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Related Concept Videos

Physiological Pharmacokinetic Models: Blood Flow-Limited Versus Diffusion-Limited Models00:57

Physiological Pharmacokinetic Models: Blood Flow-Limited Versus Diffusion-Limited Models

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Physiological pharmacokinetic models, often called flow-limited or perfusion models, typically assume a swift drug distribution between tissue and venous blood, creating a rapid drug equilibrium. This premise is based on the idea that drug diffusion is extremely fast, and the cell membrane presents no barrier to drug permeation. In this scenario, where no drug binding occurs, the drug concentration in the tissue equals that of the venous blood leaving the tissue. This greatly simplifies the...
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Diffusion01:21

Diffusion

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Diffusion is a type of passive transport. In passive transport, a substance tends to move from an area of high concentration to an area of low concentration until the concentration is equal across the space. For example, take the diffusion of substances through the air. When someone opens a perfume bottle in a room filled with people, the perfume is at its highest concentration in the bottle and is at its lowest at the edges of the room. The perfume vapor will diffuse, or spread away, from the...
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Diffusion01:12

Diffusion

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Diffusion is the passive movement of substances down their concentration gradients—requiring no expenditure of cellular energy. Substances, such as molecules or ions, diffuse from an area of high concentration to an area of low concentration in the cytosol or across membranes. Eventually, the concentration will even out, with the substance moving randomly but causing no net change in concentration. Such a state is called dynamic equilibrium, which is essential for maintaining overall...
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Passive Diffusion: Overview and Kinetics01:17

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Passive diffusion is a critical process that allows small lipophilic drugs to cross the cell membrane along a concentration gradient. This mechanism's efficiency depends on four primary factors: the membrane's surface area, the drug's lipid-water partition coefficient, the concentration gradient, and the membrane's thickness.
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Model Approaches for Pharmacokinetic Data: Distributed Parameter Models01:06

Model Approaches for Pharmacokinetic Data: Distributed Parameter Models

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Pharmacokinetic models are mathematical constructs that represent and predict the time course of drug concentrations in the body, providing meaningful pharmacokinetic parameters. These models are categorized into compartment, physiological, and distributed parameter models.
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Protein Diffusion in the Membrane01:24

Protein Diffusion in the Membrane

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Proteins show rotational as well as lateral diffusion across the membrane. The lateral diffusion of proteins was confirmed through the cell fusion experiment where mouse and human cells were fused, resulting in hybrid cells. When the human and mouse cells fused, the specific membrane proteins on human and mouse cells were marked with the red and green-fluorescent markers, respectively. Initially, the red and green fluorescence was located on the respective hemisphere of the cell. As time...
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Related Experiment Video

Updated: Oct 29, 2025

Image Processing Protocol for the Analysis of the Diffusion and Cluster Size of Membrane Receptors by Fluorescence Microscopy
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Bayesian inversion of a diffusion model with application to biology.

Jean-Charles Croix1, Nicolas Durrande2, Mauricio A Alvarez3

  • 1School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, UK. j.croix@sussex.ac.uk.

Journal of Mathematical Biology
|July 6, 2021
PubMed
Summary

This study introduces a Bayesian method to model protein transcription from messenger RNA (mRNA). It successfully estimates diffusion, self-regulation rates, and source functions for complex biological systems.

Keywords:
Bayesian inverse problemsDiffusion equationFunctional MCMCGaussian processes

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Area of Science:

  • Mathematical Biology
  • Computational Biology
  • Biophysics

Background:

  • Modeling complex dynamical systems like partial differential equations is crucial for understanding natural phenomena.
  • Inverse problems in experimental sciences often present challenges due to ill-posedness.
  • Protein transcription from messenger RNA (mRNA) involves complex regulatory processes.

Purpose of the Study:

  • To estimate jointly the differential operator coefficients (diffusion and self-regulation rates) and a functional source term for a linear parabolic equation modeling protein transcription from mRNA.
  • To apply a recent Bayesian methodology for infinite-dimensional inverse problems to provide a unique posterior distribution.
  • To summarize the posterior distribution using a Maximum a Posteriori (MAP) estimator.

Main Methods:

  • Utilized a linear parabolic partial differential equation as a model for protein transcription from mRNA.
  • Applied Bayesian inference for infinite-dimensional inverse problems to estimate model parameters.
  • Employed a Maximum a Posteriori (MAP) estimator to summarize the resulting posterior distribution.
  • Illustrated the theoretical solution using a Markov Chain Monte Carlo (MCMC) algorithm adapted for non-Gaussian settings.

Main Results:

  • Developed a robust Bayesian framework for inverse problems in biological modeling.
  • Successfully estimated diffusion coefficients, self-regulation rates, and functional source terms from data.
  • Demonstrated the effectiveness of the MAP estimator in summarizing the posterior distribution.
  • Validated the approach with a state-of-the-art MCMC algorithm.

Conclusions:

  • The proposed Bayesian methodology offers a powerful tool for analyzing complex inverse problems in biological systems.
  • Accurate estimation of model parameters enhances the understanding of biological phenomena like protein transcription.
  • The integration of MAP estimation and advanced MCMC algorithms provides a reliable approach for parameter inference.