A circuit theory of protein structure
Summary (3 min read)
1. Overview
- Helices are mapped to inductors, strand pairs to capacitors, turns/loops to resistors, and helix pairs to transformers (coupled inductors).
- Cys-Cys bonds are capacitors that cause the circuit to fold on itself like the protein modeled.
- The resulting linear circuit is fully described by its input impedance Z(s), a positive real (p.r.) function of the form P(s)/Q(s), where s is the complex frequency, or equivalently a pole-zero map.
- The result is a mathematical representation of protein structure with systematic procedures for analysis, synthesis, classification, and design, augmented by an electrical-circuit-based alternative to ribbon diagrams.
- Section 7 concludes with a brief discussion of the potential applications of this approach.
2. Protein modeling: analytical and synthetic methods
- Also, since form often determines function, knowledge of the relationship of tertiary structure to function is of fundamental importance [2, 3].
- The identification of secondary structure consisting of alpha helices, beta sheets, and turns/loops from the primary amino acid sequence of a protein is now fairly routine [4].
- In the present work, protein structure is modeled via passive analog lumped electrical circuits [12].
- As is customary in electrical engineering, the terms ‘circuit’ and ‘network’ are used interchangeably in what follows.
3.1 Secondary level structure and its circuit analogues
- At the secondary level, local chemical constraints and physical forces cause the linear sequence to form helices, strands (which themselves come together to form sheets), and turns that connect strands and/or helices.
- One helix turn corresponds to about 3.5 residues in the primary sequence, and a sheet has two or more strands.
- Element values are chosen so that every element contributes to the circuit impedance without being swamped out by the others in an appropriate frequency range.
- The resulting RLC circuit for secondary structure is named 'p-RLC-s circuit'.
- Table 1 shows the mapping from protein secondary elements to electrical circuit elements.
3.2 Tertiary structure
- There are two ways in which tertiary structure is obtained from secondary structure:.
- This is modeled here as coupled coils with mutual inductance M. The resulting p-RLC-t circuit (with capacitive bridges) or p-RLCM-t circuit (with coupled coils) represents tertiary structure.
- Thioredoxin Data from the public domain protein database PDB [15] are used to fix circuit element values, with tertiary elements determined by visual inspection of the ribbon diagram and the schematic for the protein’s entry in PDB, also known as Example.
- As an example, the p-RLCM-t circuit for thioredoxin (PDB accession id: 1SRX) is shown in Figure 2.
- This manual process can be replaced with a computer program that generates 'protein circuits' from PDB data and computes their pole-zero maps using a combination of symbolic and numerical computing [16].
3.3 Constraints on 'protein circuit' topology
- As a consequence, the electrical analogue for secondary structure has an approximate chain or ladder structure (which is modified by the addition of other circuit elements to add more tertiary structure, see below).
- This places the following fundamental constraints on the circuit topology for the secondary circuit and any other tertiary additions to it: an inductor cannot be a shunt element in the ladder bridges cannot occur, which also means that the ladder cannot be a series of lattices.
- Other consequential constraints are discussed below in the section on synthesis.
4.1 Circuit properties of a p-RLC(M) circuit
- In addition to loop and node equations ('Kirchhoff's laws') there are some other considerations.
- Thus the protein’s primary sequence has an implicit direction associated with it because of the order in which the protein is synthesized in the cell, which is N-terminal to C-terminal.
- This is in contrast with passive electrical circuits whose electrical behavior is usually independent of the terminals of the port (with one exception, that of a polarized capacitor, but this behavior is largely a d.c. behavior).
- It is nevertheless useful to retain the directionality property of the protein's primary sequence when coupled coils (representing helix pairs) are present since the coupling between them is dependent on the direction of current flow.
- The dot rule captures [12] this property in a natural way.
4.2 Input impedance and spectral properties
- For proteins without any helices or all-helix proteins, the poles and zeros are all on the negative real axis.
- Several techniques to reduce the effort required are available [16].
- When Z(s) is available, the amplitude function |Z(jω)| and the phase function φ(jω) can be computed for s = jω in a routine manner.
- The pole-zero distribution and the phase plot for thioredoxin are shown below for the secondary and tertiary structure circuits.
- The change in the pole-zero pattern (going from all poles and zeros on the negative real axis for secondary to negative real and some complex poles and zeros for tertiary) can be used to characterize the protein.
5. Protein pairs: transfer function analysis
- Many of the interactions occur because of pairs of proteins coming together (‘docking’) and forming an aggregate shape that causes specific biophysical and/or biochemical reactions to take place.
- In the model presented here this can be represented by capacitive contacts and/or formation of helix pairs by helices in the two proteins.
- The protein interaction can be effectively studied through the transfer function T(s) for the two-port circuit.
6. Network synthesis methods and an example of 'protein circuit' synthesis
- The last is not as useful in the present context because it is not easy to design circuits with transformers, which means that proteins with helix pairs are excluded.
- A specified positive real Z(s) is implemented with a one-terminal RLCM network.
- They include Foster I and II forms, Cauer I and II forms, Brune ladders, Darlington’s method, Bott-Duffin synthesis, and Miyata’s and Kuh’s methods [13, 14].
- Starting with an impedance function Z(s) (or equivalently a set of poles and zeros) a secondary structure can be derived and modified to yield a tertiary structure.
7. Discussion
- The model presented here provides an electrical-circuit-based alternative to ones based in chemical topology [19] or lattice structures [5, 6].
- This approach has several potential uses [1], such as modeling of protein folding (based on sensitivity analysis of the 'protein circuit' [10]), searching for proteins that are similar in some sense, drug design and discovery, and the electrical properties of proteins (leading possibly to the use of proteins as nano-level circuits).
- Circuit simulation provides correlates to chemical structure and behavior of existing proteins, and drug design may be viewed as circuit synthesis using 'protein circuit' libraries followed by biochemical synthesis using libraries of designed motifs.
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"A circuit theory of protein structu..." refers background or methods in this paper
...Standard analysis and synthesis methods [10-14] may then be used to analyze and synthesize 'protein structures'....
[...]
...They include Foster I and II forms, Cauer I and II forms, Brune ladders, Darlington’s method, Bott-Duffin synthesis, and Miyata’s and Kuh’s methods [13, 14]....
[...]
...Different forms can be used in different stages to form mixed ladders, leading to a variety of implementations [12, 14]....
[...]
...This essentially results in a two-port network [13, 14] in which one of the ports is represented by the N and C terminals of one of the proteins and the other port by the N and C terminals of the second protein....
[...]
...Network synthesis methods and an example of 'protein circuit' synthesis An RLCM circuit (one-port or two-port) can be synthesized using frequency domain methods [10, 11, 13, 14] or timedomain-based ones [17]....
[...]
473 citations
"A circuit theory of protein structu..." refers background in this paper
...The identification of secondary structure consisting of alpha helices, beta sheets, and turns/loops from the primary amino acid sequence of a protein is now fairly routine [4]....
[...]