Understanding Cell Motion and Electrotaxis with Computational Methods Blake Cook 15th of February, 2018
Outline 1 Biological context 2 Image analysis 3 Modelling membrane dynamics 4 Discussion
Outline 1 Biological context Chemotaxis Model organism 2 Image analysis 3 Modelling membrane dynamics 4 Discussion
Chemotaxis Movement of a motile cell or organism, or part of one, in a direction corresponding to a gradient of increasing or decreasing concentration of a particular substance. Directed movement of cells tends to be in response to signalling molecules, released by other cells in minuscule amounts E.g. Development of tissue and organs, Immune system cell response to pathogens Do cells respond in a similar way to electrical fields?
Chemotaxis E.g. E. Coli run and tumble motion Figure: E. Coli chemotaxis Cell swims in a direction and randomly change direction after tumbling at random times Direction chosen is biased towards positive nutrient gradients Source: http://biologicalexceptions.blogspot.co.uk/2014/09/
Chemotaxis But not all motile cells have flagella.. (a) White blood cells (b) Amoeba Source: Wikipedia entries, Neutrophil, and Chaos (genus), https://en.wikipedia.org/wiki/file:neutrophils.jpg https://en.wikipedia.org/wiki/file:chaos_carolinense.jpg
Chemotaxis Cellular cytoskeleton Polymer chain complexes inside cells Responsible for cellular shape and motion Segments of membrane attached to ends of the microfibre complex pull or push to move the surface in a particular direction Source: Wikipedia entry Cytoskeleton, https://commons.wikimedia.org/wiki/file:fluorescentcells.jpg
Model organism Dictyostelium discoideum Source: Wikipedia entry on Dictyostelium discoideum, https://en.wikipedia.org/wiki/file:dicty_life_cycle_h01.svg
Model organism Dictyostelium discoideum Eukaryotic organism with well known genetic composition Grow quickly and have limited cell types Has both unicellular and multi cellular stages Used for many studies in cell biology (chemotaxis, cell death, cell sorting, pattern formation,...)
Biological context Image analysis Modelling membrane dynamics Discussion Model organism Initiation of Slime Mold Aggregation Viewed as an Instability (Keller and Segel, 1970) [1] Classic model in mathematical biology describing aggregation Source: J. Cejkova, University of Chemistry and Technology, Prague (2006), https://www.youtube.com/watch?v=7w-wcp7-wew
Outline 1 Biological context 2 Image analysis Watershed transform Tracking cells 3 Modelling membrane dynamics 4 Discussion
Biological context Image analysis Modelling membrane dynamics Data Figure: Typical recording of Dictyostelium cells Discussion
Watershed transform Technique used for image segmentation Plot greyscale image as surface, identify local minima ( catchment basins ) and saddles ( ridge-lines ) Use basins as internal markers Use ridge-lines as segment boundaries Figure: Illustration of watershed method Source: [2] https://uk.mathworks.com/company/newsletters/articles/thewatershed-transform-strategies-for-image-segmentation.htm
Watershed transform Explanation of Watershed in 1D Original image Cell A Cell B External mask Complement Watershed Internal mask Combine
Watershed transform Normalise contrast (1% saturated at high and low intensities) Get external mask (by threshold or edge detection) Get internal mask (local maximums then de-noise) Figure: Image analysis process (1/2)
Watershed transform Complement frame adjusted, impose complement of ext mask and int mask as minimum values Apply watershed transform Overlay on original data Figure: Image analysis process (2/2)
Tracking Figure: Typical input and output from image analysis
Tracking Cells uniquely labelled, writes following data: Centroid Velocity Orientation Area Number of arms Trunk Acc. branch Rej. branch Figure: Arm counting process
Tracking Chemotaxis Control Figure: Spatial results from control and chemotaxis clips
Tracking Electrotaxis Control Figure: Spatial results from control and electrotaxis clips
Tracking Figure: Comparison of cell behaviour across all experiments
Outline 1 Biological context 2 Image analysis 3 Modelling membrane dynamics Membrane model Geometry Forces 4 Discussion
Membrane model Let {r 1,..., r n} be the position of nodes on membrane Connect nodes anticlockwise to form a regular polygon Apply forces on each node and solve ODEs r i+1 r i r i 1 y x Figure: Sketch of off-lattice membrane model
Membrane model Let {r 1,..., r n} be positions of nodes on cell surface and let N i (t) denote the neighbouring nodes of node i at time t. Assume inertial terms are small enough to be inconsequential compared to dissipative terms in equation of motion: η d r i dt = Bi(t) + j N i (t) F ij(t), where η is a drag coefficient, F ij denotes the force on node i from node j and Bi(t) is the sum of any other forces on node i at time t.
Membrane model Equation of motion: η d r i dt = Bi(t) + j N i (t) F ij(t), Discretise ODEs and solve numerically using a forward Euler scheme with timestep t, r i(t + t) = r i(t) + t B i(t) + F ij(t). η j N i (t)
Geometry d max d min Figure: Node geometry (removed nodes in red, new nodes in green) Connect points in anticlockwise direction to form regular polygon, define neighbourhood N i (t) = {i 1, i + 1}. Merge nodes that get too close together, add new nodes between two nodes if they get too far apart.
Forces Tension: neighbouring nodes have linear spring interactions. Conserves surface area of cell. Each connection has a resting spring length New connections assigned spring lengths such that total tension in system is conserved after adding new nodes i 1 i F i,i+1 l rest i + 1 d(i, i + 1) Figure: Tension interactions
Forces Curvature: connections in triplet of nodes will form a natural angle Node triplet will tend towards forming an obtuse angle, counteracting smaller angles and larger angles Volume conservation: cell will resist extension and compression i i + 1 θ i 2 B i θ i i 1 Figure: Curvature responses and volume conservation
Forces Cytoskeleton: system of activators and inhibitors at each node d A dt = f (A, B) + D A 2 A, d B dt = g(a, B) + D B 2 B, D A D B. Outward force proportional to concentration of activator Inward force proportional to concentration of inhibitor
Outline 1 Biological context 2 Image analysis 3 Modelling membrane dynamics 4 Discussion Future work Applications
Future work Image analysis Tune parameters for more datasets Extract cell boundary data Model development Fit parameters to cell boundary data Implement simple cytoskeletal forces Potential for extension to 3D model
Applications Wound closure and electrotaxis Small electric fields at wound sites Hypothesised that cells are directed by electrical fields generated by disruptions in epithelial layer Is this a process that can be manipulated or accelerated? Figure: Generation of electric fields at wound sites Source: [3] M. Zhao, Electrical fields in wound healing - An overriding signal that directs cell migration, 2009
References E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, vol. 26, no. 3, pp. 399 415, 1970. S. Eddins, The watershed transform: strategies for image segmentation, The Mathworks Journal, Matlab, News and Notes, pp. 47 48, 2002. M. Zhao, Electrical fields in wound healing-an overriding signal that directs cell migration, Seminars in Cell and Developmental Biology, vol. 20, no. 6, pp. 674 682, 2009. J. M. Osborne, A. G. Fletcher, J. M. Pitt-Francis, P. K. Maini, and D. J. Gavaghan, Comparing individual-based approaches to modelling the self-organization of multicellular tissues, PLoS Computational Biology, vol. 13, no. 2, pp. 1 34, 2017. D. M. Richards and T. E. Saunders, Spatiotemporal Analysis of Different Mechanisms for Interpreting Morphogen Gradients, Biophysical Journal, vol. 108, no. 8, pp. 2061 2073, 2015. L. Tweedy, B. Meier, J. Stephan, D. Heinrich, and R. G. Endres, Distinct cell shapes determine accurate chemotaxis., Scientific reports, vol. 3, p. 2606, 2013. Thank you for your time!