Tag Archives: diffusion

Modelling from the cell through the individual to the host population

During the lock-down in the UK due to the coronavirus pandemic, I have been reading about viruses and the modelling of them.  It is a multi-disciplinary and multi-scale problem; so, something that engineers should be well-equipped to tackle.  It is a multi-scale because we need to understand the spread of the virus in the human population so that we can control it, we need to understand the process of infection in individuals so that we can protect them, and we need to understand the mechanisms of virus-cell interaction so that we can stop the replication of the virus.  At each size scale, models capable of representing the real-world processes will help us explore different approaches to arresting the progress of the virus and will need to be calibrated and validated against measurements.  This can be represented in the sort of model-test pyramid shown in the top graphic that has been used in the aerospace industry [1-2] for many years [see ‘Hierarchical modelling in engineering and biology’ on March 14th, 2018] and which we have recently introduced in the nuclear fission [3] and fusion [4] industries [see ‘Thought leadership in fusion engineering’ on October 9th, 2019].  At the top of the pyramid, the spread of the virus in the population is being modelled by epidemiologists, such as Professor Neil Ferguson [5], using statistical models based on infection data.  However, I am more interested in the bottom of the pyramid because the particles of the coronavirus are about the same size as the nanoparticles that I have been studying for some years [see ‘Slow moving nanoparticles’ on December 13th, 2017] and their motion appears to be dominated by diffusion processes [see ‘Salt increases nanoparticle diffusion’ on April 22nd, 2020] [6-7].  The first step towards virus infection of a cell is diffusion of the virus towards the cell which is believed to be a relatively slow process and hence a good model of diffusion would assist in designing drugs that could arrest or decelerate infection of cells [8].  Many types of virus on entering the cell make their way to the nucleus where they replicate causing the cell to die, afterwhich the virus progeny are dispersed to repeat the process.  You can see part of this sequence for coronavirus (SARS-COV-2) in this sequence of images. The trafficking across the cytoplasm of the cell to the nucleus can occur in a number of ways including the formation of a capsule or endosome that moves across the cell towards the nuclear membrane where the virus particles leave the endosome and travel through microtubules into the nucleus.  Holcman & Schuss [9] provide a good graphic illustrating these transport mechanisms.  In 2019, Briane et al [10] reviewed models of diffusion of intracellular particles inside living eukaryotic cells, i.e. cells with a nuclear enclosed by a membrane as in all animals.  Intracellular diffusion is believed to be driven by Brownian motion and by motor-proteins including dynein, kinesin and myosin that enable motion through microtubules.  They observed that the density of the structure of cytoplasm, or cytoskeleton, can hinder the free displacement of a particle leading to subdiffusion; while, cytoskeleton elasticity and thermal bending can accelerate it leading to superdiffusion.  These molecular and cellular interactions are happening at disparate spatial and temporal scales [11] which is one of the difficulties encountered in creating predictive simulations of virus-cell interactions.  In other words, the bottom layers of the model-test pyramid appear to be constructed from many more strata when you start to look more closely.  And, you need to add a time dimension to it.  Prior to the coronavirus pandemic, more modelling efforts were perhaps focussed on understanding the process of infection by Human Immunodeficiency Virus (HIV), including by a multi-national group of scientists from Chile, France, Morocco, Russia and Spain [12-14].  However, the current coronavirus pandemic is galvanising researchers who are starting to think about novel ways of building multiscale models that encourage multidisciplinary collaboration by dispersed groups, [e.g. 15].


[1] Harris GL, Computer models, laboratory simulators, and test ranges: meeting the challenge of estimating tactical force effectiveness in the 1980’s, US Army Command and General Staff College, May 1979.

[2] Trevisani DA & Sisti AF, Air Force hierarchy of models: a look inside the great pyramid, Proc. SPIE 4026, Enabling Technology for Simulation Science IV, 23 June 2000.

[3] Patterson EA, Taylor RJ & Bankhead M, A framework for an integrated nuclear digital environment, Progress in Nuclear Energy, 87:97-103, 2016.

[4] Patterson EA, Purdie S, Taylor RJ & Waldon C, An integrated digital framework for the design, build and operation of fusion power plants, Royal Society Open Science, 6(10):181847, 2019.

[5] Verity R, Okell LC, Dorigatti I, Winskill P, Whittaker C, Imai N, Cuomo-Dannenburg G, Thompson H, Walker PGT, Fu H, Dighe A, Griffin JT, Baguelin M, Bhatia S, Boonyasiri A, Cori A, Cucunubá Z, FitzJohn R, Gaythorpe K, Green W, Hamlet A, Hinsley W, Laydon D, Nedjati-Gilani G, Riley S, van Elsland S, Volz E, Wang H, Wang Y, Xi X, Donnelly CA, Ghani AC, Ferguson NM, Estimates of the severity of coronavirus disease 2019: a model-based analysis., Lancet Infectious Diseases, 2020.

[6] Coglitore D, Edwardson SP, Macko P, Patterson EA, Whelan MP, Transition from fractional to classical Stokes-Einstein behaviour in simple fluids, Royal Society Open Science, 4:170507, 2017.

[7] Giorgi F, Coglitore D, Curran JM, Gilliland D, Macko P, Whelan M, Worth A & Patterson EA, The influence of inter-particle forces on diffusion at the nanoscale, Scientific Reports, 9:12689, 2019.

[8] Gilbert P-A, Kamen A, Bernier A & Garner A, A simple macroscopic model for the diffusion and adsorption kinetics of r-Adenovirus, Biotechnology & Bioengineering, 98(1):239-251,2007.

[9] Holcman D & Schuss Z, Modeling the early steps of viral infection in cells, Chapter 9 in Stochastic Narrow Escape in Molecular and Cellular Biology, New York: Springer Science+Business Media, 2015.

[10] Braine V, Vimond M & Kervrann C, An overview of diffusion models for intracellular dynamics analysis, Briefings in Bioinformatics, Oxford University Press, pp.1-15, 2019.

[11] Holcman D & Schuss Z, Time scale of diffusion in molecular and cellular biology, J. Physics A: Mathematical and Theoretical, 47:173001, 2014.

[12] Bocharov G, Chereshnev V, Gainov I, Bazhun S, Bachmetyev B, Argilaguet J, Martinez J & Meyerhans A, Human immunodeficiency virus infection: from biological observations to mechanistic mathematical modelling, Math. Model. Nat. Phenom., 7(5):78-104, 2012.

[13] Bocharov G, Meyerhans A, Bessonov N, Trofimchuk S & Volpert V, Spatiotemporal dynamics of virus infection spreading in tissues, PLOS One, 11(12):e)168576, 2016.

[14] Bouchnita A, Bocharov G, Meyerhans A & Volpert V, Towards a multiscale model of acute HIV infection, Computation, 5(6):5010006, 2017.

[15] Sego TJ, Aponte-Serrano JO, Ferrari-Gianlupi J, Heaps S, Quardokus EM & Glazier JA, A modular framework for multiscale spatial modeling of viral infection and immune respons in epithelial tissue, bioRxiv. 2020.

Size matters

Most of us have a sub-conscious understanding of the forces that control the interaction of objects in the size scale in which we exist, i.e. from millimetres through to metres.  In this size scale gravitational and inertial forces dominate the interactions of bodies.  However, at the size scale that we cannot see, even when we use an optical microscope, the forces that the dominate the behaviour of objects interacting with one another are different.  There was a hint of this change in behaviour observed in our studies of the diffusion of nanoparticles [see ‘Slow moving nanoparticles‘ on December 13th, 2017], when we found that the movement of nanoparticles less than 100 nanometres in diameter was independent of their size.  Last month we published another article in one of the Nature journals, Scientific Reports, on ‘The influence of inter-particle forces on diffusion at the nanoscale‘, in which we have demonstrated by experiment that Van der Waals forces and electrostatic forces are the dominant forces at the nanoscale.  These forces control the diffusion of nanoparticles as well as surface adhesion, friction and colloid stability.  This finding is significant because the ionic strength of the medium in which the particles are moving will influence the strength of these forces and hence the behaviour of the nanopartices.  Since biological fluids contain ions, this will be important in understanding and predicting the behaviour of nanoparticles in biological applications where they might be used for drug delivery, or have a toxicological impact, depending on their composition.

Van der Waals forces are weak attractive forces between uncharged molecules that are distance dependent.  They are named after a Dutch physicist, Johannes Diderik van der Waals (1837-1923).  Electrostatic forces occur between charged particles or molecules and are usually repulsive with the result that van der Waals and electrostatic forces can balance each other, or not depending on the circumstances.


Giorgi F, Coglitore D, Curran JM, Gilliland D, Macko P, Whelan M, Worth A & Patterson EA, The influence of inter-particle forces on diffusion at the nanoscale, Scientific Reports, 9:12689, 2019.

Coglitore D, Edwardson SP, Macko P, Patterson EA, Whelan MP, Transition from fractional to classical Stokes-Einstein behaviour in simple fluids, Royal Society Open Science, 4:170507, 2017. doi: .

Patterson EA & Whelan MP, Tracking nanoparticles in an optical microscope using caustics. Nanotechnology, 19 (10): 105502, 2009.

Image: from Giorgi et al 2019, figure 1 showing a photograph of a caustic (top) generated by a 50 nm gold nanoparticle in water taken with the optical microscope adjusted for Kohler illumination and closing the condenser field aperture to its minimum following method of Patterson and Whelan with its 2d random walk over a period of 3 seconds superimposed and a plot of the same walk (bottom).

Slow moving nanoparticles

Random track of a nanoparticle superimposed on its image generated in the microscope using a pin-hole and narrowband filter.

A couple of weeks ago I bragged about research from my group being included in a press release from the Royal Society [see post entitled ‘Press Release!‘ on November 15th, 2017].  I hate to be boring but it’s happened again.  Some research that we have been performing with the European Union’s Joint Research Centre in Ispra [see my post entitled ‘Toxic nanoparticles‘ on November 13th, 2013] has been published this morning by the Royal Society Open Science.

Our experimental measurements of the free motion of small nanoparticles in a fluid have shown that they move slower than expected.  At low concentrations, unexpectedly large groups of molecules in the form of nanoparticles up to 150-300nm in diameter behave more like an individual molecule than a particle.  Our experiments support predictions from computer simulations by other researchers, which suggest that at low concentrations the motion of small nanoparticles in a fluid might be dominated by van der Waals forces rather the thermal motion of the surrounding molecules.  At the nanoscale there is still much that we do not understand and so these findings will have potential implications for predicting nanoparticle transport, for instance in drug delivery [e.g., via the nasal passage to the central nervous system], and for understanding enhanced heat transfer in nanofluids, which is important in designing systems such as cooling for electronics, solar collectors and nuclear reactors.

Our article’s title is ‘Transition from fractional to classical Stokes-Einstein behaviour in simple fluids‘ which does not reveal much unless you are familiar with the behaviour of particles and molecules.  So, here’s a quick explanation: Robert Brown gave his name to the motion of particles suspended in a fluid after reporting the random motion or diffusion of pollen particles in water in 1828.  In 1906, Einstein postulated that the motion of a suspended particle is generated by the thermal motion of the surrounding fluid molecules.  While Stokes law relates the drag force on the particle to its size and fluid viscosity.  Hence, the Brownian motion of a particle can be described by the combined Stokes-Einstein relationship.  However, at the molecular scale, the motion of individual molecules in a fluid is dominated by van der Waals forces, which results in the size of the molecule being unimportant and the diffusion of the molecule being inversely proportional to a fractional power of the fluid viscosity; hence the term fractional Stokes-Einstein behaviour.  Nanoparticles that approach the size of large molecules are not visible in an optical microscope and so we have tracked them using a special technique based on imaging their shadow [see my post ‘Seeing the invisible‘ on October 29th, 2014].


Coglitore D, Edwardson SP, Macko P, Patterson EA, Whelan MP, Transition from fractional to classical Stokes-Einstein behaviour in simple fluids, Royal Society Open Science, 4:170507, 2017. doi: