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Open Access Publications from the University of California

Scholarly Works (6 results)

  • Article
  • Peer Reviewed
UC Irvine Previously Published Works (2019)
    Cover page: Confederated modular differential equation APIs for accelerated algorithm development and benchmarking
    • Article
    • Peer Reviewed
    UC Irvine Previously Published Works (2017)
    Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.
      Cover page: ADAPTIVE METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS VIA NATURAL EMBEDDINGS AND REJECTION SAMPLING WITH MEMORY.
      • Article
      • Peer Reviewed
      UC Irvine Previously Published Works (2017)
      Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.
        Cover page: Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory
        • Thesis
        • Peer Reviewed
        UC Irvine Electronic Theses and Dissertations (2018)

        Stochastic models of biochemical interactions elucidate essential properties of the network which are not accessible to deterministic modeling. In this thesis it is described how a network motif, the proportional-reversibility interaction with active intermediate states, gives rise to the ability for the variance of biochemical signals to be controlled without changing the mean, a property designated as mean-independent noise control (MINC). This noise control is demonstrated to be essential for macro-scale biological processes via spatial models of the zebrafish hindbrain boundary sharpening. Additionally, the ability to deduce noise origin from the aggregate noise properties is shown.

        However, these large-scale stochastic models of developmental processes required significant advances in the methodology and tooling for solving stochastic differential equations. Two improvements to stochastic integration methods, an efficient method for time stepping adaptivity on high order stochastic Runge-Kutta methods termed Rejection Sampling with Memory (RSwM) and optimal-stability stochastic Runge-Kutta methods, are combined to give over 1000 times speedups on biological models over previously used methodologies. In addition, a new software for solving differential equations in the Julia programming language is detailed. Its unique features for handling complex biological models, along with its high performance (routinely benchmarking as faster than classic C++ and Fortran integrators of similar implementations) and new methods, give rise to an accessible tool for simulation of large-scale stochastic biological models.

          Cover page: Simulation and Control of Biological StochasticityCreative Commons 'BY' version 4.0 license
          • Article
          • Peer Reviewed
          UC Irvine Previously Published Works (2018)
          Stochasticity affects accurate signal detection and robust generation of correct cell fates. Although many known regulatory mechanisms may reduce fluctuations in signals, most simultaneously influence their mean dynamics, leading to unfaithful cell fates. Through analysis and computation, we demonstrate that a reversible signaling mechanism acting through intermediate states can reduce noise while maintaining the mean. This mean-independent noise control (MINC) mechanism is investigated in the context of an intracellular binding protein that regulates retinoic acid (RA) signaling during zebrafish hindbrain development. By comparing our models with experimental data, we find that the MINC mechanism allows for sharp boundaries of gene expression without sacrificing boundary accuracy. In addition, this MINC mechanism can modulate noise to levels that we show are beneficial to spatial patterning through noise-induced cell fate switching. These results reveal a design principle that may be important for noise regulation in many systems that control cell fate determination.
            Cover page: Mean-Independent Noise Control of Cell Fates via Intermediate StatesCreative Commons 'BY' version 4.0 license
            • Article
            • Peer Reviewed
            UC Irvine Previously Published Works (2016)
            Morphogen gradients induce sharply defined domains of gene expression in a concentration-dependent manner, yet how cells interpret these signals in the face of spatial and temporal noise remains unclear. Using fluorescence lifetime imaging microscopy (FLIM) and phasor analysis to measure endogenous retinoic acid (RA) directly in vivo, we have investigated the amplitude of noise in RA signaling, and how modulation of this noise affects patterning of hindbrain segments (rhombomeres) in the zebrafish embryo. We demonstrate that RA forms a noisy gradient during critical stages of hindbrain patterning and that cells use distinct intracellular binding proteins to attenuate noise in RA levels. Increasing noise disrupts sharpening of rhombomere boundaries and proper patterning of the hindbrain. These findings reveal novel cellular mechanisms of noise regulation, which are likely to play important roles in other aspects of physiology and disease.
              Cover page: Noise modulation in retinoic acid signaling sharpens segmental boundaries of gene expression in the embryonic zebrafish hindbrainCreative Commons 'BY' version 4.0 license
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