Non-Markovian models of intracellular transport in a heterogeneous environment

Lead Research Organisation: University of Manchester
Department Name: Mathematics


Transport of all kinds of components within the cell - from vesicles along the cytoskeleton through to transcription factors along DNA - is fundamental to cell function and health. Neurons are particularly susceptible to small changes in vesicle transport, which underlie motor neuron disease and may also contribute to neuronal degeneration seen in Alzheimer's disease and during ageing. Despite experimental facts that intracellular transport is heterogeneous and non-Markovian with subdiffusive and superdiffusive regimes most mathematical models for vesicles trafficking are Markovian and homogeneous.

The main challenge for our Manchester interdisciplinary team is to obtain new non-Markovian models of heterogeneous intracellular transport supported by experiments. These models will provide a tool set for analysing transport processes in a much more realistic way, opening the way for greatly improved analysis and ultimately understanding of these highly complex cellular behaviours. This will allow other researchers to formulate and test new hypotheses. In the long term, therefore, non-Markovian models have the potential to lead to insight into neurological diseases, ageing and other processes that involve intracellular transport such as bacterial and viral infection. Such knowledge will be important for developing new treatments.

Our project combines three different approaches: mathematical modelling, numerical modelling and experimental validation, which complement each other. This strategy will provide multidisciplinary study of the intracellular transport problem and ensure maximum impact across and within several disciplines. Our project will allow applied mathematicians (PI and RA), cell biologists and biophysicists (Co-Is and Project Partners) to collaborate thus making significant advances in intracellular transport research and support a cross-disciplinary dissemination.

Planned Impact

The main impact of our project will be in the international development of cell research. We are going to achieve this by setting up a multidisciplinary team of researchers from the University of Manchester, Curie Institute in Paris, Inria Serpico team in Rennes, LIRIS laboratory in Lyon and the University of California at Merced. Our team will consist of applied mathematicians (PI, RA), experimental cell biologists and computational biophysicists (Co-Is and Project Partners). This project has a potential social impact through enhancing of quality of life and health because it can give important insights into intracellular transport which is of fundamental importance to cell biology. Malfunctions in protein and vesicles trafficking inside living cells are major contributory factors in a number of diseases. Defects of intracellular transport contribute to Huntington's disease and Altzheimer's. A deeper understanding of intracellular transport can lead to more effective treatment strategies of cancers, viral infection and bacterial infections. As our Project Partners stated, our models will be used in the Curie Institute in Paris which is the leading world centre specializing on cell biology and the hospital treatment of cancer.

A major strength of our proposal is that it allows direct testing of our new mathematical models on data sets already obtained by the Co-Is. We envisage further experiments (Co-Is and Project Partners) to validate our non-Markovian models. This strengthens the position of the PI and Co-Is, but also improve the UK's knowledge base in non-Markovian intracellular transport. Our new non-Markovian heterogeneous transport models will likely have a broad range of potential applications in many fields such as agriculture, bioengineering and pharmaceutics in the longer term. For example, the transport of substances in random porous media involves a combination of subdiffusion and superdiffusion. Therefore, the pharmaceutical industry might benefit from our new techniques to improve drug delivery.

Together with our Project Partners from France and USA, we intend to organize a international multidisciplinary workshop on stochastic modelling of intracellular transport and to invite key experts in the field. The purposes of this meeting are: (1) to promote the exchange ideas between UK applied mathematicians and USA and European cell biologists; (2) identify potential areas of further collaboration on important aspects of stochastic intracellular transport; (3) to provide training in non-Markovian transport and fractional partial differential equations for PhD students and biologists. Running the workshop will allow to disseminate our results on new transport models.

We are also committed to engaging with non-specialists about the need for high quality basic science and the specific goals and outcomes of this research project. We will engage school science students by writing an article about vesicle movement and how mathematical modelling is crucial for understanding this process, for publication in the Biological Sciences review, which is sent to secondary schools throughout the country. We will take part in Faculty and University activities that showcase our science to the public and schools, such as Faculty open days, the Schools open day, and National Science week, which happen annually. The aim will be to engage with students by expanding their knowledge through describing exciting new research discoveries.


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Description We propose a model of superdiffusive Levy walk as an emergent nonlinear phenomenon in systems of ´interacting individuals. Our interacting run-and-tumble model leads to the superdiffusive growth of the mean-squared displacement and the power-law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a cooperative effect without using the standard assumption of the power-law distribution of run distances from the inception.

We demonstrate the phenomenon of cumulative inertia in intracellular transport involving multiple motor proteins in human epithelial cells by measuring the empirical survival probability of cargoes on the microtubule and their detachment rates. We found the longer a cargo moves along a microtubule, the less likely it detaches from it. As a result, the movement of cargoes is non-Markovian and involves a memory. We observe memory effects on the scale of up to 2 s. We provide a theoretical link between the measured detachment rate and the superdiffusive Lévy-walk-like cargo movement.

We examined experimental trajectories of organelles inside a living cell and propose a mathematical model that describes the previously reported transition from sub-diffusive to super-diffusive motion. In order to explain super-diffusive behaviour at long times, we introduce non-Markovian detachment kinetics of the cargo: the rate of detachment is inversely proportional to the time since the last attachment. Recently, we observed the non-Markovian detachment rate experimentally in eukaryotic cells. We suggested different scenarios of how this effective non-Markovian detachment rate could arise. The non-Markovian model is successful in simultaneously describing the time averaged variance (the time averaged mean squared displacement corrected for directed motion), the mean first passage time of trajectories and the multiple peaks observed in the distributions of cargo velocities. We argue that non-Markovian kinetics could be biologically beneficial compared to the Markovian kinetics commonly used for modelling, by increasing the average distance the cargoes travel when a microtubule is blocked by other filaments. In turn, sub-diffusion allows cargoes to reach neighbouring filaments with higher probability, which promotes active motion along the microtubules.

We find the asymptotic representation of the solution of the variable-order fractional diffusion equation, which remains unsolved since it was proposed by Chechkin, Gorenflo, and Sokolov [J. Phys. A, 38, L679 (2005)]. We identify a new advection term that causes ultraslow spatial aggregation of subdiffusive particles due to dominance over the standard advection and diffusion terms in the long-time limit. This uncovers the anomalous mechanism by which nonuniform distributions can occur.

We developed a deep learning feedforward neural network trained on fractional Brownian motion, providing a novel, accurate and efficient method for resolving heterogeneous behavior of intracellular transport in space and time. The neural network requires significantly fewer data points compared to established methods. This enables robust estimation of Hurst exponents for very short time-series data, making possible direct, dynamic segmentation and analysis of experimental tracks of rapidly moving cellular structures such as endosomes and lysosomes. By using this analysis, fractional Brownian motion with a stochastic Hurst exponent was used to interpret, for the first time, anomalous intracellular dynamics, revealing unexpected differences in behavior between closely related endocytic organelles.

We introduce a persistent random walk model with finite velocity and self-reinforcing directionality, which explains how exponentially distributed runs self-organize into truncated Lévy walks observed in active intracellular transport by Chen et al. [Nature Mater., 14, 589 (2015)]. We derive the nonhomogeneous in space and time, hyperbolic partial differential equation for the probability density function (PDF) of particle position. This PDF exhibits a bimodal density (aggregation phenomena) in the superdiffusive regime, which is not observed in classical linear hyperbolic and Lévy walk models. We find the exact solutions for the first and second moments and criteria for the transition to superdiffusion.
Exploitation Route Our models lead to new experiments on cargo movement inside living cells.
Our deep learning feedforward neural network trained on fractional Brownian motion provides a novel, accurate and efficient method for resolving heterogeneous behavior of intracellular transport in space and time.
Sectors Healthcare

Description BioProton: Biologically relevant dose for Proton Therapy Planning
Amount £1,394,633 (GBP)
Funding ID EP/S024344/1 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 04/2019 
End 03/2024
Description Modelling anomalous transport of nanoparticles and DNA repair to improve radiotherapy
Amount £702,575 (GBP)
Funding ID EP/V008641/1 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 01/2021 
End 12/2023