Persistent homology analysis of protein structure, flexibility, and folding

A correlation matrix based filtration gives rise to an accurate prediction of the optimal characteristic distance used in protein B-factor analysis.

“This approach gives rise to an accurate prediction of the optimal characteristic distance used in protein B-factor analysis.”

Further increase of rc lowers CC values until it reaches the bottom at around rc = 5Å.

“Further increase of rc will lower CC values until it reaches the bottom at around rc = 5Å”

In the β2 panel for the icosahedron, a single bar represents a two-dimensional void enclosed by the surface of the icosahedron.

“Finally, in the β2 panel, there is a single bar, which represents a two-dimensional void enclosed by the surface of the icosahedron.”

There exists a unique β0 bar with length about 4.1 Å, reflecting the shortest distance between two closest adjacent Cα atoms from two individual beta strands.

“However, there exists a unique β0 bar which has a length about 4.1 Å. This bar reflects the shortest distance between two closest adjacent two Cα atoms from two individual beta strands.”

With the increase of η, both CC and A1 reach their maximum around η = 6Å.

“With the increase of η, both CC and A1 reach their maximum around η = 6Å.”

The positions where the bars terminate are exactly the corresponding bond lengths.

“The positions where the bars terminate are exactly the corresponding bond lengths.”

25 hexagon rings in C70 further evolve to two-dimensional holes, represented by 25 bars in the β2 panel.

“Additionally, 25 hexagon rings further evolve to two-dimensional holes, which are represented by 25 bars in the β2 panel.”

The persistent homology results obtained with η = 2Å, 6Å, and 16 Å share certain similarity with the β1 pattern of the distance based filtration.

“First, it can be seen that our persistent homology results obtained with η = 2Å, 6Å, and 16 Å in b, c, and d share certain similarity with the β1 pattern of the distance based filtration in a.”

No β1 bar is generated for the icosahedron because no one-dimensional circle has ever formed.

“As no one-dimensional circle has ever formed, no β1 bar is generated.”

When the cutoff distance increases to 5Å, a large number of β1 bars is produced and persists beyond the filtration size limit.

“When the cutoff distance increases to 5Å, a large number of β1 bars is produced and persists beyond the filtration size limit.”

In the β1 panel of the beta-sheet structure of the beta barrel, there is a unique bar that lasts from around 4 to 16 Å.

“In the β1 panel, there is a unique bar that lasts from around 4 to 16 Å.”

The central void structure of C70 is captured by the persisting β2 bar.

“The central void structure is captured by the persisting β2 bar.”

For the icosahedron, 11 of the 12 β0 bars disappear simultaneously with only one surviving to the end.

“As the filtration continues, 11 of them disappear simultaneously with only one survived and persisting to the end.”

In the CG model of the 19-residue alpha helix, there are 16 bars in the β1 panel.

“Additionally there are 16 bars in the β1 panel.”

19 residues in the alpha helix produce exactly 16 loops.

“Finally, 19 residues in the alpha helix produce exactly 16 loops as seen in Fig. 6.”

Each four Cα atoms in the alpha helix form a one-dimensional loop, corresponding to a β1 bar.

“It can be seen clearly that each four Cα atoms in the alpha helix form a one-dimensional loop, corresponding to a β1 bar.”

For the icosahedron, originally 12 β0 bars coexist, indicating 12 isolated vertices.

“For β0 number, originally 12 bars coexist, indicating 12 isolated vertices.”

There are 6 distinct groups in β0 bars of C70 due to 6 types of bond lengths in the C70 structure.

“Obviously, there are 6 distinct groups in β0 bars due to the factor that there are 6 types of bond lengths in the C70 structure.”

Once the cutoff distance is larger than 5Å, the CC values begin to increase dramatically until they reach the peak value around rc ≈ 7Å.

“Once the cutoff distance is larger than 5Å, the CC values begin to increase dramatically until it reach the peak value around rc ≈ 7Å”

For both 1GVD and 3MRE, when the cutoff distance is smaller than 3.8Å, no CC is obtained because pseudo-spring/pseudo-bond is not constructed and GNM is not properly set.

“For both cases, when the cutoff distance is small than 3.8Å, no CC is obtained because pseudo-spring/pseudo-bond is not constructed and GNM is not properly set.”

The CC values fluctuate as the cutoff distance increases further, due to improper balances in local connectivity and global connectivity.

“The CC values fluctuate as the cutoff distance increases further, due to improper balances in local connectivity and global connectivity.”

The average correlation coefficients obtained from FRI and MND behave similarly as the characteristic distance η increases.

“It can be seen that the average CC values obtained from FRI and MND behave similarly as η increases.”

When η is less than 3Å, the correlation coefficient is much lower than other values.

“Specifically, when η is less than 3Å, the correlation coefficient is much lower than other values.”

There is excellent consistence between persistent homology prediction and molecular dynamics simulation.

“An excellent consistence between our persistent homology prediction and molecular dynamics simulation is found.”

Different parameter values induce dramatically distinguished topological patterns.

“Different parameter values induce dramatically distinguished topological patterns.”

Optimal characteristic distances are revealed from our persistent homology analysis.

“Optimal characteristic distances are revealed from our persistent homology analysis.”

MTFs quantitatively predict protein folding stability.

“MTFs are employed to characterize protein topological evolution during protein folding and quantitatively predict the protein folding stability.”

As rc increases to around 4Å, relatively small CC values are acquired.

“As rc increases to around 4Å, we acquire relatively small CC values.”

In the CG model of two parallel beta strands from protein 2JOX, the barcode is much simpler with only 7 individual β1 bars.

“The barcode of the CG model, on the other hand, is much simpler with only 7 individual β1 bars.”

With the increase of the cutoff distance rc, the number of persistent β1 bars drops.

“With the increase of the cutoff distance rc, the number of persistent β1 bars drops.”

In the β0 panel of the CG model of two parallel beta strands, the lengths of most bars are around 3.8 Å, the average length between two adjacent Cα atoms.

“From the β0 panel we can see that the lengths of most bars are still around 3.8 Å, i.e., the average length between two adjacent Cα atoms as discussed above.”

Each symmetric turn generates 3 β1 bars.

“Each of symmetric turns also generates 3 β1 bar.”

The proposed approach gives rise to an accurate prediction of the optimal characteristic distance used in the FRI method for protein flexibility analysis.

“This approach gives rise to an accurate prediction of optimal characteristic distance used in the FRI method for protein flexibility analysis.”

There is a total of 36 β1 bars in C70 corresponding to 12 pentagon rings and 25 hexagon rings.

“There is a total of 36 β1 bars corresponding to 12 pentagon rings and 25 hexagon rings.”

The accumulated bar length, which represents the total connectivity, continuously decreases during the simulated protein unfolding process.

“Through the analysis of their topological invariants, we found that the accumulated bar length, which represents the total connectivity, continuously decreases in the protein unfolding process.”

When rc is set to about 12 Å, almost no persistent β1 bar can be found for both 1GVD and 3MRE.

“When rc is set to about 12 Å, almost no persistent β1 bar can be found for both 1GVD and 3MRE.”

For protein 2GI9, the linear correlation between energy prediction using negative accumulation bar length of the first Betti number and total energy is confirmed.

“Again the linear correlation between the energy prediction using negative accumulation bar length of the first Betti number and total energy is confirmed.”

For protein 2GI9, the CC values are as high as 0.972 and 0.971 for distance based filtration and correlation matrix based filtration, respectively.

“The CC values are as high as 0.972 and 0.971, for distance based filtration and correlation matrix based filtration, respectively.”

A striking linear relation between total energy and A1− can be found.

“Indeed, a striking linear relation between the total energy and A1− can be found.”

As a protein unfolds, its stability decreases.

“Additionally, as a protein unfolds, its stability decrease.”

The average accumulation bar length A1 behaves similarly to the average correlation coefficients as η increases.

“The average accumulation bar length A1 behaves in a similar manner.”

During the SMD simulation, protein total energy becomes higher as it unfolds.

“its total energy becomes higher during the SMD simulation.”

During the unfolding process, the hydrogen bonds that support the basic protein configuration are continuously broken.

“It is found that during the unfolding process, the hydrogen bonds that support the basic protein configuration are continuously broken.”

The deviation between the optimal characteristic distance in the protein flexibility analysis and the optimal filtration parameter η is about 2Å to 3Å.

“The deviation between optimal characteristic distance in protein the flexibility analysis and the “optimal filtration parameter η” is about 2Å to 3Å.”

As protein 1I2T unfolded, its related β1 value decreased.

“We found that as the protein unfolded, their related β1 value decreases.”

The average accumulation bar length A1 has a peak value around 6Å, consistent with the earlier finding with protein 1YZM.

“However, its peak value is around 6Å, which is consistent with our earlier finding with protein 1YZM.”

The location of the maximum accumulated bar length gives an accurate prediction of the optimal characteristic distance for the flexibility and rigidity index (FRI) analysis of protein temperature factors.

“It is found that the location of the maximum accumulated bar length gives an accurate prediction of the optimal characteristic distance for the flexibility and rigidity index (FRI) analysis20 of protein temperature factors.”

• cites [20] Xia KL Opron K Wei GW Multiscale multiphysics and multidomain models — Flexibility and rigidity Journal of Chemical Physics 2013 139 194109 24320318 PMC3855066 sourceNot checked yet

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• AssumptionThe authors state this as an assumption: The accumulated bar length correlates linearly to protein rigidity due to hydrogen bond strength, hydrophobic effects, electrostatic and van der Waals interactions.

Topological connectivities, i.e., 2-complexes and 3-complexes, reduce dramatically from conformation Fig. 19a to conformation Fig. 19g.

“Obviously, topological connectivities, i.e., 2-complexes and 3-complexes, reduce dramatically from conformation Fig. 19a to conformation Fig. 19g.”

Using linear regression, a correlation coefficient about 0.947 can be obtained for 31 configurations of protein 1I2T.

“Using a linear regression algorithm, a correlation coefficient about 0.947 can be obtained for 31 configurations of protein 1I2T.”

The linear relation between protein total energy and first Betti number is not highly accurate, with a correlation coefficient about 0.89 for 31 configurations of 1I2T.

“However, using the least square fitting, we found that this linear relation is not highly accurate with a correlation coefficient about 0.89 for 31 configurations of 1I2T.”

For four protein configurations in Fig. 20, their β1 values are 47, 19, 13 and 5, respectively.

“For four protein configurations in Fig. 20, their β1 values are 47, 19, 13 and 5, respectively.”

This aspect implies the robustness of the correlation matrix based filtration.

“This aspect is different from that of the cutoff distance based filtration discussed previously and implies the robustness of the correlation matrix based filtration.”

The stronger the interaction is, the more unified dynamics will be.

“The stronger the interaction is, the more unified dynamics will be.”

The negative accumulated bar length can be used to describe the stability of a protein.

“As the relative stability of a protein is proportional to its topological connectivity, which is a topology-function relationship, the negative accumulated bar length can be used to describe the stability of a protein.”

• AssumptionRests on an assumption: The relative stability of a protein is proportional to its topological connectivity.

This β1 bar is due to the global hole of the beta barrel.

“Obviously, this β1 bar is due to the global hole of the beta barrel.”

The interaction among particles leads to collective motions in a protein.

“The interaction among particles leads to collective motions in a protein.”

This method has also been successfully applied to the prediction of Debye-Waller factors.

“This method has also been successfully applied to the prediction of Debye-Waller factors.20”

• cites [20] Xia KL Opron K Wei GW Multiscale multiphysics and multidomain models — Flexibility and rigidity Journal of Chemical Physics 2013 139 194109 24320318 PMC3855066 sourceNot checked yet

  Not checked in the free run.

If the cutoff distance is larger than 14Å, excessive global connections are included, which leads to a reduction in prediction accuracy, especially for small proteins.

“At the same time, if the cutoff distance is larger than 14Å, excessive global connections are included, which leads to a reduction in prediction accuracy, especially for small proteins.”

This work reveals the topology-function relationship of proteins.

“This work reveals the topology-function relationship of proteins.”

• AssumptionRests on an assumption: There is a topology-function relationship in many biomolecular systems.

The paper's findings shed light on the topological interpretation of the optimal cutoff distance in GNM.

“Our findings shed light on the topological interpretation of the optimal cutoff distance in GNM.”

Our persistent homology analysis explains the optimal range of cutoff distances (i.e., rc = 7 Å to 8Å) for GNM in the literature very well.

“Therefore, our persistent homology analysis explains the optimal range of cutoff distances (i.e., rc = 7 Å to 8Å) for GNM in the literature73 very well.”

• cites [73] Yang L Song G Jernigan RL Protein elastic network models and the ranges of cooperativity Proceedings of the National Academy of Sciences 2009 106 30 12347 12352 PMC2718344 19617554 sourceNot checked yet

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If the cutoff distance used in elastic network models is smaller than 5Å, the constructed network is over-simplified without any global connectivity contribution.

“From the above analysis, it can seen that if the cutoff distance used in elastic network models is smaller than 5Å, the constructed network is over-simplified without any global connectivity contribution.”

The relaxation time is associated with protein flexibility.

“Therefore, the relaxation time is associated with protein flexibility.”

The persistent homology analysis of alpha helices and beta sheets reveals their topological features which are useful for deciphering protein fingerprints.

“The persistent homology analysis of alpha helices and beta sheets reveals their topological features which are useful for deciphering protein fingerprints as demonstrated in the above example.”

All-atom model based filtration can preserve more local scale topological information.

“However, all-atom model based filtration can preserve more local scale topological information.”

Since the same kernel and the same η are used in the FRI model and the persistent homology model, it is natural for the η corresponding to the maximum A1 to be the optimal characteristic distance in the FRI prediction.

“Since the same kernel and the same η are used in the FRI model and the persistent homology model (i.e., accumulation bar length), it is natural for the η corresponding to the maximum A1 to be the optimal characteristic distance in the FRI prediction.”

Barcode representation is able to incorporate certain geometric information.

“Therefore, barcode representation is able to incorporate certain geometric information.”

Our results unveil the topology-function relationship of proteins.

“Our results unveil the topology-function relationship of proteins.”

• AssumptionThe authors state this as an assumption: Protein structures are uniquely determined by various internal and external interactions, while protein functions such as stability and flexibility are solely determined by the structure.

These bars are the fingerprints of PHE or TYR types of amino acid residues.

“These bars are the fingerprints of “PHE” or “TYR” types of amino acid residues.”

The FRI theory has been intensively validated by comparing with the experimental data, especially the Debye-Waller factors.

“The FRI theory has been intensively validated by comparing with the experimental data, especially the Debye-Waller factors.20”

• cites [20] Xia KL Opron K Wei GW Multiscale multiphysics and multidomain models — Flexibility and rigidity Journal of Chemical Physics 2013 139 194109 24320318 PMC3855066 sourceNot checked yet

  Not checked in the free run.

Each β2 bar represents a void formed by one turn of three helices due to the high symmetry of the structure.

“Using our slicing technique, it can be found that each β2 bar represents a void formed by one turn of three helices due to the high symmetry of the structure.”

Without consideration of cofactors, the models may offer optimal B-factor prediction at wrong characteristic distances.

“Without any consideration of cofactors, our models may offer optimal B-factor prediction at wrong characteristic distances.”

• AssumptionRests on an assumption: Cofactors like metal ions, ligands, and small sugar molecules play an important role in protein stability and flexibility.

The birth and death of each k-complex uniquely represent either a local or a global geometric feature.

“Therefore, the birth and death of each k-complex uniquely represent either a local or a global geometric feature.”

The resulting topological diagrams shed light on the optimal cutoff distance used in the protein B-factor prediction with the Gaussian network model.

“The resulting topological diagrams shed light on the optimal cutoff distance used in the protein B-factor prediction with the Gaussian network model.”

Topological invariants, such as the first Betti number, give rise to a natural description of protein rigidity and flexibility.

“Consequently, topological invariants, such as the first Betti number, give rise to a natural description of protein rigidity and flexibility.”

Protein folding and unfolding involve massive changes in local topology and global topology.

“Protein folding and unfolding involve massive changes in its local topology and global topology.”

Considering only relatively small sized proteins may result in a lack of representation for relatively large proteins.

“This may results in a lack of representation for relatively large proteins.”

We are able to reveal the topology-function relationship.

“We therefore are able to reveal the topology-function relationship.”

Well-folded proteins, especially well-folded globular proteins, have abundant non-covalent bonds due to hydrogen bonds and van der Waals interactions, which translates into higher numbers of topological invariants, particularly a large measurement of the first Betti number.

“Our basic observation is that well-folded proteins, especially well-folded globular proteins, have abundant non-covalent bonds due to hydrogen bonds and van der Waals interactions, which translates into higher numbers of topological invariants, particularly, a large measurement of the first Betti number.”

All these topological features make the MTF unique for each biomolecule.

“It is all of these topological features that make the MTF unique for each biomolecule.”

The persistent homology based model provides an excellent quantitative prediction of protein total energy during the protein folding/unfolding process.

“Our persistent homology based model is found to provide an excellent quantitative prediction of protein total energy during the protein folding/unfolding process.”

• AssumptionThe authors state this as an assumption: The accumulated bar length correlates linearly to protein rigidity due to hydrogen bond strength, hydrophobic effects, electrostatic and van der Waals interactions.

In spite of the stated issues, the persistent homology analysis successfully captures the basic correlation coefficient behavior.

“In spite of these issues, our persistent homology analysis successfully captures the basic correlation coefficient behavior.”

The benefit of using the coarse-grained model is that the number of β1 and β2 is zero for a straight-line like peptide generated by SMD.

“The benefit of using the coarse-grained model is that the number of β1 and β2 is zero for a straight-line like peptide generated by SMD.”

There is a strong anti-correlation between protein total energy and its first Betti number during the protein unfolding process.

“Therefore, there is a strong anti-correlation between protein total energy and its first Betti number during the protein unfolding process.”

Topology and persistent homology are potential tools for protein folding characterization.

“Therefore, topology and persistent homology are potential tools for protein folding characterization.”

Consequently, the number of high order complexes that may be formed during the filtration decreases because of protein unfolding.

“Consequently, the number of high order complexes that may be formed during the filtration decreases because of protein unfolding.”

Persistent homology should provide an efficient tool for both qualitative characterization and quantitative analysis of protein folding or unfolding.

“Therefore, persistent homology should provide an efficient tool for both qualitative characterization and quantitative analysis of protein folding or unfolding.”

There is an obvious correlation between protein folding/unfolding and its topological trait.

“A comparison between configurations in Fig. 19 and their corresponding barcodes in Fig. 20 shows an obvious correlation between protein folding/unfolding and its topological trait.”