A circuit theory of protein structure

Protein secondary and tertiary structure is modeled as a linear passive analog lumped electrical circuit. Modeling is based on the structural similarity between helix, sheet, turn/loop, and helix pair in proteins and inductor, capacitor, resistor, and transformer in electrical circuits; it includes methods from circuit analysis and synthesis. A 'protein circuit' is a one-port with a restrictive circuit topology (for example, the circuit for a secondary structure cannot be a Foster II ladder or a Wheatstone-like bridge). It has a rational positive real impedance function whose pole-zero distribution serves as a compact descriptor of secondary and tertiary structure, which is reminiscent of the Ramachandran plot. Standard circuit analysis methods such as node/loop equations and pole-zero maps may be used to study differences at the secondary and tertiary levels within and across proteins. Pairs of interacting proteins can be modeled as two-ports and studied via transfer functions. Similarly circuit synthesis methods can be used to construct 'protein circuits' whose real counterparts may or may not exist. An analysis example shows how a 'protein circuit' is constructed for thioredoxin and its pole-zero map obtained. A synthesis example shows how an electrical circuit with a single Brune section is obtained from a specified set of poles and zeros and then mapped to an artificial protein with a helix pair (corresponding to the transformer in the Brune section). Possible applications to folding, drug design, and visualization are indicated. 1. Overview A model of protein structure based on electrical circuits is described. 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. The following is a summary of this report. Section 2 gives a brief review of protein structure modeling and a summary of the current approach. Section 3 discusses the derivation of RLCM 'protein circuits' from secondary and tertiary structure. Restrictions on 'protein circuit' topology resulting from the sequential nature of the protein’s primary sequence are noted. Section 4 looks at the application of circuit analysis methods to 'protein circuits' based on impedance functions and to pairs of ‘protein circuits’ using transfer functions. In Section 5, modeling of protein pairs using transfer functions is briefly examined. In Section 6, synthesis methods for the design of protein ‘circuits’ are described. Section 7 concludes with a brief discussion of the potential applications of this approach. An earlier version of this report is available at [1]. 2. Protein modeling: analytical and synthetic methods Proteins structure can be considered at three levels: 1) primary, in which a protein is a sequence of amino acids (or equivalently a string of characters drawn from an alphabet of twenty characters); 2) secondary, in which subsequences form three types of geometric shapes: helices, sheets, and turns/loops; and 3) tertiary, in which the secondary structure folds on itself to form complex three-dimensional shapes, within which a number of recognizable ‘motifs’ such as jelly roll, helix pairs, etc. are often seen. One of the main objectives in the study of proteins is to map the primary sequence of a protein to tertiary structure. 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 mapping secondary to tertiary structure there are several approaches, including: 1) Analytical methods, which use some kind of minimization of an energy function based on covalent and non-covalent interactions among the side chains and the backbone; some of them are based on lattice models that use cubes [5] or cylinders [6] as structural elements; 2) Synthetic methods, which are aimed at the opposite: deriving a primary sequence that leads to a desired tertiary shape; this reverse process is studied in drug discovery and design and is largely ad hoc [7]; and 3) Visualization studies, which seek to represent graphically the interactions of secondary structure that lead to discernible tertiary substructures seen in classes of naturally occurring proteins [2, 3]; they are often based on diagrammatic representations, such as Richardson’s schematics [2], skeletal structures [3], and TOPS diagrams [8] (which look similar to class diagrams in object-oriented design [9]), and the conventional stick-ball model [2, 3]. In the present work, protein structure is modeled via passive analog lumped electrical circuits [12]. Helix (H), sheet (E), and turn (T) in proteins are mapped to inductor (L), capacitor (C), and resistor (R). By adding capacitive bridges to represent bonds between distant residues and transformers (with mutual inductance M between the coils) to represent helix pairs the resulting RLCM circuit can be used to represent tertiary structure. The equivalence is shown in Figure 1. Standard analysis and synthesis methods [10-14] may then be used to analyze and synthesize 'protein structures'. As is customary in electrical engineering, the terms ‘circuit’ and ‘network’ are used interchangeably in what follows. . CC-BY-NC-ND 4.0 International license peer-reviewed) is the author/funder. It is made available under a The copyright holder for this preprint (which was not . http://dx.doi.org/10.1101/023994 doi: bioRxiv preprint first posted online Aug. 5, 2015;