through. this proposition fits as elongation can reduce the auto-inhibition of the unfolded protein’s random coiled structure. in general, the elongation form induces the phosphorylation sites to become more exposed so that kinases may easily interact. alternative hypothesis: the table shows t-test results on the rgyr distributions and end-to-end distance distributions of docking and undocking structural ensembles of the constructed 100mer peptide.  we compared the rgyr distributions of structure ensembles of the lrp intracellular domain before and after applying spatial constraints. we observe that the whole structure becomes open or extended and find that the near-membrane region appears to be elongated with given horizontal and vertical spatial constraints. during the initiation, the spatial constraints caused by the plasma membrane and nearby assemblies or molecules force an elongation form to dominate the conformational space of the lrp intracellular domain. we demonstrated that such an elongation process is required for unfolded protein structures because it can relieve the auto-inhibitory effect and grant kinases easy access. the near-membrane lrp intracellular domain extension could expose the s/t cluster phosphorylation site for ck which subsequently promotes downward ppppx phosphorylation events. this study elaborates details on the activation of lrp through its conformational change in the current wnt/β-catenin pathway initiation models. trades provides a new way to investigate signal transduction mechanisms through computational simulations and bioinformatics methods. more importantly, it demonstrates a way to study the conformational behavior of other proline-rich unfolded protein regions including those in signaling pathways and mechanobiological systems. the wnt/β-catenin signaling pathway plays important roles in cancer and diseases. the lrp5/ mutation is responsible for bone density disorders, ocular disorders and disorders of cholesterol and glucose metabolism. the findings in this study could pave the way to the development of new therapeutics through structure based drug design with the consideration of spatial constraints imposed by cellular components. experiments proposed to validate the lrp elongation model are single-molecule fluorescence resonance energy transfer and time-resolved fluorescence resonance energy transfer. trades was originally validated with successful comparison to tr-fret distribution. sm-fret and tr-fret have been applied to study the conformations of full-length p which has both folded and intrinsically disordered domains. sm-fret can measure the radius of gyration of the lrp intracellular domain. tr-fret can measure the end-to-end distance distribution within the lrp intracellular domain. collectively these experiments will provide significant validation of the findings presented in this study. 
to test whether this observed ensemble conformational change is sequence dependent, we did a control simulation with a hypothetical sequence with 50% proline and 50% serine in alternating residues. this elongation could relieve conformational auto-inhibition of the ppppx motif binding sites and allow gsk and ck to approach their phosphorylation sites, thereby activating lrp and the downstream pathway. lrp is a membrane protein crucial in the initiation of canonical wnt/β-catenin signalling. in our study, we use a computational simulation tool to sample the conformational space of the lrp intracellular domain, under the spatial constraints imposed by the membrane and the close approach of the neighboring intracellular molecular complex, which is assembled on frizzled when wnt binds to both lrp and frizzled on the opposite side of the membrane.
 lrp is a membrane protein crucial in the initiation of canonical wnt/β-catenin signalling. its function is dependent on its proline-serine rich intracellular domain. lrp has five pppp motifs that are phosphorylated during activation, starting with the site closest to the membrane. like all long proline rich regions, there is no stable 3d structure for this isolated, contiguous region.  in our study, we use a computational simulation tool to sample the conformational space of the lrp intracellular domain, under the spatial constraints imposed by the membrane and the close approach of the neighboring intracellular molecular complex, which is assembled on frizzled when wnt binds to both lrp and frizzled on the opposite side of the membrane. we observe that an elongated form dominates in the lrp intracellular domain structure ensemble. this elongation could relieve conformational auto-inhibition of the ppppx motif binding sites and allow gsk and ck to approach their phosphorylation sites, thereby activating lrp and the downstream pathway.  we propose a model in which the conformation of the lrp intracellular domain is elongated before activation. this is based on the intrusion of the frizzled complex into the ensemble space of the proline rich region of lrp which alters the shape of its available ensemble space. to test whether this observed ensemble conformational change is sequence dependent, we did a control simulation with a hypothetical sequence with 50% proline and 50% serine in alternating residues. we confirm that this ensemble neighbourhood-based conformational change is independent of sequence and conclude that it is likely found in all proline rich sequences. these observations help us understand the nature of proline rich regions which are both unstructured and which seem to evolve at a higher rate of mutation, while maintaining sequence composition. november- december asia pacific bioinformatics network tenth international conference on bioinformatics – first iscb asia joint conference kuala lumpur, malaysia 