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Conceived and designed the experiments: YS AS XS. Performed the experiments: YS. Analyzed the data: YS AS XS. Wrote the paper: YS AS XS.

The authors have declared that no competing interests exist.

Biological systems are often treated as time-invariant by computational models that use fixed parameter values. In this study, we demonstrate that the behavior of the p53-MDM2 gene network in individual cells can be tracked using adaptive filtering algorithms and the resulting time-variant models can approximate experimental measurements more accurately than time-invariant models. Adaptive models with time-variant parameters can help reduce modeling complexity and can more realistically represent biological systems.

In science and engineering, computational models are needed to describe the relationship between input and output data of a system as well as to estimate future outputs based on inputs. One common approach for constructing models from measured input/output data is system identification (SI), which uses computational techniques to build models of dynamical systems using the data

Gene regulatory networks are dynamical systems. Biologists regularly attempt to infer gene regulatory networks and build mathematical models based on measured signaling (protein, messenger RNA, microRNA, etc.) levels. Recent technological advancement has made it possible to perform time-lapse microscopy to track dynamical signaling states in individual cells using fluorescent reporters (reviewed in

However, models of gene regulatory networks derived by SI have to cope with various sources of uncertainty _{1}). Second, the behavior of the network is influenced by environmental factors (_{2}), which are often difficult to model. Third, the observed data are subject to measurement errors (_{3}). All these sources of uncertainty contribute to the perceived stochasticity of gene networks preventing the model estimates from better matching the data.

(_{1}_{2}_{3}_{a}_{b}_{k}

To achieve a better “fit” between models and measurements, researchers often resort to increasing the order or complexity of their models

In this study, we demonstrate that adaptive filtering (in engineering, the term

Before a model is constructed from data using SI, three choices should be made: the model structure, model order, and parameter estimation method by which a candidate model structure/order combination is assessed _{a}_{b}_{k}_{a}_{b}^{st} order discrete-time model can be:_{1}_{1}

We first assume the ARX model is time-invariant, so the model has constant parameters. We proceed to find the model order that gives best estimates. For each model order, the best parameter values that fit the measured data are identified using the Least Squares estimation method (see _{a}_{b}_{k}_{a}_{b}_{k}_{a}_{b}_{k}

The poor model estimates are probably caused by many factors. The p53-MDM2 dynamics are known to be influenced by other genes and proteins _{1}_{2}_{3}

Can a time-variant p53-MDM2 model improve the model performance? If so, it will indicate that the measured dynamics of the p53-MDM2 negative feedback in individual cells has a time-variant component. To test this hypothesis, we implement and compare three adaptive filtering algorithms, NLMS (Normalized Least Mean Squares), RLS (Recursive Least Squares), and Kalman filter (see _{2}_{3}

Using the previous 4^{th} order ARX model (_{a}_{b}_{k}^{rd} order grey-box ARX model (_{a}_{b}_{k}^{nd} order ARX model (_{a}_{b}_{k}^{rd} and 2^{nd}) adaptive models (time-varying models using adaptive filtering) to achieve comparable performance to the high-order (4^{th}) model. These observations suggest that the measured dynamics of the p53-MDM2 gene network has a time-variant component (_{1}_{2}_{3}

(^{th} order ARX model (_{a}_{b}_{k}^{rd} order grey-box ARX model (_{a}_{b}_{k}^{nd} order ARX model (_{a}_{b}_{k}^{th} order ARX model (_{a}_{b}_{k}_{a}_{b}_{k}^{rd} order ARX model (_{a}_{b}_{k}

Tracking the parameters over time provides an intuitive way for evaluating the time-variant component of the measured p53-MDM2 dynamics. ^{rd}-order ARX model (_{a}_{b}_{k}

In this work, we demonstrate that time-variant models using adaptive filters can provide more accurate estimates of single cell measurements than time-invariant models. Taking time variation into consideration allows lower-order, simpler models to outperform higher-order, time-invariant models. SI with adaptive filters can provide a useful modeling methodology thanks to the increasing number of time-series and single cell measurements that are becoming available these days. The exact mechanisms of these systems are often not completely understood, making grey- and black-box SI models a convenient tool for estimating system behaviors. Although we introduced adaptive filtering as an estimation technique for better fitting a model to data, the same approach may be used to elucidate the adaptive behavior of biological systems. In that respect, tools from adaptive networks

285 Image frames were extracted from the video file

For a single-input/single-output system, the ARX model structure is represented as

For SI we used the MATLAB System Identification Toolbox (Mathworks, USA) and the LabVIEW System Identification Toolkit (National Instruments, USA). For Least Squares-based time-invariant parameter estimation, the input and output data were divided into two sets of data, estimation and validation sets. Estimation data (from image frames 1 to 142) is the data set used to fit a model to the data, while validation data (from image frames 143 to 285) is the data set used for model validation purposes. For the adaptive filter implementations, the input and output data were not divided into estimation and validation sets because this division is not necessary; instead, the filters were iteratively and continuously applied to the data set.

The performance was measured using the Best Fit score and the equation for computing the score is:

In the equation-error approach, the data vector

The parametric vector to be estimated is denoted by

The parametric vector to be estimated is denoted by

The parametric vector

The self-adjustable step size

The estimated parametric vector

Similar to RLS, the estimated parametric vector _{3}_{2}

_{a}_{b}_{k}

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