Draft:Triangulation sensing

• Comment: Issues have not been fixed since the last decline. LR.127 (talk) 01:18, 18 September 2024 (UTC)

• Comment: Requires complete rewrite and more references to prove notability. The Herald (Benison) (talk) 07:01, 4 February 2024 (UTC)

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Triangulation sensing in biology refers to the use of multiple signals or sources of information to accurately determine the location, orientation, or movement of a biological object, stimulus, or entity in space. This concept borrows from the principle of triangulation in geometry, where the position of an unknown point is determined by measuring angles or distances from multiple known points. The theory can be described the computational steps for a cell containing on its surface small windows, to estimate the location of a source emitting random particles in a medium. The concept of triangulation sensing suggests that the source of an external gradient can be localized by a cell by evaluating the number of binding events at each receptor. This approach allows the reconstruction of the gradient source's location using molecular fluxes of diffusing particles.

Navigation in Space

• Migratory birds and marine animals use environmental cues like Earth's magnetic field, celestial navigation, and olfactory landmarks, often integrating multiple sources of information to navigate long distances.

Chemical Gradients

• Cells, such as bacteria and immune cells, perform chemotaxis by sensing chemical gradients in their environment. By detecting differences in concentration between multiple points on their surface, they "triangulate" the direction of the gradient. Bacteria like E. coli move toward higher concentrations of attractants (e.g., nutrients) by detecting temporal changes in chemical gradients.

Mechanical Sensing

• Organisms or cells can use mechanical cues from their environment to determine spatial orientation or stress distribution. For example, plants sense the direction of light (phototropism) or gravity (gravitropism) by triangulating signals within specialized cells.

Neuronal Triangulation

• Neurons in the brain, especially in animals like rodents, use triangulation-like processes to determine spatial location using specialized cells such as place cells, grid cells, and head direction cells in the hippocampus and entorhinal cortex.

Physical Model

In biological systems, particles such as morphogens or transcription factors bind to cellular receptors to mediate processes like neuronal navigation in Brain. During neuronal development, the growth cone of a neuron must navigate across long distances to establish connections between distant brain regions. This navigation relies on external chemical gradients interacting with cellular receptors to provide positional information. The transformation of these external gradients into actionable positional data is a crucial step in the guidance process.

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Reconstruction of Gradient Source Location

The reconstruction of a gradient source from diffusing particles involves the following steps:

1. Arrival of Diffusing Particles Brownian particles are released from the source and diffuse through the medium, arriving at small absorbing receptors.
2. Counting Particle Arrivals Receptors count the number of Brownian particles that bind within specific time windows, providing an estimate of the flux at each receptor.
3. Source Position Estimation The source position is determined by combining the fluxes recorded at multiple receptors, which is mathematically equivalent to solving the inverse problem of the Laplace's equation.
4. Reduction of Fluctuations The accuracy of source localization improves as additional receptors (or windows) are incorporated into the system, reducing flux fluctuations.

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Mathematical Formulation

The mathematical model is based on solving the Laplace equation asymptotically. It considers molecules diffusing toward N narrow absorbing windows distributed on the surface of a three-dimensional object, typically modeled as a sphere (in three dimensions) or a disk (in two dimensions).

1. Diffusion and Binding Individual Brownian particles are released from a source at position x0​ outside the object and diffuse through the medium. Narrow absorbing windows serve as receptors with boundary conditions representing fast binding.
2. Flux Estimation The steady-state flux at each receptor is proportional to the density of particles arriving at that window.
3. Source Localization To reconstruct the source position x0​, at least three receptors are required. The process involves inverting a system of equations derived from the fluxes. For N>3, numerical methods are employed to solve the overdetermined system, increasing accuracy.The procedure can be accelerated by hybrid stochastic simulations.

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Numerical and Computational Enhancements

When the number of receptors N is large (e.g., N>10), hybrid stochastic simulations can accelerate the computation of the source position. These methods integrate deterministic approaches with stochastic particle simulations, reducing computational time while maintaining accuracy.