Enhance 34 to 46: DNA replication (short version) @NilsBerglund

Nils Berglund | Enhance 34 to 46: DNA replication (short version) @NilsBerglund | Uploaded May 2024 | Updated October 2024, 1 hour ago.
Today's simulation is released both in this shorter version, and in the longer version youtu.be/vYfMEr0182g , both in 60 fps. It shows a simplified model for DNA replication. Compared to previous similar simulations, the DNA strand to be replicated is shorter, allowing the process to get closer to completion. There remain some defects in the replicated strands, however.
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.
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 near the right border), 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.
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.

Render time: 19 minutes 45 seconds
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Finals" by Silent Partner

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

$Waves of two different frequencies crossing a biconvex lens Someone asked about the effect of the lights frequency on the working of a lens. The theory of geometrical optics, or ray optics, considers an idealized situation in which the wavelength is negligibly small, and much of basic optics relies on this approximation. Increasing the wavelength results in effects such as diffraction, which may degrade the quality of the lens. In this simulation, two sources at different frequencies emit light crossing a biconvex lens. The frequency of the lower source is three times the frequency of the upper one. While the difference of frequencies has an effect on the waves interference, the energy profile in the focal plane does not show any important difference. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:15 In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows the square of the signal along the focal plane, averaged over a time window. The focal plane and the plane containing the sources are roughly symmetrical with respect to the vertical symmetry axis of the lens. Render time: 34 minutes 41 seconds Compression: crf 23 Color scheme: Part 1 - Twilight by Bastian Bechtold https://github.com/bastibe/twilight Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Bright Idea by Geographer@geographermusic See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on a few previous simulations of wave equations. The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf C code: https://github.com/nilsberglund-orleans/YouTube-simulations https://www.idpoisson.fr/berglund/software.html Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code! #wave #lens #focus$ $3D representation of a gradient index lens This is a 3D representation of the simulation https://youtu.be/KB_Roqs38F8 showing the light of two point sources crossing a gradient index lens. The simulation is still two-dimensional, but the z-coordinate is used to represent a combination of the wave height or energy, and of the refractive index. Several recent videos on this channel, such as https://youtu.be/CdpTq3dSrXA and https://youtu.be/aYvsBgyJgIE , showed planar waves crossing a gradient index lens. In this video, we look instead at two light sources that are located quite close to the lens. The lens is indeed able to focus the light on two spots located in the focal plane. In practice, one seldom has light sources located so close to the lens (except perhaps in macro photography), but this set-up was better suited to the simulation method used here. To be able to get such a small focal length, the refractive index has been chosen rather large in the center of the lens, with a value of 1.82. The videos are inspired by Huygens Optics recent short https://youtube.com/shorts/VGd3Ajnp6e0 showing the principle of a gradient index lens. Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation https://youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens. In this simulation, the index decreases like sqrt(n0² - a*r²), where r is the distance to the axis of symmetry. This results in the incoming waves being focused at two points in the (estimated) focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:23 In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged from the beginning of the simulation. The z-coordinate shows the sum of the wave height or energy and of the local refractive index. There are absorbing boundary conditions on the exterior borders of the simulation. Render time: 1 hour 35 minutes Compression: crf 23 Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Cloud Patterns by Silent Partner See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on a few previous simulations of wave equations. The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf C code: https://github.com/nilsberglund-orleans/YouTube-simulations https://www.idpoisson.fr/berglund/software.html Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code! #wave #lens #gradient_index$ $Classics revisited: A hexagonal parabolic resonator This is a remake of the video https://youtu.be/lVDwH5PfEGY of a parabolic resonator with six sides, in much higher resolution, and with a pulsing source. The resonator is made of six parts of parabolas, which share the central point as a common focus. Whenever a circular wave centered at the focus hits a parabola, it is transformed into a linear wave. When this linear wave hits another parabola, it is transformed back into a circular wave, converging at the focal point. This pattern repeats almost periodically, except that diffraction effects (and probably also some numerical dispersion) slowly degrade the pattern. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:52 In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, slightly averaged over a sliding time window. Render time: 22 minutes 14 seconds Compression: crf 23 Color scheme: Part 1 - Twilight by Bastian Bechtold https://github.com/bastibe/twilight Part 2 - Turbo, by Anton Mikhailov https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a Music: Into the Void by TrackTribe@TrackTribe See also https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ for more explanations (in French) on a few previous simulations of wave equations. The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf C code: https://github.com/nilsberglund-orleans/YouTube-simulations https://www.idpoisson.fr/berglund/software.html Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code! #wave #resonator$

A simplistic model for tides - periodic forcing

$Waves of two different frequencies crossing a randomized square lattice This simulation is similar to the one shown in the video https://youtu.be/pObtx2xmZDA , but instead of being on a regular square lattice, the position of the scatterers has been slightly randomized. The radius of the scatterers is also random. This has a dramatic effect on the waves, which have much more trouble crossing the lattice, a phenomenon related to Anderson localization. The frequency of the lower source is three times the frequency of the upper one. The resulting wavelengths are such that the open intervals in the grating are roughly between both wavelengths. As a result, waves of the lower source pass the grating a bit more easily, as can be best seen on the energy plot. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:27 In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window. The contrast has been enhanced by a shading procedure, similar to the one I have used on videos of reaction-diffusion equations. The process is to compute the normal vector to a surface in 3D that would be obtained by using the third dimension to represent the field, and then to make the luminosity depend on the angle between the normal vector and a fixed direction. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows a time-averaged version of the signal near the right boundary of the simulated rectangular area. More precisely, it shows the square root of an average of squares of the respective field value (wave height or energy). Render time: 34 minutes 25 seconds Compression: crf 23 Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Where She Walks by Everet Almond See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on a few previous simulations of wave equations. The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf C code: https://github.com/nilsberglund-orleans/YouTube-simulations https://www.idpoisson.fr/berglund/software.html Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code! #wave #diffraction #grating$

Speed and vorticity of a fluid on a rotating sphere

An asteroid impact in the Indian Ocean, with tsunami-induced flooding

Shallow water flowing over an immersed dodecahedron

$Exciting spirals in a circle with sixteen out of phase sources Like several previous videos on this channel , this one shows waves in a circular cavity excited by pulsing sources in other, connected cavities. In this case, there are sixteen sources, pulsing a fraction 1/32 of a period out of phase. The resulting wave patterns start to resemble spirals, as suggested in comments to a previous video. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:27 In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, slightly averaged over a sliding time window. The color dependence on wave height has been chosen quite important, so that the darker regions concentrate near the nodal lines of the waves, that is, the curves where the wave height is zero. Render time: 17 minutes 51 seconds Compression: crf 23 Color scheme: Part 1 - Twilight by Bastian Bechtold https://github.com/bastibe/twilight Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Asleep by Verified Picasso@verifiedpicasso7260 See also https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ for more explanations (in French) on a few previous simulations of wave equations. The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf C code: https://github.com/nilsberglund-orleans/YouTube-simulations https://www.idpoisson.fr/berglund/software.html Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code! #wave #resonator$