Some commenters asked me to try and simulate DNA replication. This video shows a first step in that direction. The initial state is chosen such that a strand of DNA forms. After a while, small particles representing enzymes are released (they are generated randomly near the right border), that break the connection between base pairs, thereby trying to separate the strands. A later step will be to add nucleotides that recombine with the two strands of DNA.
Each T-shaped molecule in this simulation represents a nucleotide, consisting of a phosphate-deoxyribose backbone, and one nucleic base among adenine (A), thymine (T), guanine (G) and cytosine (C). Atoms belonging to different molecules interact via a Lennard-Jones potential, while atoms within the same molecule interact with a stiff harmonic potential. Whenever two ends of T-bars come close to each other, they "react" to become attached. In a similar way, adenine and thymine can attach to each other, as well as cytosine and guanine.
Each reacting extremity of a nucleotide consists of two atoms. This provides more rigidity to the larger molecules formed after reactions. The left and right ends of each backbone are considered as being different, and can only attach to a backbone end of the other type. This is to avoid that base pairs point in different directions (a problem that is specific to the 2D nature of this simulation). The temperature is slowly increased during the simulation. This is because otherwise the system tends to "freeze", probably due to energy being absorbed in small vibrations within the molecules. Coulomb-like interactions between base pairs and backbone ends have been added, which are attractive for pairs that can combine, and repulsive otherwise.
The particles' color hue depends on their type. The base types appear in the following colors: A red, T yellow, C green, G cyan. The enzymes, represented by yellow triangles, interact via a weak electrostatic repulsion, and are attracted by the bases. When an enzyme approaches a pair of bases, these detach, and become "inactive" in order to prevent them from attaching again.
To save on computation time, particles are placed into a "hash grid", each cell of which contains between 3 and 10 particles. Then only the influence of other particles in the same or neighboring cells is taken into account for each particle.
The temperature is controlled by a thermostat, implemented here with the "Nosé-Hoover-Langevin" algorithm introduced by Ben Leimkuhler, Emad Noorizadeh and Florian Theil, see reference below. The idea of the algorithm is to couple the momenta of the system to a single random process, which fluctuates around a temperature-dependent mean value. Lower temperatures lead to lower mean values.
The Lennard-Jones potential is strongly repulsive at short distance, and mildly attracting at long distance. It is widely used as a simple yet realistic model for the motion of electrically neutral molecules. The force results from the repulsion between electrons due to Pauli's exclusion principle, while the attractive part is a more subtle effect appearing in a multipole expansion. For more details, see https://en.wikipedia.org/wiki/Lennard...

Render time: 14 minutes 16 seconds
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-wor...

Music: "When Johnny Goes Marching" by E's Jammy Jams@ethaneubanks749

Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). https://doi.org/10.1007/s10955-009-97...
http://www.maths.warwick.ac.uk/~theil...

Current version of the C code used to make these animations:
https://github.com/nilsberglund-orlea...
https://www.idpoisson.fr/berglund/sof...
Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund...
(in French, some with a Spanish translation)

#molecular_dynamics #dna