Statistical framework for transient transfection

In what follows, we define a gene circuit as a set of N genes (1) and their corresponding gene products (2) in which a subset of components (3) is defined as input and a subset of components (4) is defined as output.

Further, consider a cell that harbors a gene circuit, either in a stably-integrated or transiently-delivered fashion, such that a gene g_i is present in k_i copies in that cell, and the entire set of copy numbers is a vector (5)

Hereafter, we consider only interactions between circuit components that have been intentionally engineered (i.e., chromatin-related effects do not interfere with circuit function in the stable case), and assume that the biochemical parameters describing individual interactions do not change between stably integrated and transiently-delivered components. Even though individual cells in a population of stable clones may behave differently, e.g., through stochastic effects [32], we expect the aggregate statistics of different clones containing identical circuit copy number to be similar to the aggregate statistics of cells transiently transfected with the same circuit copy number. Therefore, when considering stable clones, we imply an idealized "averaged" clone in which the integrated circuit is governed by the same parameters as the transiently transfected circuit. It then follows that if we apply the same input I to a population of cells that all harbor the circuit with the copy number vector k, and allow the cells to arrive at a steady state in the stable case and to the quasi-steady state in the transient case (see S1 Text “In-silico time-courses”), then the outputs O will form the same statistical distribution, which can be mono- or multimodal [33], in both cases. Reporting the distribution of O for various inputs I would conclude the characterization of a stably-integrated circuit, because all cells harbor the exact same vector k, which can be engineered or experimentally determined post factum after clonal isolation.

In the transient transfection experiment, while the values of I and O could be collected for individual cells, the underlying values of k are unknown because the process of transient delivery is extremely noisy. The only way to derive useful data from transient transfections is to deduce, at least for a subset of cells, their k values, and group together data from cells with similar values of k. If this can be accomplished, the input and output values measured in these cells will be similar to the values one would have obtained with a circuit stably integrated at k copies. Below, we develop a statistical description of a transient co-transfection process, which leads us to identify cells residing in binned modes of input and output distributions as cells for which the copy number vector k can be estimated.

We start with the statistical description of a multi-plasmid co-transfection of N constitutively expressed and mutually independent genes g₁,g₂,…,g_N, generating (fluorescent) protein products O₁,O₂,…,O_N. Note that there is no input in this system, so every protein product can be called an “output”. Available data [6] suggest that experimentally-observed distributions of a protein level expressed from a constitutive promoter are lognormal. The mean of the distribution is proportional to the gene copy number k_i with the promoter-dependent global proportionality coefficient β_i being independent of k_i; the standard deviation σ_i of the log-transformed protein level distribution may, in principle, depend on a copy number, but we assume it to be constant in the following equations. Let us define a random variable Y_i as the log-transformed protein output of the gene g_i.

(6)

Y_i is distributed normally and its mean/mode μ_i, and standard deviation, σ_i, relate as follows to the mean of the underlying pre-transformed distribution (7) and therefore (8)

The conditional probability density function (pdf) of Y_i given a gene copy number k_i and parameter β_i is then described by (9)

For a vector of gene copy numbers k = [k₁,k₂,…,k_N], a conditional multivariate pdf of the log-transformed protein expression values Y = [Y₁,Y₂,…,Y_N], provided that each gene generates its own protein output independently of each other, is described by a multivariate normal distribution without covariances (for simplicity, we assume σ_i to be the same for all genes and use a symbol σ in what follows): (10)

To describe the distribution of gene copy numbers in a transient transfection, we introduce an independent parameter m that we call “multiplicity of transfection”. Indeed, there is no experimental data that concerns the probability distribution of genes in a co-transfection, and it likely depends on the exact transfection protocol. Therefore, we make a baseline assumption about the pdf of the gene copy number vector k as a multivariate normal distribution without covariance that depends on the multiplicity of transfection. The standard deviation of each gene copy number distribution scales linearly with multiplicity, with the scaling factor ε. To account for gene combinations that deviate from an equimolar ratio, a parameter a_i describes the relative abundance of a gene. In this case, one gene is assigned as the "reference" with a_i = 1.

(11)

Lastly, m itself can be distributed non-uniformly according to its pdf p(m). Distributions such as Poisson [34], Gamma [35], lognormal [36] or even a combination of them [37], have been used to describe the process of DNA or viral vector delivery to cells. For transient lipofection of DNA, lognormal distributions approximate experimental data well, and therefore (12)

One of the genes and its protein product is assigned the role of, respectively, a reference gene and a reference protein (sometimes called “transfection marker”); let us assume it is k₁, with gene product O₁ and its log-transformed counterpart Y₁. Thus, by definition a₁ = 1. To derive the conditional marginal pdf p(Y_i|Y₁), which is the probability to find the value Y_i of the log-transformed protein O_i expression in a cell in which the log-transformed reference protein expression equals Y₁, we first drop irrelevant variables from Eq 10 to evaluate joint probability distribution of log-transformed protein levels [Y_i,Y₁] given the underlying gene copy numbers [k_i,k₁]: (13)

In turn the gene copy numbers are conditionally dependent on the multiplicity parameter m: (14)

The global joint probability distribution p([Y_i,Y₁]) of Y_i and Y₁ over all values of [k_i,k₁] is obtained by integrating over all values of [k_i,k₁]: (15)

It is customary, as already done earlier [6,38], to bin cells that share the same Y₁, the log-transformed value of O₁, because this is the only readout independent of the other components, as it is a self-contained gene expressed from a constitutive promoter. We follow this approach here: cells binned according to their Y₁ value will exhibit certain log-transformed distributions of the other proteins Y₂,…,Y_N. Knowing the joint pdf (Eq 15), one can derive the conditional probability of Y_i given Y₁ (that is, the pdf of Y_i among cells that express Y₁ log-transformed copies of the reference protein), as follows: (16)

The mode of this distribution, i.e., the most probable value of Y_i given the reference Y₁, can be found by solving the equation (17)

Let us denote this most probable value as . The value of can be determined experimentally as a mode of Y_i distribution after binning the cells according to their Y₁ value. The equation may have more than one solution, corresponding to multimodal probability density function from Eq 16.

This bring us to the most relevant question of this section: What is the distribution of the gene copy number k_i for the cells that reside in the mode(s) of Y_i, and what is the most probable value of k_i? To answer this question, we evaluate the conditional probability p(k_i|Y_i) according to Bayes’ theorem: (18)

In order to find the most probable copy number we solve the equation (or equations, when takes more than one value) (19)

Knowing the most probable gene copy numbers in the cells residing in the modes of log-transformed protein distributions allows us to correlate the data to what might be obtained in cells with stably integrated constructs harboring similar gene copy numbers.

Numerical analysis of transient co-transfection of constitutively expressed genes

An analytical solution of Eq 19 does not exist, and we solve it using numerical simulations. To this end, we performed in-silico simulations of a transient co-transfection containing multiple (N = 5) independent genetic constructs (Methods). The change in protein expression over time, , of each gene g_i can be described by an ordinary differential equation (ODE) with kinetic parameter , gene copy number k_i and degradation rate δ_i (20)

In the steady state, i.e. , the steady-state level of O_i is proportional to k_i with the global coefficient of proportionality and identical to Eq 7: (21)

Iterating multiple times to simulate multiple single cells j (1≤j≤C, where C is the total number of simulated "cells"), we draw the multiplicity m_j from a lognormal distribution (Eq 12) with parameters that roughly fit experimental data (see below) and initialize gene copy number vectors (22) according to Eq 11 with pre-set parameters (Methods). To create log-normal protein distributions given k_j according to Eq 10, for each k_ji in k_j, local proportionality factor b_ji is drawn from a log-normal distribution: (23) with fixed β_i values (Methods, S1 Table); the values of σ are fixed for a given simulation run and systematically varied between 0.00 and 0.32 in different runs. A value (24) is the level of protein O_i in cell j (S1 Fig).

The generated in-silico dataset (S2A Fig) is similar to a flow cytometry dataset what one would obtain in a transient co-transfection experiment of constitutively-driven genes. In order to confirm that the parameters, and in particular the values of σ are realistic, we transiently co-transfected five plasmids, each expressing constitutively different fluorescent protein (O₁: SBFP2, O₂: Cerulean, O₃: Citrine, O₄: mCherry and O₅: iRFP; Methods) (S2B Fig). We find that the standard deviation of the log-transformed protein expression distribution in cells pre-binned on similar values of the reference protein, which we denote σ*, (25) depends on Y₁, and indeed ranges between 0.1–0.3 (S2C Fig). Higher values of σ*are observed as very low Y₁ values, and they plateau towards 0.1 for larger Y₁. Accordingly, the range of σ values used in the simulations, and given that σ<σ*, constitutes a realistic range for the gene expression variability due to "intrinsic noise".

Next, we simulate transient co-transfections using two different gene ratios; (i) equimolar and (ii) a ratio of 1.0:1.3:0.8:0.5:0.4, the latter following some fine-tuning in a parallel experimental project (manuscript under preparation), to generate a joint pdf p(Y) (Figs 1A, S3A, S3B, S4A and S4B). We use these datasets to solve Eqs 17 and 19 numerically, that is, determine the and . To do so, we bin cells that share a log-transformed reference protein value Y₁, evaluate the conditional pdf p(Y_i|Y₁) and, first, determine numerically the value of . Second, we retrospectively look up the values of k_i in cells whose Y_i and Y₁ expression levels lie in the vicinity of a vector []. The empirical distribution of k_i (Figs 1B, S3C and S4C) is from Eq 19, and the mode of the gene copy number distribution, , or , is determined numerically (Figs 1C, S3D and S4D, Methods).

According to Eq 7 there is a simple, linear relation between O_i and the gene copy number k_i, linking them via the global coefficient of proportionality β_i. The global coefficient of proportionality can be determined experimentally using e.g., a calibrated Western blot to measure the absolute amount of protein and calibrated qPCR to measure absolute mean internalized gene copy numbers. For the transfection reference protein, we introduce the variable that corresponds to the gene copy number that one would "naïvely" anticipate to be the most probable value leading to a particular Y₁, given β₁: (26)

For the other log-transformed outputs Y_i the naïvely anticipated copy number in cells that express certain level of the reference protein Y₁, is defined by a similar relationship: (27)

Given that the value of Y₁ and β₁ are the only “knowable” parameters, it is of interest to ask how the actual copy numbers relate to these anticipated values. Using our simulated datasets, we compute the ratio between numerically found and the anticipated copy number from Eq 27 (Fig 1D) as a function of Y₁. We find that the deviation from the anticipated value is a decreasing monotonous function of Y₁ with the following properties: (1) The deviation is always positive for values of Y₁<ln E[O₁]; (2) the deviation is essentially zero when Y₁ = ln E[O₁], and (3) it is negative for Y₁>ln E[O₁]. Further, the absolute magnitude of the deviation increases with increasing σ (S3E and S4E Figs). However, for all noise levels, the deviation is zero at the global mean of the O₁ distribution, E[O₁], and (28)

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Fig 1. In-silico simulation of multiple gene co-transfections.

In-silico simulations shown for two cases with parameter settings σ = 0.08, ε = 0.04 and m = 10: equimolar (a₁:a₂:a₃:a₄:a₅ = 1.0: 1.0: 1.0: 1.0: 1.0; top row) and "nominal" (a₁:a₂:a₃:a₄:a₅ = 1.0: 1.3: 0.8: 0.5: 0.4; bottom row) ratio of gene mix. (A) Density plots show the amount of expressed proteins O₂ versus O₁. Solid black lines indicate the edges of an example bin for the transfection reference protein O₁. (B) The plots show the distribution of gene copy numbers in cells whose O₁ values fall into the bin shown in panel A. Copy number distributions corresponding to different genes g_i are shown using different colors (legend on the very right of the figure). In the equimolar case, gene copy number distributions overlap, while in the "nominal" case they are separated. (C) The modes of the copy number distributions are plotted versus the median signal of the transfection reference protein O₁ in all bins (colored lines). The dash-dotted line marks the mean (E[O₁]) of the global O₁ distribution. (D) For each bin of the transfection reference protein O₁, the modes of the gene copy number distributions from each gene g_i are determined numerically. The ratio of the numerically-determined mode of the copy number distribution and the anticipated copy number are computed and plotted versus the corresponding O₁ values in individual bins. The global mean of O₁ is shown with a dash-dotted line. (E) Ratios of gene copy number modes relative to the gene copy number mode of the transfection reference protein , as a function of the O₁ median value in the bins.

https://doi.org/10.1371/journal.pcbi.1008389.g001

We further analyzed the ratio of the modes of gene copy number distributions to in cells that reside in the close vicinity of the log-transformed expression vector . The ratio stays constant for almost the entire range of Y₁ values (Figs 1E, S3F and S4F). Since the naïve estimate and the numerical mode of the absolute gene copy number coincide at the global mean of the O₁, both the relative and the absolute abundance of the gene copy numbers can be deduced with high certainty in cells that express O₁ around its global mean. This is true regardless of the chosen distribution of m and β_i. Simulations that employ Poisson, Gamma or lognormal distributions show a strikingly similar effect (S5 Fig). Appropriate experimental techniques allow measuring both the protein copy number [39] and the gene copy [6] number in the cells residing at the global mean of O₁, making it possible to determine the value of β₁ experimentally and therefore extrapolate directly to the ground truth expected in the stable cell line with the similar gene copy number.

Numerical analysis of transient co-transfection of non-trivial gene circuits

Next, we consider the case when the same genes, apart from the transfection reference protein gene g₁, encode a set of genes interacting in a circuit. Depending on the circuit, log-transformed distribution Y_i of protein O_i in cells pre-binned on the value of Y₁ may exhibit mono-, bi- or multimodality. We may consider the joint probability distribution of the vector of independent constitutive genes and their gene products, k∙Y, as a baseline state of any circuit. When the genes are interconnected (not including the reference gene g₁ and its log-transformed product Y₁), this baseline distribution is transformed because the values of Y are no longer independent. However, the values of k remain the same, because they represent the exact same underlying process of DNA delivery, and only the Y values change relative to the independent, constitutive values. We hypothesize that despite the fact that the values of Y are no longer independent of k_j for i≠j, k vectors corresponding to the (possible multiple) multivariate modes of Y|Y₁, would not deviate far from the k vectors obtained in the case of independent co-transfection. We further hypothesize that this deviation will decrease as the noise in the system increases to biologically-plausible levels.

To test these hypotheses, we simulated two three-node gene circuit architectures (currently being investigated experimentally in a related project, see S6 Fig for the experimental results of the fan-out circuit); a simple monomodal fan-out circuit (FO; Fig 2A) and a non-trivial pitchfork bifurcation circuit, also known as reinforced incoherent feed forward motif RIFFM [4,30,40,41] (Fig 2B). The input to the circuit is a transcriptional activator PIT2 [42], whose level is tuned by Doxycycline via an bi-directional TRE promoter that also drives a fluorescent protein mCherry as a proxy for input expression. The first PIT2 target promoter (P1) drives the D. melanogaster derived transcriptional repressor Knirps (kni) and translationally-linked fluorescent protein Cerulean, constituting the first circuit output. The second PIT2 target promoter (P2) drives the transcriptional repressor LacI fused to a KRAB domain and translationally linked to a fluorescent protein Citrine, representing the second circuit output. In RIFFM circuit, kni is able to repress P2 while LacI is able to repress P1; in FO, the mutual repression is eliminated via mutations.

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Fig 2. Schematics of gene circuits FO and RIFFM and analysis outline.

(A)| Monomodal/fan-out (FO) and (B) bi-modal (RIFFM) gene circuit. The circuits are composed of five independent genes. Constitutively-expressed transcription factor rtTA co-induces PIT2 and the fluorescent protein mCherry in a Doxycycline-dependent fashion. PIT2 in turn activates two promoters P1 and P2, which express a transcriptional repressor lacI-KRAB and Kni, respectively, co-expressed via P2A linkers with Citrine and Cerulean fluorescent reporters. In the FO circuit the repressors do not interact due to mutated promoter binding sites, while in the RIFFM circuit they repress each other, establishing a mutual inhibition. (C) Graphical illustration of all steps to find the modes within our simulated data sets. The raw flow cytometry like data (1) is binned by the transfection marker (i.e. Y₁: SBFP2). The binned data is isolated and subsequent analysis is done only on this subset (2). Afterwards the distribution of the log-transformed input signal (mCherry) within the binned subset is determined and at least one Gaussian is fitted to the distribution. A narrow bin(s) around the mode(s) (black arrows) of the fitted distribution(s) is determined (pink bars), thus obtaining a subset of the originally-binned dataset. The subset around the peaks ( and ) are now analyzed individually. For illustrative purposes, we show the input (mCherry) and output (Cerulean) signal of the subset around (3) and (4). The modes of the log-transformed output distributions are identified using a similar peak finding procedure as for the input signal (mCherry). The output histograms and their modes (black arrows) are shown on the sides of plot. Within these modes, we look at the pdf of the gene copy number distributions for all circuit genes as well as the transfection reference gene and identify the vectors and , respectively.

https://doi.org/10.1371/journal.pcbi.1008389.g002

We built mechanistic kinetic models of the circuits FO and RIFFM (S2 Text "Simple Fan-Out Model" and S3 Text "Detailed Models") and simulated the flow cytometry dataset for multiple transiently transfected cells j (1≤j≤C, where C is the total number of simulated "cells"). As above, every gene is encoded on a separate plasmid. We also compared this to a single-plasmid setup with all five genes are located on a single DNA backbone, but saw only a marginal difference in outcomes (S7 Fig). The multiplicity of transfection and the gene copy numbers are simulated as above (S1A Fig); the gene copy numbers become the initial conditions for running a simulation. Differently from that case of constitutive co-transfection, we directly simulate circuit dynamics governed by kinetic parameters p; to simulate the effects of intrinsic gene expression noise, the parameters that govern protein translation rates are sampled independently from a lognormal distributions with nominal parameter values π_i and preset "noise" levels ranging, as above, from 0.00 to 0.32: (29)

For every cell j, the drawn parameter values p_ij are used in a dynamic simulation ran to a steady state, with the simulated steady state input and output protein levels corresponding to the readouts from that cell.

First, we simulate mono- and bi-modal gene circuits for a single Doxycycline/input level. Similar to the data analysis above, we bin the cells according to Y₁ value. In the bin, we first focus on the input protein Y_input and identify its mode. Importantly, in the general case the distribution of Y_input|Y₁ can be bi-modal, leading to two numerically-found values and . In this case, we consider separately the cells residing close to the expression vectors and . Next, for every circuit output we consider the distributions and . These distributions can also be multimodal; in what follows we assume they are bi-modal. We denote the modes of the output distribution corresponding to the high mode of the input and , and use similar notation for the output modes corresponding to the low mode of the input, if the latter is present. Lastly, we consider all cells in the vicinity of the expression vector and and evaluate the copy number distribution of every circuit gene as well as the reference gene. These are monomodal distributions, with the modes denoted respectively as and (see Fig 2C for schematic description of the process). These numerically evaluated values are then compared to the naively anticipated values calculated according to the Eq 27. Note that in these simulations, the transfection reference gene expression is modelled explicitly as a transcription/translation/degradation cascade with corresponding kinetic parameters; the value of β₁ for Eq 27 is calculated according to Eqs 20 and 21.

The analysis of the simulated data reveals the following: for the monomodal FO circuit, the behavior of the copy number modes of the input and the output genes is quantitatively identical to what is observed in the simulation of multiple constitutive gene co-transfection (Fig 3A–3E). The bi-modal circuit (Fig 3F) shows its bi-modal behavior at the lower intensities of the transfection reference protein O₁ (10³–10⁵). In this range, distributions of gene copy numbers for the high and low modes of output expression are slightly diverging (Figs 3G–3K). They are, however, almost fully overlapping, and their modes differ only by a few percent respectively upwards or downwards relative to the monomodal case, despite large difference in the corresponding protein modes. We quantify the divergence in gene copy number modes between high and low protein output modes by introducing a metric (30) with being the mode of the transfection marker copy number distribution found in the constitutive co-transfection simulation (Fig 1). We observe a steady increase in upon an increase in noise level σ (Fig 3L), however, it is less that 10% for realistic levels of noise. Furthermore, we quantify bi-modality-dependent deviations of gene copy number ratios between the high and low output modes and introduce the metric (31)

Unlike , this deviation of gene copy number ratios, Δϕ_i, decreases with an increase in the noise level σ (Fig 3M). These observations confirm our hypothesis that even in multimodal circuits, the cells that share the same amount of the reference protein, also share very similar gene copy numbers, both in absolute and especially, in relative terms (S8–S10 Figs). Moreover, deviations from the nominal gene copy number ratio decrease with the increase of noise levels σ.

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Fig 3. In-silico simulation of transiently-transfected gene circuits FO and RIFFM and the copy number analyses at various σ noise levels.

In-silico simulations (global parameters σ = 0.08, ε = 0.04, μ_m = 1.4979, σ_m = 1.2686 and a₁:a₂:a₃:a₄:a₅ = 1.0: 1.3: 0.8: 0.5: 0.4) shown for two circuits: A-E monomodal/fan-out (FO) and F-M bi-modal (RIFFM). (A) Raw data of the simulated transiently transfected FO circuit. The amount of expressed proteins O₂ (left: Cerulean) or O₃ (right: Citrine) versus O₁ (transfection marker: SBFP2) is shown as a density plot. Solid lines indicate the edges of a transfection marker bin. (B) Gene copy number distributions of cells binned by a particular value of log-transformed O₁ in a bin shown in panel A. (C) The modes of the copy number distributions, , are plotted versus the median signal of the transfection reference protein O₁ for all bins (colored lines). The dash-dotted line marks the mean (E[O₁]) of the global O₁ distribution. (D) The ratio of the numerically-determined mode of the copy number distribution and the anticipated copy number plotted versus the O₁ values in individual bins. The global mean of O₁ is shown with a dash-dotted line. (E) Modes of the gene copy number distributions normalized by the mode of the transfection reference gene are shown as a function of O₁ values in individual bins. (F) Raw data of the simulated RIFFM circuit with bi-modal output O₂ (Cerulean; left) or O₃ (Citrine; right). The black lines indicate a bin within the bimodal range. (G) The modes of the copy number distributions, and , are plotted versus the median signal of the transfection reference protein O₁ of all bins (colored lines). Dashed segments indicate the range of O₁ in which the high and low modes do not coincide; the black dash-dotted line indicates the mean (E[O₁]) of the global O₁ distribution. (H) The ratios of the numerically-determined mode of the copy number distribution and and the anticipated copy number are plotted versus the corresponding O₁ values in individual bins. Dashed segments indicate the range of O₁ in which the ratios corresponding to high and low modes do not coincide. The global mean of O₁ is shown with a straight dash-dotted line. (I) Modes of the gene copy number distributions and , normalized by the mode of the transfection reference gene are shown as a function of O₁ values in individual bins. Dashed lines indicate the range of O₁ where the values do not coincide. (J) The ratios depicted in panel I are shown in greater detail for the outputs O₂ (Cerulean; top) and O₃ (Citrine; bottom). (K) Fitted gene copy number distributions of the indicated bin in F are shown for all genes (left to right) and noise levels σ. Black curves indicate the fitted distributions to the low output mode, , and colored curves the fitted distributions of the high output mode . (L) Divergence in gene copy number modes between high and low protein output modes normalized to the mode of copy number distribution of the transient co-transfection for all noise levels σ as a function of O₁ values in individual bins. (M) Difference of copy number modes’ ratios Δϕ_i in the high and low protein output modes for all noise levels σ as a function of O₁ values in individual bins.

https://doi.org/10.1371/journal.pcbi.1008389.g003

The rationale for extracting input/output relationship from transient transfection data

The analyses above suggest a workflow for analyzing and deducing input/output relationships from transiently-transfected circuits. To summarize the findings so far, we show that cells, which express a certain level of reference protein O₁ and reside at multivariate modes of log-transformed input and output expression, harbor both input and output genes (or plasmids) with the following properties: (1) even for multimodal outputs with large differences in protein level modes, the distribution of the input and output genes copy numbers corresponding to the different output protein modes are almost overlapping, with the copy number modes varying by about 10% for biologically-realistic noise values, and thus can be treated as the same copy number for all practical purposes; (2) the copy number distribution modes’ ratio almost exactly corresponds to the nominal ratio used in a transfection for all reference protein bin values, for both low and high output modes, and the deviation from the nominal ratio decreases with increased noise; (3) the distribution modes’ absolute values exactly match the naïve anticipation (Eq 27) when the reference protein is expressed at the level E[O₁]; (4) the distribution modes’ absolute values (for both high and low output modes) deviate from the naïve expectation in a predictable linear fashion as a function of log-transformed reference protein level Y₁, with the magnitude of the deviation increasing with the overall noise level. However, because the actual noise level and thus the degree of deviation can be quantified experimentally, even in cells that lie away from E[O₁] the copy numbers can be estimated not only in relative but also in absolute terms. We show that this holds for a case of a complex circuit that generates bi-modal output distribution, with only slight deviations of the copy number modes from the expectation. Accordingly, by analyzing the input and output values in the cells that reside in the multivariate modes of circuit inputs’ and outputs’ distributions (after binning by the log-transformed reference protein value Y₁), we should be able to extract the information about the input/output response of the circuit that is comparable to the stable cell line harboring the circuit at the copy number derived from Y₁ according to Eq 26 and corrected by the measure of deviation that depends on the noise level.

Overview of the workflow validation procedure

To validate the workflow suggested above, we simulate transient transfection and stable integration for FO and RIFFM circuits for a wide range in circuit input levels using the exact same ODE model, with the input modulated via varying Dox level; otherwise the parameter randomization is performed as described above according to Eq 29. To simulate a stable integration dataset, we initialize a copy number vector k such that the ratio between individual genes corresponds to the nominal ratio of the transient transfection, and the absolute copy number of the reference gene is set to different fixed values corresponding to the bins used for transient transfection data analysis (S11 Fig; Methods). After the datasets are simulated, we extract input/output relationships corresponding to various bins of log-transformed values of the (transfection) reference protein Y₁ from the simulated transient transfection data, as described in detail in the next section. This is compared to the results of the stable integration simulation performed for the copy numbers that correspond to those reference protein levels. The simulation of the stable integration scenario generates an input/output “cloud”, as has also been demonstrated experimentally [8,43,44]. The cloud can be used "as is" for the purpose of comparison, or it can also be processed via output mode identification for different input levels and building averaged curves. The process is illustrated schematically in S12 Fig.

The input/output relationship is generated for each level of log-transformed reference protein Y₁. We make use of the datasets simulated with different Doxycycline levels and thus different amounts of input expressed per gene copy. (Note that Doxycycline does not affect the gene copy number or the expression of the transfection reference protein, it is not a direct input and is not a part of the input/output relationships that we seek.) For a given Y₁ bin, a single transfection experiment simulation with a fixed Doxycycline value generates (for a bimodal case) at most four points on the input/output curve: []; ; ; and []. In most cases, the input will only exhibit a single mode that we denote for uniformity and thus only two points will be generated for a bi-modal case, and one for a monomodal case. For this same reference protein bin, we repeat the procedure defined earlier (Fig 2C) for every Doxycycline level and generate multiple [input; output] pairs that cover the entire input range. The procedure can be done for any desired reference protein bin, thus showing circuit behavior for different absolute gene copy number of its components (Fig 4).

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Fig 4. Peak Finder Algorithm For Flow cytometry (PFAFF) analysis strategy.

The steps of finding peaks in distributions of a binned data as shown in Fig 2C are repeated for every input induction level (i.e. Doxycycline level; here three representative cases for low (1), medium (2) and high (3) input modulation levels are depicted) for each output (here, Cerulean and Citrine). Initially, a bin of the transfection reference protein (here, SBFP2) is determined and all downstream analyses only deals with cells residing in this bin. The plots in the dashed box show the example workflow of applying the peak finding to the individual input modulation levels. The density plots depict the binned data and the adjacent histograms show the distributions of their respective input and output proteins. Black wedges indicate the modes of the (convoluted) distributions and black markers indicate their location on the density plots. The [input; output] mode pairs, identified in this workflow (markers), derived from the raw data corresponding to the different input modulation level are plotted on the input/output mapping charts the output Cerulean (4) and Citrine (5).

https://doi.org/10.1371/journal.pcbi.1008389.g004

The fact that every transient transfection performed with a certain Doxycycline level generates only up to four, and usually one or two, points on the input/output relationship curve is slightly counterintuitive because a flow cytometry plot would reveal wide distribution of the input values. However, this distribution results from the variability in the copy number of the input gene in the cells and is therefore irrelevant to the determination of the input/output relationship. In order to characterize an entire curve, there must be a practical way to modulate input expression per gene copy and repeat the experiment multiple times, every time with a different degree of modulation. This can be done with Doxycycline as in our case; when this is not feasible, one can mimic input modulation by systematically changing the relative dosage of a constitutive input-expressing gene, or use a series of constitutive promoters of varying strengths. In another observation, when extracting the modes, the high output modes corresponding to both the low and the high input modes fall on the same curve when plotted against the input values; the same is true for the low output modes (S13 Fig). This is not surprising, because the input value is the only determinant of the output. Therefore, we pool high and low output modes, respectively, and interpret them as the averaged input/output relationship of a circuit; when the behavior is bimodal, two curves are generated.

Validation using direct simulation data

We applied our data generation tool for transient transfections to FO and RIFFM circuit architectures and simulated 500,000 cells at twelve different Doxycycline input concentrations and six noise levels of σ. In Fig 5A we show representative examples of the raw data from transient circuit simulations at a single noise level (σ = 0.16) and various Doxycyline modulations. Note the shift in the scatter plots in response to Doxycycline increase. For each noise level, we extract the corresponding input/output relations of the data set with our analysis strategy that we call PeakFinder Analysis For Flow cytometry, or PFAFF. The algorithm bins simulated cells according to the expression level of the transfection reference protein SBFP2 each bin containing an equal number of cells (9.5% of total population, 10 bins in total). Next, we determine the modes of log-transformed input and output protein distributions of cells residing in each bin for the different Doxycycline levels as described above, and build the input/output relationships corresponding to that bin (S14–S19 Figs). In the stable integration scenario, we build datasets that correspond to different fixed sets of gene copy numbers. Specifically, the copy numbers are set to correspond to the median copy numbers of the bins used to process the simulated transient transfection data (Methods). We simulate 5,000 cells per Doxycycline value and repeat this for twelve different Doxycycline values to cover the entire input range. This simulation is repeated for each σ. These data serve as the gold standard to evaluate the performance of our method for transient transfection data processing, by how well the input/output relationships match the stable integration simulation for matching bin/stable copy number.

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Fig 5. Results of PFAFF analysis on simulated FO and RIFFM flow cytometry data set.

(A) Raw simulated transient transfection data for circuits FO and RIFFM at noise level σ = 0.16, modulated by different input levels of Doxycycline (columns). Each circuit (FO, RIFFM) is represented by two rows of charts depicting the input versus their respective output signals. (B) Simulated flow cytometry data set of the bin that lies at the transfection reference protein's global mean (transfection reference: SBFP2) at different gene expression noise levels σ (columns) ranging from 0.00–0.32. The plots show the input/output curves extracted by PFAFF (colored crosses; top row: Cerulean, bottom row: Citrine) atop of the simulated stably integrated circuit at the indicated gene expression noise level σ (grey density plot). FO undergoes an activation in both output colors. The RIFFM circuit shows a bi-modal behavior already at low noise levels for both output colors.

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In Fig 5B we show the stable integration simulation and the analysis results from PFAFF, the latter extracted from a transfection reference bin that lies close to the global mean of the reference protein (i.e. SBFP2 bin 5; see S14–S19 Figs for all bins, input and noise levels); the former simulated for the gene copy number that corresponds to this reference protein level. Note that the number of transfection reference bins does not influence the outcome (S20 Fig). We plot the stable integration outputs at various input levels as density plots in the background (grayscale). For the lowest noise case (σ = 0.00), the density plots for the stable integration collapse to curves, as expected. Gradually increasing σ leads to increasingly diffuse input/output relationships. Atop these density plots we superimpose the mode values extracted by PFAFF from the corresponding simulated transient transfection data and binned for the cells that express the same level of the transfection marker as the stable integration. The analysis suggests that the input/output relationship extracted using PFAFF superimposes with the input/output “cloud” simulated for the stable integration, when both reflect similar underlying absolute gene copy number.

In order to expand the number of circuits for analyses, we simulated another commonly studied circuit family–the type-1 incoherent feed-forward motifs. Our simulations include two versions of the I1-FFL (one for each repressor; I1-FFL1 and I1-FFL2; Fig 6A and 6B). We applied the same analyses as before to both I1-FFLs and show the comparison of input/output from stable integrated and transiently transfected circuits (Fig 6C; see S21–S26 Figs for all bins, inputs and noise levels). As is the case for other two circuits, there is excellent correspondence between the input/output curves extracted from simulated transient transfection data, and the input/output behavior of the comparable simulated stable case. This motivated us to expand the number of circuits by a coherent feed-forward, a negative feedback and lastly a positive feedback motif; all of them showing similar, excellent agreement (S27–S29 Figs).

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Fig 6. Results of PFAFF analysis on simulated I1-FFLs flow cytometry data sets.

(A) and (B) Circuit architecture of two additional I1-FFLs. (C) Simulated flow cytometry data set of the bin that lies at the transfection reference protein's global mean (transfection reference: SBFP2) at different gene expression noise levels σ (columns) ranging from 0.00–0.32 is processed by PFAFF. The plots show the input/output curves extracted by PFAFF (colored crosses; top row: Cerulean, bottom row: Citrine) atop of the simulated stably integrated circuit at the indicated gene expression noise level σ (grey density plot). Both I1-FFLs show adaptive behaviors in their respective outputs.

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The initial qualitative analysis uncovers excellent overlap between the input/output relationships found by PFAFF, and the input/output clouds from the corresponding stable integration simulations. Only at the highest simulated levels of σ, i.e. 0.32, the PFAFF algorithm has minor difficulties with extracting expected input/output relationships. Indeed, a σ of 0.32 is much larger than variations observed typically in nature [9,45]. To obtain a quantitative measure of the correspondence, we extracted modes from the log-transformed protein expression distributions of input (mCherry) and outputs (Cerulean and Citrine) from the stable integration data sets with the same peak finder algorithm that we employ in PFAFF (Methods). We correlated the obtained modes with the modes that were found by PFAFF in the transient transfection case (Fig 7). In the pooled modes from all data sets, meaning all external input levels and bins for each noise level, we find a high correlation between the modes from both simulation scenarios for all expression noise levels (mean of Pearson correlation coefficient ρ>0.91±0.01 for output modes; σ: 0.00–0.32).

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Fig 7. Quantitative analysis of the stable integration and transient transfection data sets.

Correlation between extracted output modes of stable integration simulation and PFAFF results from transient transfection simulations. The mode values for both output colors (Cerulean and Citrine) for all input modulator levels and for all bins/stable copy number sets are plotted, and Pearson correlation coefficient (ρ_Cer or ρ_Cit) is shown for each plot.

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Cloning

Standard cloning techniques were used to clone all plasmids. We used E. coli DH5a and DH10B as the cloning strains, cultured in LB Broth Miller Difco (BD; Cat. no. 244610) and Ampicillin (100ug/ml, Sigma-Aldrich; Cat. no. A0166-5G) as selection medium.

Cell culture and reagents

All experiments were done with HEK 293 cells (Life Technologies) and were grown at 37°C, 5% CO₂ in complete medium (DMEM (Thermo Fischer; Cat no. 11965092) supplemented with 10% fetal bovine serum (FBS; Sigma-Aldrich; Cat. no. F9665) and 1% Penicillin/Streptomycin (Sigma-Aldrich; Cat. no. P4333)). They were sub-cultured by seeding 10⁶ cells into T75 flasks every 3–4 days.

Transfection

One day prior transfection, cells were passed through a 40um cell strainer (Falcon; Cat. No 352340) and counted with Bio Rad TC10. In each well (uncoated 6-well plates, Thermo Scientific Nunc; Cat. No. 2020–10) 300,000 cells were seeded and incubated for another 24 hours. On the day of transfection DNA was diluted in 250ul Opti-MEM I Reduced Serum (Gibco, Life Technologies Cat no. 31985–962) and mixed with a 244ul Opti-MEM I/6ul Lipofectamin 2000 Transfection Reagent (Thermo Fischer; Cat. no. 11668019). After a 20 minutes’ incubation step at room temperature, the transfection mix was added drop wise to the wells. The cells were incubated for another 72 hours before being measured by flow cytometry.

Flow cytometry

All samples were measured with a BD LSR Fortessa cell analyzer. The medium was removed and cells were incubated with 300ul StemPro Accutase Cell Dissociation Reagent (Thermo Fischer; Cat. no. A1110501) at 37°C, 5% CO₂ for 10 minutes. Reporter specific combinations were used to measure all four fluorescent proteins independently, but still providing a setup of little bleed over. In particular, we used for: SBFP2 a 405nm laser with 445/15, Cerulean 445nm laser with 473/10, Citrine 488nm laser with 542/27, mCherry 561nm laser with 610/20 and iRFP 640nm laser with 710/50 emission filter sets. We used the same PMTs (FSC: 350, SSC: 350, SBFP2: 220, Cerulean: 242, Citrine: 220, mCherry: 245, iRFP: 460) throughout all measurements and controlled for consistency of the instrument by using SPHERO RainBow Calibration particles (Cat no. 559123, BD).

Co-transfection experiment

We experimentally co-transfected five different fluorescent protein genes (SBFP2 [46], Cerulean [47], Citrine [48], mCherry [49], iRFP [50]; S30 Fig), individually driven by an Ef1a promoter and analyzed them via flow cytometry. The amount of transfected DNA (ng) was adjusted according to each plasmid’s size (nominal ratio: SBFP2: Cerulean: Citrine: mCherry: iRFP = 504: 634: 400: 239: 249). We collected more than 1,000,000 events and stringently gated the live population (~750,000 cells). This experiment created a five-dimensional distribution of fluorescent values.

Fan-out gene circuit experiment

We transfected five gene cassettes on individual plasmids as depicted in Fig 2A into HEK293 cells and activated the circuit through the addition of Doxycycline at eight different input modulation levels (0nM, 0.90nM, 3.15nM, 0.01uM, 0.05uM, 0.13uM, 0.45uM and 1.35uM). After 72h post-transfection we analyzed the induced cells using flow cytometry and collected more than 1,000,000 events per replicate (n = 3). The obtained data were subjected to our analysis pipeline as outlined in section Data Analysis.

Model

Parameters

In-silico flow cytometry simulations

Data analysis