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Assortative mixing in Protein Contact Networks and protein folding kinetics
Ganesh Bagler, Somdatta Sinha
TL;DR
The paper examines whether topological properties of native-state protein structures relate to folding kinetics. It analyzes protein contact networks and long-range interaction networks, finding assortativity at both scales and different correlations between clustering and folding rate.
Problem
The paper asks whether general network parameters derived from native-state protein structures reveal relationships between structure and folding kinetics.
Source: §I · p. 1Method
The authors analyze native-state structures of 30 single-domain, two-state proteins as PCNs and as LINs that exclude short-range interactions.
Source: §I · p. 1, Controls · p. 2Results
Both PCNs and LINs are assortative, and assortativity is positively correlated with folding rate at both scales; LIN clustering is negatively correlated with folding rate.
Source: §I · p. 2Takeaways & Limitations
The authors conclude that general network parameters can provide insight into protein structural properties and serve as indicators of biophysical processes such as folding.
Source: §I · p. 2
Abstract
from arXiv · showhide
Starting from linear chains of amino acids, the spontaneous folding of proteins into their elaborate three-dimensional structures is one of the remarkable examples of biological self-organization. We investigated native state structures of 30 single-domain, two-state proteins, from complex networks perspective, to understand the role of topological parameters in proteins' folding kinetics, at two length scales-- as ``Protein Contact Networks (PCNs)'' and their corresponding ``Long-range Interaction Networks (LINs)'' constructed by ignoring the short-range interactions. Our results show that, both PCNs and LINs exhibit the exceptional topological property of ``assortative mixing'' that is absent in all other biological and technological networks studied so far. We show that the degree distribution of these contact networks is partly responsible for the observed assortativity. The coefficient of assortativity also shows a positive correlation with the rate of protein folding at both short and long contact scale, whereas, the clustering coefficients of only the LINs exhibit a negative correlation. The results indicate that the general topological parameters of these naturally-evolved protein networks can effectively represent the structural and functional properties required for fast information transfer among the residues facilitating biochemical/kinetic functions, such as, allostery, stability, and the rate of folding.
I. INTRODUCTION
The paper asks whether network properties of native-state protein structures relate to folding kinetics. It analyzes PCNs and LINs, finding assortativity at both scales and distinct relationships between network parameters and folding rate.
I. INTRODUCTION
- Protein folding kinetics matter because protein structure specifies biochemical function, and misfolding can lead to disorders.Source: §I · p. 1
- PCNs model amino-acid units linked by noncovalent interactions; LINs retain long-range contacts while excluding short-range interactions.The comparison is intended to examine long-range interactions and their relation to folding rate.Source: §I · p. 1
- The study tests whether general network parameters from native-state structures reveal relationships between protein structure and folding kinetics.It focuses on single-domain, two-state proteins with available folding-rate measurements.Source: §I · p. 1
- The degree distribution of PCNs partly accounts for their assortativity, as shown using control networks.Source: §I · p. 2
- Assortativity in both PCNs and LINs is positively correlated with folding rate, while LIN clustering is negatively correlated and PCN clustering shows negligible correlation.The authors suggest short-range interactions may reduce the negative association seen for LINs.Source: §I · p. 2
- Earlier measures including CO, LRO, and TCD have negative correlations with folding rate; the paper evaluates assortativity and clustering as additional indicators.LRO and TCD prediction accuracy is reported as unchanged when short-range interactions are excluded.Source: §I · p. 2
II. METHODS
The methods represent protein structures as contact networks, derive long-range-only subnetworks, and compare both with randomized controls.
II. METHODS
- A PCN represents each amino acid by its Cα atom and links residues whose Cα atoms are within the stated contact-distance threshold.The threshold is Rc ≤ 8 Å.Source: Controls · p. 2
- A LIN is a subset of its PCN that retains backbone links and contacts between residues separated by 12 or more amino acids along the backbone.Removing short-range contacts leaves the LIN with the same nodes but fewer links.Source: Controls · p. 2
- Two control types preserve backbone connectivity while randomizing noncovalent contacts; 100 instances of each type were generated per protein and averaged.Source: Controls · p. 2
- Type I controls preserve node and link counts, while Type II also preserves the PCN connectivity distribution and randomizes pair-connectivity 2,000 times.Further construction details are stated to be in Supplementary Data.Source: Controls · p. 2
Data
The kinetic analysis covers 30 single-domain, two-state globular proteins with available experimental folding rates, spanning three structural classes.
Data
- The folding-rate analysis uses 30 single-domain, two-state globular proteins with experimental ln(kF) values, comprising 5 all-α, 13 all-β, and 12 αβ proteins.Source: Data · p. 2
- Across the 30 proteins, ln(kF) ranges from −1.48 to 9.8, and folding times are on the order of 10^5 seconds.Source: Data · p. 2
- Protein sizes range from 43 to 126 amino acids; preliminary network analysis covers 110 proteins spanning 43 to 2,359 residues.Structural data came from the Protein Data Bank.Source: Data · p. 2
Network parameters
The section defines network path length, clustering, degree, and assortativity as parameters for characterizing PCNs, LINs, and their controls.
Network parameters
- Characteristic path length averages shortest-path lengths and indicates network compactness.Shortest-path length is the number of links traversed along the shortest route between two nodes.Source: Network parameters · pp. 2–3
- Clustering coefficient measures network cliquishness by averaging each node’s proportion of connected neighbor pairs.Source: Network parameters · p. 3
- Degree is a node’s number of neighbors, and remaining degree is one less than total degree.The section also lists maximum degree, average degree, and average nearest-neighbor degree as degree-based measures.Source: Network parameters · p. 3
- Assortative mixing means high-degree nodes tend to connect to other high-degree nodes; the coefficient r measures degree correlation across link endpoints.The coefficient is defined using endpoint degrees and remaining-degree distributions.Source: Network parameters · p. 3
III. RESULTS
PCNs and LINs differ in clustering and characteristic path length, with the LINs retaining a range of clustering values despite removal of short-range contacts.
Clustering coefficients of PCNs and LINs
- PCNs have higher clustering (CP CN = 0.562±0.029) than their random controls, while LIN clustering averages 0.259 ± 0.109.The L-C plot compares PCNs and LINs with Type I random controls.Source: Clustering coefficients of PCNs and LINs · p. 3, Figure 1 · p. 3
- PCNs have high clustering (C_PC N = 0.562 ± 0.029) compared with their random controls, while LINs show clustering distributed across a range (C_LIN = 0.259 ± 0.109).Random controls had L and C values of 2.621 ± 0.411 and 0.0557 ± 0.0476 for PCNs, and 3.256 ± 0.056 and 0.075 ± 0.012 for LINs.Source: Clustering coefficients of PCNs and LINs · p. 3
- LINs have a longer characteristic path length than PCNs (LLIN = 8.72 ± 4.564; LPCN = 5.818 ± 2.826), attributed to fewer contacts.The passage suggests that the range of LIN clustering may give these networks specificity that short-range contacts obscure in PCNs.Source: Clustering coefficients of PCNs and LINs · p. 3
Degree distributions of PCNs and LINs
The normalized degree distributions differ between PCNs and LINs: PCNs are Gaussian-like, while LINs are concentrated at low degrees and fit a single-scale exponential.
Degree distributions of PCNs and LINs
- The cited passage reports that PCNs have a Gaussian degree distribution, with best-fit parameters A = 5.538, w = 6.265, and xc = 9.373.It also says P(k) is obtained for a given protein so proteins of different sizes can be compared.Source: Degree distributions of PCNs and LINs · p. 4
- LIN degree distributions concentrate most nodes at low degree, unlike the Gaussian-shaped PCN distributions.Figure 2 compares normalized P(k) for PCNs and LINs across the 30 proteins.Source: Degree distributions of PCNs and LINs · p. 4, Figure 2 · p. 4
- LIN degrees best fit a single-scale exponential, P(k) ∼ k−γ exp (−k/kc), with γ = 0.24 and kc = 4.4.Degree-one nodes in LINs are the N- and C-terminal amino acids.Source: Degree distributions of PCNs and LINs · p. 4
- Type I random controls of both PCNs and LINs have Poisson degree distributions.The passage reports these distributional properties across proteins irrespective of function or structural classification.Source: Degree distributions of PCNs and LINs · p. 4
Assortative nature of PCNs and LINs
PCNs and LINs show positive degree correlations, with assortativity present at both short and long contact scales and reported across diverse protein structures.
Assortative nature of PCNs and LINs
- In PCNs and LINs, average nearest-neighbor degree increases with node degree; Type I controls remain nearly constant.The curves tend to saturate at larger degrees, which the authors suggest may reflect steric hindrance limiting maximum node degree.Source: Assortative nature of PCNs and LINs · p. 5
- Assortativity coefficients are positive in all 30 proteins, ranging from 0.09 < r < 0.52 for PCNs and 0.12 < r < 0.58 for LINs.Type I controls cluster around zero, indicating no degree correlations.Source: Assortative nature of PCNs and LINs · p. 5
- Positive assortativity was also observed across a larger, diverse set of protein structures, which the authors identify as a generic feature.Those additional observations are reported as data not shown.Source: Assortative nature of PCNs and LINs · p. 5
assortativity
Preserving PCN degree distributions while randomizing pair-connectivities partially restores assortativity in both PCNs and their LINs.
assortativity
- Type II controls preserve PCN degree distributions while randomizing pair-connectivities, unlike Type I controls.Their average nearest-neighbor degree increases with degree in both PCNs and LINs.Source: assortativity · p. 6
- Type II controls partially recover assortativity in both PCNs and their LINs.The recovery in LINs occurs despite their degree distribution differing substantially from that of PCNs.Source: assortativity · p. 6
- The partial recovery indicates that degree distribution contributes to, but does not fully explain, observed assortative mixing.The passage notes that rewiring can produce assortativity or disassortativity without changing degree distribution.Source: assortativity · p. 6
protein folding rates
The section relates general network parameters of protein contact networks and long-range interaction networks to folding rates. Figure 5 shows a negative correlation between LIN clustering and folding rate.
protein folding rates
- The analysis relates C and r in PCNs and LINs to experimental folding rates for 30 proteins.The proteins are single-domain, two-state folders.Source: protein folding rates · p. 6
- Figure 5 shows a negative correlation between LIN clustering coefficient and folding rate.The figure plots ln(kF) against the clustering coefficient of LINs.Source: Figure 5 · p. 6
Average Clustering Coefficient and Rate of Folding
LIN clustering is strongly negatively correlated with folding rate, whereas PCN clustering has no significant relationship. Assortativity correlates positively with folding rate in both network representations.
Average Clustering Coefficient and Rate of Folding
- For 30 single-domain, two-state proteins, LIN clustering correlates negatively with ln(kF) (r = −0.7712; p < 0.0001), while PCN clustering has no significant relationship.Neither Type I nor Type II random controls show a correlation between LIN clustering and folding rate.Source: Average Clustering Coefficient and… · p. 6
- LIN clustering counts three-residue loops among distant contacts, and its negative correlation with folding rate is comparable to that of Contact Order (r = −0.7574; p < 0.0001).The passage interprets both relationships as indicating the importance of long-range contact formation in folding rate.Source: Average Clustering Coefficient and… · pp. 6–7
Coefficient of Assortativity and Rate of Folding
- LIN assortativity correlates positively with ln(kF) (r = 0.6981; p < 0.0005), excluding five α proteins; PCN assortativity also correlates positively (r = 0.5943; p < 0.005).Figure 6 plots folding rate against LIN assortativity; random controls show no definite trend.Source: Coefficient of Assortativity and Rate… · p. 7, Figure 6 · p. 7
IV. DISCUSSION
The discussion distinguishes short- and long-range contact organization and relates both to folding kinetics. It presents assortativity at both scales as a feature that correlates positively with folding rate.
IV. DISCUSSION
- Assortativity is presented as an exceptional feature of naturally evolved protein networks at both short and long contact scales, and as positively correlated with folding kinetics.The authors say it contributes positively to speeding folding at both scales.Source: §IV · p. 7
- PCNs have high clustering, whereas LINs have lower, distributed clustering that correlates negatively with folding rate; the authors suggest long-range cliques may slow folding.They propose backbone connectivity and steric factors as possible reasons.Source: §IV · p. 7
- Short-range interactions may reduce the negative contribution of LIN properties, while native-state long-range network parameters reflect the importance of long-range contacts in folding rate.The discussion notes that transition-state and native-state Contact Order correlate similarly with folding rate.Source: §IV · p. 8
- The discussion frames clustering and assortativity as operating differently across scales: clustering may support function and allostery, while assortativity is associated with faster folding at both scales.Reduced LIN clustering may facilitate communication among distant residues, but may also increase folding time.Source: §IV · p. 8