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Abstract: The Abelian Higgs model in two-dimensional spacetime is an example of a gauge theory with spontaneous symmetry breaking. In our study, we show that, with lattice regularization, this model is applicable to solving the problem of modeling the three-dimensional folding of polypeptide chains.
In this work, we consider only the protein backbone ($C_{\alpha}$ atoms) rather than the full-atom protein molecule. Such a chain is a curve that can be described using discrete Frenet coordinates (curvature and torsion). In [1], the magnitude of the complex scalar field in the Abelian Higgs model is interpreted as the local curvature of the chain, and the phase of the field determines the torsion. The action of this model involves interaction with an external field $\alpha_{x,\mu\nu}$, which models defines the non-uniform influence of the solvent (water molecules) on the protein chain and has the following form:
$$S = \beta \sum_x\sum_{\mu<\nu} (1-\cos({\theta_{x,\mu\nu} + \alpha_{x,\mu\nu})}) + m^2\theta_{x,\mu} + $$ $$+ \sum_{x,\mu} \left| \phi_{x} - e^{i( \theta_{x,\mu})} \phi_{x+\hat{\mu}} \right|^2 + \sum_{x} ( - \kappa |\phi_{x}|^2 + \lambda |\phi_{x}|^4 ) + \frac{\kappa^2}{4\lambda}. $$ The numerical simulation [2] is performed using a hybrid Monte Carlo algorithm, in which a new field configuration is accepted with a probability proportional to $\exp(-S)$. During the algorithm, the external field $\alpha_{x,\mu\nu}$ is optimized to reproduced the desired protein structure. The simulation results show that the external field obtained in this way allows for the complete reconstruction of the curvature and torsion profiles. Using myoglobin (PDB id: 1ABS) as an example, a structure corresponding to the native conformation with an RMSD of 1.2 Å was reproduced [3].
Acknowledgments: This work was supported within the grant of the Ministry of Science and Higher Education of the Russian Federation (FEFU Program «PRIORITY 2030», topic No. ASP-25-02-1.03-0022).
References:
1. Niemi, A.J. Gauge fields, strings, solitons, anomalies, and the speed of life. Theor Math Phys 181, 1235–1262 (2014). https://doi.org/10.1007/s11232-014-0210-x
2. Heitger J. Numerical Simulations of Gauge-Higgs Models on the Lattice. 1997. 220 p.
3. Liubimov, S.D., Gerasimeniuk, N.V., Korneev, A.A. et al. Modeling the Structure of Myoglobin within the Abelian Higgs Model. Biochem. Moscow Suppl. Ser. A, 2025, vol. 19, pp. 449--455. DOI: https://doi.org/10.1134/S1990747825700436.
