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Mathematical Models of Cell Invasion Track Spatial Position And Newsjournal of the Society for Industrial and Applied Mathematics sinews.siam.org Volume 52/ Issue 1 January/February 2019 Life Sciences Special Issue Understanding Knowledge In this special issue, explore the application of mathematics and computational science to numerous topics in the life sciences. Networks in the Brain By Matthew R. Francis be tested experimentally. Fundamental research on brain networks can potentially ne strength of the human mind is facilitate the understanding and treatment O its ability to find patterns and draw of conditions as diverse as schizophrenia connections between disparate concepts, and Parkinson’s disease. a trait that often enables science, poetry, In mathematical terms, a network is a visual art, and a myriad of other human type of graph: a set of points connected by endeavors. In a more concrete sense, the lines. The particular model for knowledge brain assembles acquired knowledge and networks is based on the assumption that links pieces of information into a network. humans essentially experience phenomena Knowledge networks also seem to have as discrete events or concepts arranged a physical aspect in the form of intercon- sequentially in time. Each of these is mod- nected neuron pathways in the brain. eled as a node in a graph, with lines called During her invited address at the 2018 “edges” linking them together. The edges SIAM Annual Meeting, held in Portland represent possible transitions between the Ore., last July, Danielle Bassett of the events or concepts; a particular graph thus University of Pennsylvania illustrated how describes how knowledge networks inter- brains construct knowledge networks. Citing connect ideas. The first big question for this early 20th century progressive educational model is whether optimal pathways of con- reformer John Dewey, she explained that the nectivity that maximize learning exist. goal of a talk—and learning in general—is to map concepts from the speaker/teacher’s Hot Thoughts and Modular Thinking Figure 2. Scout bees apply a head-butt stop signal to dancers promoting alterna- mind to those of his or her listeners. When To address this problem, Bassett and tive nest sites. Image courtesy of James Nieh. the presenter is successful, the audience her collaborators constructed a knowledge In an article titled “Multi-Agent Decision-Making in Design” on gains new conceptual networks. network consisting of nodes that represent page 6, Paul Davis reports on Naomi Leonard’s use of nonlinear More generally, Bassett explored how random stimuli. Each node was connected dynamics to design multi-agent control systems, taking cues from humans acquire knowledge networks, to the same number of other nodes—cre- collective animal behavior. whether that process can be modeled math- ematically, and how such models may See Knowledge Networks on page 4 Mathematical Models of Cell Invasion Track Spatial Position and Progression through the Cell Cycle AID Postage P S. rmit 360 No ity and proliferation are often based on the In 2008, the introduction of a novel exper- U. Bellmawr, NJ Bellmawr, By Matthew J. Simpson Nonprofit Org Nonprofit Pe Fisher-Kolmogorov equation [3, 4, 6]. We imental methodology called the Fluorescent patial spreading of cell populations is can write this equation in one dimension as Ubiquitination-based Cell Cycle Indicator S crucial for embryonic development, (FUCCI) [5] enabled real-time analysis of tissue repair, and malignant invasion. 2 the cell cycle. FUCCI provides spatial and ∂s ∂ s s During embryonic development, various = D +−ls 1 , (1) temporal information about the cell cycle’s ∂t 2 K types of tissues and organs with different ∂x progression through fluorescent probes that functions evolve from an initially homoge- glow red when the cell is in the first gap neous population of cells in the very early where sx(,t)> 0 represents cell density, (G1) phase of the cycle and green when embryo. Normal development requires that D> 0 is cell diffusivity, l > 0 signifies the in the synthesis (S), second gap (G2), or these structures arise with the correct cell cell proliferation rate, and K > 0 denotes mitotic/cell division (M) phases [5, 9]. types in the appropriate three-dimensional the environment’s carrying capacity den- Figure 1a depicts still images from a time- geometry, a process that necessitates simul- sity. This simple mathematical model sup- lapse movie of a FUCCI-transduced mela- taneous migration and growth of cell popu- poses that cells undergo an unbiased ran- noma cell that illustrate these colour changes lations. Abnormal cell migration during dom walk and proliferate logistically so that during the cell cycle. A visual comparison of adulthood is associated with pathological some initial density will evolve—through the phase contrast image of a C8161 melano- conditions like cancer invasion and metas- a combination of diffusion and carrying- ma spheroid (see Figure 1b) and a correspond- tasis. While motility of individual cells is capacity limited growth—to produce mov- ing FUCCI-transduced spheroid (see Figure crucial for population-level spreading, cell ing fronts. Such fronts lead to the coloni- 1c) confirms that the FUCCI system reveals a proliferation plays an important and some- zation of initially-vacant regions that will great deal about the relationship between spa- times overlooked role in driving spatial eventually reach carrying capacity density tial location and the cell cycle. For example, expansion of cell populations [4]. K. For certain initial and boundary condi- freely cycling cells—indicated by the mixture Previous researchers have employed both tions, (1) can yield travelling wave solutions of red and green fluorescence—are visible on continuum differential-equation models and [4]. This type of modelling framework has the FUCCI spheroid’s outer shell. Upon look- discrete models that use lattice-based random successfully described a range of phenom- ing deeper into the spheroid, we observe cells walks, lattice-free random walks, and cellu- ena related to collective cell spreading and that are largely arrested in the G1 (red) phase, lar automata to mathematically model spatial cell invasion, from the study of simple in presumably due to a lack of nutrients such as spreading in cell populations. Continuum vitro scratch assays to in vivo malignant mathematical models of combined cell motil- brain tumour progression. See Cell Cycle on page 2 and APPLIED MATHEMATICS INDUSTRIAL Figure 1. Experimental images and conceptual model for the cell cycle as revealed by FUCCI labelling. 1a. Time-lapse images depicting the progression of three generations of FUCCI-transduced C8161 melanoma cells through the cell cycle. The numbers in each subfigure indicate for the time in hours. 1b. Phase contrast image of a C8161 melanoma spheroid with an approximate diameter of 0.5 millimeters. 1c. Image of a slice of a C8161 melanoma spheroid of similar size to 1b where the cells are transduced with FUCCI. 1d. Conceptual model of FUCCI technology. 1e. Experimental data presenting quantification of the rate of red-to-green and green-to-red transitions for the C8161 melanoma cell line. 1a courtesy of [2], 1b and 1c courtesy of [1], 1d and 1e courtesy of [8]. SOCIETY 3600 Market Street, 6th Floor Philadelphia, PA 19104-2688 USA 2 • January/February 2019 SIAM NEWS Cell Cycle Continued from page 1 Volume 52/ Issue 1/ January/February 2019 oxygen. The spheroid’s centre completely 5 Politics, Policy, lacks fluorescence; this indicates the presence Science, and Risk of a necrotic core. Unsurprisingly, FUCCI Paul Davis reviews Michael technology has revolutionized the analysis Lewis’s The Fifth Risk, which of cell proliferation and has applications in describes politics’ impact experimental models of cancer, stem cell, and on federal science, specifi- developmental biology [9]. cally at the Departments of We have recently sought to develop and Energy, Agriculture, and validate continuum models of cell invasion Commerce. Lewis interviews for use with FUCCI technology [7, 8]. We experts in the field who employ FUCCI-transduced C8161 mela- describe potential risks in noma cells and perform a series of scratch the electric grid and radioac- assays to observe cell migration; these tive waste storage, as well as disappearing data on climate assays involve creating a “scratch” in a cell change and natural disasters. monolayer and capturing images close to the scratch at regular intervals. Next we attempt 6 Forward-Looking Panel to quantitatively describe these experiments at MPE18 Examines the with a new continuum model — a generali- Intersection of Mathematics sation of the Fisher-Kolmogorov equation. and Environmental Science Figure 2a shows a population of FUCCI- How do mathematicians con- transduced C8161 melanoma cells on a two- duct successful interdisciplinary dimensional substrate following the creation work and foster collaborations of a 0.5 millimeter artificial scratch in the within various areas of climate population. Figure 2d displays the same science? What are some of the experiment after 18 hours; now we see that future’s biggest challenges? the cells have moved into the initially-vacant Climate scientists pondered these and other questions dur- region while continuing to progress through ing a forward-looking panel at the cell cycle. To model these experiments, we last year’s SIAM Conference suppose that the cell population—with total on Mathematics of Planet density sx(,t)—consists of two subpopula- Figure 2. Comparison of experiential images, experiential data, and solutions of the new Earth. Hans Engler and Emil tions, yielding sx(,tv)(=+rgxt,) vx(,t). mathematical model for scratch assays with FUCCI-transduced melanoma cells. 2a-2c. Initial Constantinescu offer a recap. Here vx(,t) is the red cell density in the G1 condition for experimental modelling comparison th= 0. 2d-2f. Comparison of experimental r observation and modelling prediction th= 18 . 2a and 2d display images of a scratch assay phase and vx(,t) is the green cell density in g with C8161 melanoma cells where the scratch’s initial width is approximately 0.5 millimeters.
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