Statistical Dynamics of Spatial-Order Formation by Communicating Cells

Olimpio, Eduardo P.; Dang, Yiteng; Youk, Hyun

doi:10.1016/j.isci.2018.03.013

Cited by 23 publications

(60 citation statements)

References 54 publications

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“…The aim here is to give a technically precise, but concise reference for the model details. As such, we leave out many steps in motivating the model assumptions -the reader can refer to the main text as well as our earlier work for a detailed motivation of the model assumptions ( [1,2]).…”

Section: Supplemental Informationmentioning

confidence: 99%

“…OF F = 1 for all i and measure all concentrations in units of the OFF secretion rate (which we take to be equal for all genes, unless otherwise specified). In steady state, the concentration of signaling molecule decays with distance to the cell that is secreting it according to [2]…”

Section: Supplemental Informationmentioning

confidence: 99%

“…OF F are effective secretion rates that lump together terms of the underlying reaction-diffusion equation ( [2]), including the diffusion lengths λ (i) . However, we will not take into account this dependence and assume that the secretion rates scale accordingly with the λ (i) such that C…”

Section: Supplemental Informationmentioning

confidence: 99%

“…Next, we characterize how spatially correlated the gene expression levels using introduce an earlier defined "spatial index", extended to a system with multiple genes [1,2]:…”

Section: S13 Population-level Descriptionmentioning

confidence: 99%

“…Note that this algorithm is not guaranteed to converge to I target because at each iteration of the loop, we are not guaranteed to increase or decrease I as required. In particular, if the specified I is outside the range of possible I [2], the algorithm cannot reach the specified value of I. Therefore, we typically set a limit on the maximum number of iterations before we terminate the algorithm.…”

Section: S2 Simulation and Analysis Of The Modelmentioning

confidence: 99%

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Cellular dialogues that enable self-organization of dynamic spatial patterns

Dang

Grundel

Youk

2019

Preprint

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12Cells form spatial patterns by coordinating their gene expressions. How a group of mesoscopic 13 numbers (hundreds-to-thousands) of cells, without pre-defined morphogens and spatial 14 organization, self-organizes spatial patterns remains incompletely understood. Of particular 15 importance are dynamic spatial patterns -such as spiral waves that perpetually move and 16 transmit information over macroscopic length-scales. We developed an open-source, 17 expandable software that can simulate a field of cells communicating with any number of cell-18 secreted molecules in any manner. With it and a theory developed here, we identified all 19 possible "cellular dialogues" -ways of communicating with two diffusing molecules -and core 20 architectures underlying them that enable diverse, self-organized dynamic spatial patterns that 21 we classified. The patterns form despite widely varying cellular response to the molecules, 22 gene-expression noise, and spatial arrangement and motility of cells. Three-stage, "order-23 fluctuate-settle" process forms dynamic spatial patterns: cells form long-lived whirlpools of 24 wavelets that, through chaos-like interactions, settle into a dynamic spatial pattern. These 25 results provide a blueprint to help identify missing regulatory links for observed dynamic-26 pattern formations and in building synthetic tissues. 60 emerge in a mesoscopic population of cells. 62We sought to resolve this shortcoming by developing an open-source software that analyzing these simulations. With the software and analysis algorithms, we sought to reveal 67 quantitative mechanisms by which mesoscopic numbers of cells can use their spatially 68 heterogeneous gene-expression levels as a seed to form spatial patterns. In particular, we 69 focused on dynamic patterns -patterns that constantly change over time without ever stopping 70 such as oscillations and spiral waves [Sgro et al., 2013] -instead of static patterns that remain 71 still after forming ( Figure 1B). Through an exhaustive computational search, we discovered all 72 the ways in which cells can communicate with just two diffusing molecules to form dynamic 73 patterns, including those that have been experimentally observed. We found that just a few 74 ways of communicating, which we refer to as "cellular dialogues", can generate a large palette 75 of complex, dynamic spatial patterns such as chaotic whirlpools of wavelets and travelling 76 waves of various shapes and orientations. Viewing these simulations as exact numerical 77 experiments, we devised an analytical (pen-and-paper) approach that recapitulates the 78 simulations and used it to understand why only certain cellular dialogues can sustain dynamic 79 spatial patterns. As we will show, we discovered that cells can form dynamic spatial patterns 80 through a three-stage, "order-fluctuate-settle" process. Starting from a configuration in which 81 there is no spatial correlation among cells' gene-expression levels, we observed that cells 82 rapidly become more spatially correlated over tim...

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Section: Supplemental Informationmentioning

confidence: 99%

Section: Supplemental Informationmentioning

confidence: 99%

Section: Supplemental Informationmentioning

confidence: 99%

“…Next, we characterize how spatially correlated the gene expression levels using introduce an earlier defined "spatial index", extended to a system with multiple genes [1,2]:…”

Section: S13 Population-level Descriptionmentioning

confidence: 99%

Section: S2 Simulation and Analysis Of The Modelmentioning

confidence: 99%

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Cellular dialogues that enable self-organization of dynamic spatial patterns

Dang

Grundel

Youk

2019

Preprint

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Predictive landscapes hidden beneath biological cellular automata

Koopmans

Youk

2021

J Biol Phys

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To celebrate Hans Frauenfelder’s achievements, we examine energy(-like) “landscapes” for complex living systems. Energy landscapes summarize all possible dynamics of some physical systems. Energy(-like) landscapes can explain some biomolecular processes, including gene expression and, as Frauenfelder showed, protein folding. But energy-like landscapes and existing frameworks like statistical mechanics seem impractical for describing many living systems. Difficulties stem from living systems being high dimensional, nonlinear, and governed by many, tightly coupled constituents that are noisy. The predominant modeling approach is devising differential equations that are tailored to each living system. This ad hoc approach faces the notorious “parameter problem”: models have numerous nonlinear, mathematical functions with unknown parameter values, even for describing just a few intracellular processes. One cannot measure many intracellular parameters or can only measure them as snapshots in time. Another modeling approach uses cellular automata to represent living systems as discrete dynamical systems with binary variables. Quantitative (Hamiltonian-based) rules can dictate cellular automata (e.g., Cellular Potts Model). But numerous biological features, in current practice, are qualitatively described rather than quantitatively (e.g., gene is (highly) expressed or not (highly) expressed). Cellular automata governed by verbal rules are useful representations for living systems and can mitigate the parameter problem. However, they can yield complex dynamics that are difficult to understand because the automata-governing rules are not quantitative and much of the existing mathematical tools and theorems apply to continuous but not discrete dynamical systems. Recent studies found ways to overcome this challenge. These studies either discovered or suggest an existence of predictive “landscapes” whose shapes are described by Lyapunov functions and yield “equations of motion” for a “pseudo-particle.” The pseudo-particle represents the entire cellular lattice and moves on the landscape, thereby giving a low-dimensional representation of the cellular automata dynamics. We outline this promising modeling strategy.

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