Quantifying the Waddington landscape and biological paths for development and differentiation

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144

The brain map project aims to map out the neuron connections of the human brain. Even with all of the wirings mapped out, the global and physical understandings of the function and behavior are still challenging. Hopfield quantified the learning and memory process of symmetrically connected neural networks globally through equilibrium energy. The energy basins of attractions represent memories, and the memory retrieval dynamics is determined by the energy gradient. However, the realistic neural networks are asymmetrically connected, and oscillations cannot emerge from symmetric neural networks. Here, we developed a nonequilibrium landscape-flux theory for realistic asymmetrically connected neural networks. We uncovered the underlying potential landscape and the associated Lyapunov function for quantifying the global stability and function. We found the dynamics and oscillations in human brains responsible for cognitive processes and physiological rhythm regulations are determined not only by the landscape gradient but also by the flux. We found that the flux is closely related to the degrees of the asymmetric connections in neural networks and is the origin of the neural oscillations. The neural oscillation landscape shows a closed-ring attractor topology. The landscape gradient attracts the network down to the ring. The flux is responsible for coherent oscillations on the ring. We suggest the flux may provide the driving force for associations among memories. We applied our theory to rapid-eye movement sleep cycle. We identified the key regulation factors for function through global sensitivity analysis of landscape topography against wirings, which are in good agreements with experiments.neural circuits | free energy | entropy production | nonequilibrium thermodynamics A grand goal of biology is to understand the function of the human brain. The brain is a complex dynamical system (1-6). The individual neurons can develop action potentials and connect with each other through synapses to form the neural circuits. The neural circuits of the brain perpetually generate complex patterns of activity that have been shown to be related with special biological functions, such as learning, long-term associative memory, working memory, olfaction, decision making and thinking (7-9), etc. Many models have been proposed for understanding how neural circuits generate different patterns of activity. HodgkinHuxley model gives a quantitative description of a single neuronal behavior based on the voltage-clamp measurements of the voltage (4). However, various vital functions are carried out by the circuit rather than individual neurons. It is at present still challenging to explore the underlying global natures of the large neural networks built from individual neurons.Hopfield developed a model (5, 6) that makes it possible to explore the global natures of the large neural networks without losing the information of essential biological functions. For symmetric neural circuits, an energy landscape can be constructed that decreas...

Section: Resultsmentioning

confidence: 99%

Section: Significancementioning

confidence: 99%

Section: Significancementioning

confidence: 99%

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Nonequilibrium landscape theory of neural networks

Yan

Zhao

et al. 2013

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144

“…The probabilistic landscape approach may provide a possible answer, as the significance of each state can thereby be distinguished according to its associated weight. Functional states may correspond to higher probability of occurrence and occupy lower potential valleys (15)(16)(17)(18)(19)(20). The state space of gene regulatory networks includes states with different gene expression patterns in the cell, which further determine different cellular functions.…”

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Landscape and flux reveal a new global view and physical quantification of mammalian cell cycle

Wang

2014

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137

202

Cell cycles, essential for biological function, have been investigated extensively. However, enabling a global understanding and defining a physical quantification of the stability and function of the cell cycle remains challenging. Based upon a mammalian cell cycle gene network, we uncovered the underlying Mexican hat landscape of the cell cycle. We found the emergence of three local basins of attraction and two major potential barriers along the cell cycle trajectory. The three local basins of attraction characterize the G1, S/G2, and M phases. The barriers characterize the G1 and S/G2 checkpoints, respectively, of the cell cycle, thus providing an explanation of the checkpoint mechanism for the cell cycle from the physical perspective. We found that the progression of a cell cycle is determined by two driving forces: curl flux for acceleration and potential barriers for deceleration along the cycle path. Therefore, the cell cycle can be promoted (suppressed), either by enhancing (suppressing) the flux (representing the energy input) or by lowering (increasing) the barrier along the cell cycle path. We found that both the entropy production rate and energy per cell cycle increase as the growth factor increases. This reflects that cell growth and division are driven by energy or nutrition supply. More energy input increases flux and decreases barrier along the cell cycle path, leading to faster oscillations. We also identified certain key genes and regulations for stability and progression of the cell cycle. Some of these findings were evidenced from experiments whereas others lead to predictions and potential anticancer strategies.cell cycle phases | cell cycle checkpoints | landscape | flux T he cell cycle is a series of events that take place in a cell leading to its replication and division. Studying the cell cycle process is essential for understanding cell growth, proliferation, development, and death (1-4). Cell cycle comprises several distinct phases: G1 phase (resting), S phase (synthesis), G2 phase (interphase), and M phase (mitosis). Activation of each phase is dependent on the proper progression and completion of the previous one, which can be monitored by cell cycle checkpoints. It is now believed that all proliferation, differentiation, and cell death processes are controlled by the underlying gene regulatory networks (5), which often involve many complex feedback loops (6). The complexity of the large regulatory networks for the cell cycle makes it difficult to understand the global natures and connections between the underlying network and the cell cycle process. Furthermore, in the cell, the intrinsic fluctuations from the finite number of molecules and extrinsic fluctuations from dynamical and inhomogeneous environments (7, 8) coexist. Therefore, stochastic approaches are often required to explore the nature of the underlying network of chemical reaction soups (9-14). The challenge is how to understand the global picture and physical principles for the stability and function of the cell cycle ...

“…The epigenetic landscape, in which different phenotypic states arise, despite the fact that cells have identical gene sequences, has been discussed widely, mostly in the context of developmental biology and stem cell fate decisions (14)(15)(16)(17); it has been a useful framework, even as a qualitative description. Several studies have focused on developing the landscape concept quantitatively, where the connection between the landscape and an energy-like potential is made explicit and rigorous, but so far, these efforts have been limited to either theoretical treatments (18,19) or completely specified systems of chemical reactions (20,21) and have not been tested with experimental data.…”

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Predicting rates of cell state change caused by stochastic fluctuations using a data-driven landscape model

Sisan

Halter

Hubbard

et al. 2012

We develop a potential landscape approach to quantitatively describe experimental data from a fibroblast cell line that exhibits a wide range of GFP expression levels under the control of the promoter for tenascin-C. Time-lapse live-cell microscopy provides data about short-term fluctuations in promoter activity, and flow cytometry measurements provide data about the long-term kinetics, because isolated subpopulations of cells relax from a relatively narrow distribution of GFP expression back to the original broad distribution of responses. The landscape is obtained from the steady state distribution of GFP expression and connected to a potentiallike function using a stochastic differential equation description (Langevin/Fokker-Planck). The range of cell states is constrained by a force that is proportional to the gradient of the potential, and biochemical noise causes movement of cells within the landscape. Analyzing the mean square displacement of GFP intensity changes in live cells indicates that these fluctuations are described by a single diffusion constant in log GFP space. This finding allows application of the Kramers' model to calculate rates of switching between two attractor states and enables an accurate simulation of the dynamics of relaxation back to the steady state with no adjustable parameters. With this approach, it is possible to use the steady state distribution of phenotypes and a quantitative description of the shortterm fluctuations in individual cells to accurately predict the rates at which different phenotypes will arise from an isolated subpopulation of cells.population distribution | dynamical systems | stochastic protein expression | biological noise G enetically identical cells do not respond identically when exposed to nominally identical environmental conditions. Such nongenetic phenotypic variability has been widely observed in bacteria (1, 2), yeast (3), and mammalian cells (4-8). Population heterogeneity is thought to result from the inherently stochastic nature of intracellular events, which are subject to statistical fluctuations caused by small copy numbers of the constituent molecules, such as transcription factors (9). Many investigations into the origins and effects of stochastic gene expression have used engineered organisms and stochastic gene network models to determine the sources and magnitude of variability (10-12). These fluctuations, although causing continual change at the single-cell level, can lead to stable distributions of phenotypes within a population.The idea of a stable distribution of states in the presence of random fluctuations is reminiscent of statistical physics, where randomness results from thermal fluctuations and the stable distribution of states reflects a potential energy function. The popular concept of the epigenetic landscape suggested in the work by Waddington (13) (i.e., a surface of branching valleys and ridges on which cells explore phenotypic states) can be thought of as a series of potential energy functions. The epigenetic landscape,...