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In this view, each such pathway will add to the mutual information of the desired cellular output e. Researchers of the subject are likely to encounter both situations, and perhaps a revised form of population mutual information might be needed to quantify these effects, along with the formulation of new information theoretic metrics. As an example, for any given input x t , the mutual information gives us a sense of the diversity or spread in responses in y 2 given the cell-to-cell variability encoded in. We envision these kind of metrics to reflect the different subpopulations with similar parameters within a given population and to serve as a potential tool to quantify how cell-to-cell variability across a population might change in structure due to various time-dependent inputs.

Finally, most studies to date have focused on variability in populations of non-communicating cells. Information fidelity in cells that communicate, for example through quorum sensing for bacterial communities or cell-to-cell mechanical coupling for tissues, is still largely unstudied. How cell-to-cell communication modulates global variability and variability in initial conditions across a population, and hence mutual information of cellular pathways, is a topic that should be explored in order to determine whether and when multicellularity offers a beneficial strategy in terms of signaling fidelity.

The time-dependent mutual information is then calculated with this data. A larger number of experiments N generates a more accurate approximation of mutual information. However, we observed that convergence to accurate MI values does not increase monotonically with N for the logarithmic sampling of the doses response that we have adopted. Rather, convergence proceeds exponentially, followed by marginal gains in accuracy as N increases. Given their measured convergence rates, we can extrapolate an upper bound on the MI at an infinite number samples.

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We choose N whose calculated MI at N is within 1 percent of the extrapolated upper bound. The chemical equations for the circuit in Fig 1b are 2a 2b 2c 2d 2e where. The propensities of the reactions appear above the reaction arrows. The system is a simple cascade of reactions where the input X activates Y 1 , and subsequently the Y 1 -dependent transcription of Y 2.

The parameter values are tabulated in Table 1. Here the mean total number of Y 1 molecules, active and inactive, is. This system has only a single stationary solution. This allows us to approximate and efficiently calculate the master equation with a local affine assumption using the first two moments Eqs 6 and 7 taken from [ 24 ]. In this work, we have assumed that the stochastic global parameter, G , manifests itself in variation in translation rates. To incorporate G we define our new protein creations rates and.

The chemical equations for the synthetic circuit in Fig 3a are the same as the simple circuit except that the production of now involves an mRNA step, which does not directly affect any of our results. Also, we have added a YFP reporter of that has a half-life of 6 hours which we set the transcription factor itself to be the same. We define the estradiol dependence in mRNA as The parameters for the circuit are given in Table 2 global fluctuation model and Table 3 intrinsic fluctuation model.

The formulation that we assume in our model and data consists of a system of well-stirred chemical reactions with N molecular species. For some environmental input X t , we define the pathway state Y t to denote the vector whose integer elements Y i t are the number of molecules of the i th species at time t. The polynomial form of the propensities a j y may be derived from fundamental principles under certain assumptions [ 25 ]. Given that we solve the first two moments, we constrain our distributions to be either a negative binomial distribution or a normal distribution. For cases when , we apply the negative binomial distribution since it only requires the first two moments and is non-negative.

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The negative binomial is very close to a normal distribution for and we therefore apply the normal distribution in these regions. The value of 3 used is heuristic, but the tail of the normal distribution at negative values is negligible at this point. For linear transcriptional systems, the negative binomial is a natural steady state solution [ 28 ] which was our motivation for applying it.

Note that the negative binomial distribution is only required for our modeling of the synthetic circuit data. Our theoretical example in the first half of the paper has large enough basal levels at zero input which always keeps it in the normal distribution regime. These vectors were linearized by digestion with PmeI and transformed using standard yeast transformation techniques. The sequences for the ADH1 and GAL10 promoters were and bp upstream from the start codons for these genes, respectively.

The genotypes for these strains are listed in Table 4. Well 12 did not receive estradiol. Estradiol did not change the growth rate of the cell population. Two replicates were performed. Greater than three thousand cells were collected for each measurement. The time dependent mutual information is 8 where the second line is simply a chain-rule representation. In addition to the assumption that the quantities of y s are fluctuating extremely slowly, we will also impose that the quantities in y s are independent of x t.

The time dependent MI is approximated as 9 Finally, we can examine the mutual information between y s and y m for a given input signal s x t using the formula Strains 1 and 2 have similar growth rates that are independent of estradiol concentrations less than nM. The top panel is the normalized experimental y 2 data. Each curve is normalized by its maximum mean value in the corresponding lower panel. The lower panel is the un-normalized experimental y 2 data. The distributions presented in ascending mean values are for ascending estradiol nM values of 2. Each curve is normalized by its maximum mean value.

We would also like to thank Lucien Bogar for help in strain construction and flow cytometry measurements. Conceived and designed the experiments: MC OV. Performed the experiments: MC OV. Abstract Stochastic fluctuations in signaling and gene expression limit the ability of cells to sense the state of their environment, transfer this information along cellular pathways, and respond to it with high precision.

Author Summary This work demonstrates how different sources of variability within biochemical networks impact the interpretation of information transmission.

Noise and Fluctuations: An Introduction

This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited Data Availability: All relevant data are within the paper and its Supporting Information files.

Introduction To survive in challenging conditions, cells need to detect, transduce, and process signals from their environment. Results To compute mutual information in a given biological network, we apply simple step functions [ 14 ] of the appropriate environmental input to N populations of the same isogenic cells. Download: PPT. Fig 1. Time-dependent mutual information transmission in a simple biochemical circuit. Contribution of initial condition variability to time-dependent mutual information We first assumed that this circuit is isolated from the rest of the cell, and that any stochasticity it exhibits is only the result of its chemical reactions intrinsic variability.

Time-dependent mutual information transmission with global parameter variation Thus far, in our MI calculations, we have only accounted for variability in initial conditions given a single parameter set for the pathway. Fig 2.

Global variability has large impact on mutual information. Probing the mutual information of a simple synthetic circuit Next, we sought to probe the major determinants of mutual information for a simple synthetic transcriptional circuit Fig 3a. Fig 3. Variability in a transcriptional synthetic circuit is dominated by slowly fluctuating global variable.

Table 2. Parameters for model of synthetic circuit global model. Table 3. Parameters for model of synthetic circuit intrinsic model. Fig 4. Mutual information modeling predictions and experimental measurements of the transcriptional synthetic circuit. Discussion In this work, we illustrated how variability in initial conditions across a population, as well as slow-fluctuating extrinsic global variables can generate low values for the population mutual information in response to an input. Chemical equations for the simple in silico network The chemical equations for the circuit in Fig 1b are 2a 2b 2c 2d 2e where.

Inclusion of global parameter variability within chemical equations. Chemical equations for the synthetic circuit. Computation of first two moments using affine assumption The formulation that we assume in our model and data consists of a system of well-stirred chemical reactions with N molecular species. Synthetic circuit constructs Strains. Growth conditions and flow cytometry.

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Supporting Information. S1 Fig. Comparison of Y 2 mCherry measurements show that Strains 1 and 2 are equivalent in Y 2 output. S2 Fig. The majority of peak normalized dose-dependent distributions of Y 2 YFP for estradiol doses from 2. S3 Fig.

Comparison of alternative computational models for the transcriptional synthetic circuit demonstrate that only the slow global fluctuations model can recapitulate the experimental data. S4 Fig. Time snapshots of mutual information,. S5 Fig.

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