Nils Berglund | A and C and T and G. The alphabet of you: Almost complete DNA replication @NilsBerglund | Uploaded May 2024 | Updated October 2024, 2 hours ago.
This simulation is my first success in getting an almost perfect replica of a DNA molecule, using a very simplified model. The real process of DNA duplication is actually helped by various enzymes, such as DNA polymerase, that move nucleotides around. This mechanism is absent from the present model, which rather relies on a combination of random effects and rules for recombination. And as in the coupon's collector problem (see en.wikipedia.org/wiki/Coupon_collector%27s_problem ), in a random process, it takes longer and longer the closer one gets to completion.
The initial state is chosen such that a strand of DNA forms with one half of the nucleotides, while the other half are not allowed to react with each other. After a while, small particles representing enzymes are released (they are generated randomly), that break the connection between base pairs, thereby trying to separate the strands. The second half of the nucleotides can then recombine with the bases of the two strands, which would duplicate the DNA molecule once the process is completed.
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 are not allowed to attach again to each other.
To help the process of duplication, without the use of other enzymes, a number of artificial rules have been added. The main rule is that except for the initial DNA formation, only reactions between the two different families of nucleotides are allowed. In addition, a certain number of "repair" mechanisms have been added. The most important one is that if the newly attached nucleotides skip a base, then they detach at the backbone, which is marked by a purple disc. Another mechanism is that if a base attaches by error to the bases of two different nucleotides, then it detaches as well.
This video shows the same simulation at two different speeds:
Time lapse: 0:00
Original speed: 1:50
In the first part, the playback speed has been multiplied by 3.
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.
Render time: 42 minutes 13 seconds
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Stellar Wind" by the Unicorn Heads@dna_repaired_4K_mixed
Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). doi.org/10.1007/s10955-009-9734-0
maths.warwick.ac.uk/~theil/HL12-3-2009.pdf
Current version of the C code used to make these animations:
github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund-Nils-1343_.html
(in French, some with a Spanish translation)
#molecular_dynamics #dna
This simulation is my first success in getting an almost perfect replica of a DNA molecule, using a very simplified model. The real process of DNA duplication is actually helped by various enzymes, such as DNA polymerase, that move nucleotides around. This mechanism is absent from the present model, which rather relies on a combination of random effects and rules for recombination. And as in the coupon's collector problem (see en.wikipedia.org/wiki/Coupon_collector%27s_problem ), in a random process, it takes longer and longer the closer one gets to completion.
The initial state is chosen such that a strand of DNA forms with one half of the nucleotides, while the other half are not allowed to react with each other. After a while, small particles representing enzymes are released (they are generated randomly), that break the connection between base pairs, thereby trying to separate the strands. The second half of the nucleotides can then recombine with the bases of the two strands, which would duplicate the DNA molecule once the process is completed.
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 are not allowed to attach again to each other.
To help the process of duplication, without the use of other enzymes, a number of artificial rules have been added. The main rule is that except for the initial DNA formation, only reactions between the two different families of nucleotides are allowed. In addition, a certain number of "repair" mechanisms have been added. The most important one is that if the newly attached nucleotides skip a base, then they detach at the backbone, which is marked by a purple disc. Another mechanism is that if a base attaches by error to the bases of two different nucleotides, then it detaches as well.
This video shows the same simulation at two different speeds:
Time lapse: 0:00
Original speed: 1:50
In the first part, the playback speed has been multiplied by 3.
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.
Render time: 42 minutes 13 seconds
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Stellar Wind" by the Unicorn Heads@dna_repaired_4K_mixed
Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). doi.org/10.1007/s10955-009-9734-0
maths.warwick.ac.uk/~theil/HL12-3-2009.pdf
Current version of the C code used to make these animations:
github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund-Nils-1343_.html
(in French, some with a Spanish translation)
#molecular_dynamics #dna
