Estimating intrinsic and extrinsic noise from single-cell gene expression measurements

1 Introduction

A gene can have different expression levels in living cells that have the same genetic material and are subject to the same environment (Stegle et al., 2015). During early development of an organism, distinct expression profiles eventually lead to formation of different tissues. Moreover, complex tissues such as brain have many different subtypes of cells with different gene expression profiles. However, variation in expression between cells is reflective not only of distinct biological state, but also of stochasticity underlying many of the processes fundamental to the molecular biology of cell.

In a classic paper on the stochasticity of gene expression in single cells, Elowitz et al. (2002) introduced a clever two-reporter expression assay designed to tease apart “intrinsic” and “extrinsic” variation (also called “noise”) from the overall variability in gene expression: the intrinsic noise is the variation in the expression of the same gene in identical environment, whereas the extrinsic noise is the variation in gene expression due to cellular environment that impacts all the genes at once. The idea is as follows: two identically regulated reporter genes (cyan fluorescent protein and yellow fluorescent protein) are inserted into individual E. coli. cells, allowing for comparable expression measurements within and between cells. If n cells are assayed, this leads to expression measurements c₁, … c_n and y₁, … y_n, where the pair (c_i, y_i) represent the expression measurements for the cyan and yellow reporters in the ith cell. The goal of the experiment is to measure the variance in gene expression from the pairs (c_i, y_i) (denoted by ηtot2) and to ascribe it to two different sources: first, variability due to the different states of cells (“extrinsic noise,” denoted by ηext2), and second, inherent variability that exists even when the state of cells is fixed (“intrinsic noise,” denoted by ηint2). In Elowitz et al. (2002), these noise terms are defined as squared coefficients of variation and specific formulas are provided for estimating ηext2,ηint2 and ηtot2 (hereafter referred to as the ELSS estimates):

where c¯=1n∑i=1nci and y¯=1n∑i=1nyi.

Hilfinger and Paulsson (2011) later interpreted these estimates in terms of the “law of total variance” (explained in the next section), which sheds light on the statistical basis of the ELSS estimators but does not address questions about their statistical properties. In this paper, we derive the bias and mean squared error of the ELSS estimators and examine their optimality. We also examine the geometric and biological interpretation of the estimators.

The processes that lead to the expression of the reporters (or genes in general) are much more complex than described here, e.g. the models described in the paper ignore the effects of translation. Many studies (e.g. Rausenberger and Kollmann 2008 and Komorowski et al. 2013) have developed detailed mathematical models for these processes. While some of our results may generalize and be relevant in more general settings, we restrict our analysis to the intrinsic and extrinsic noise as examined by Elowitz et al. (2002) and accessible via static reporter expression experiments. Analyses are implemented in the R package noise available on CRAN.

2 A hierarchical model

We begin by introducing a hierarchical model that provides a formal model for the experiments of Elowitz et al. (2002) and that provides insight into the numerators of (1,2,3). They are the key components of the Elowitz et al. (2002) formulas and can be viewed as estimators of true variances. We note that lower case letters such as c_i and y_i denote observations not only in the ELSS formulas but throughout our paper; we reserve uppercase letters for random variables.

A hierarchical model for expression of the two reporters in a cell emerges naturally from the assumption that reporter expression, conditioned on the same cellular environment, is represented by independent and identically distributed random variables. To allow each cell to be different from the others, we introduce independent identically distributed random variables Z_i, for i = 1, …, n that represent the environments of cells [as in Hilfinger and Paulsson (2011)]. Consistent with Elowitz et al. (2002), we posit that the cellular conditional random variables associated to the two reporters have the same distribution F with mean M_i and variance Σi2, both parameters being unique to the i th cell:

and

Thinking of a two reporter experiment as “random,” in the sense that the states of cells Z₁, … Z_n are random, across cells we have

and

where G is the distribution of all the M_is, with mean μ and variance σμ2, and H that of all the Σi2s, with mean σ² and variance ϵ. In other words, both the mean and variance of reporter expression level is cell specific and the random variable Σi2 and its mean σ² represent the “within-cell” variation as distinguished from the parameter σμ2 which represents the “between-cell” variability in the ANOVA setting.

For any i, the mean of C_i or Y_i is μ, according to the following calculation:

The total variance in C_i (or Y_i) can be calculated using the “law of total variance”:

Using the notation of the hierarchical model described above, and dropping the subscripts for expectation because they are clear by context, we have, for any i,

With this notation equation (7) becomes

This means that the marginal (unconditional) distributions of C_i and Y_i are identical:

where the marginal distribution F′ may or may not be the same as the conditional distribution F.

In the next sections, we will derive the estimators for extrinsic and intrinsic noise, and examine the bias and MSE of each estimator. Specifically, for any estimator S, the MSE of S with respect to the true parameter τ is calculated as follows:

where E[S] − τ is the bias of S.

3 Extrinsic noise

To examine estimators for extrinsic noise, we start with the law of total variance, noting that the within-cell variability Var[E[C_i|Z_i]] can be written as:

This connection between the extrinsic noise, the law of total variance and the covariance of C_i and Y_i was noted in Hilfinger and Paulsson (2011).

Formula (11) leads to the following unbiased estimator for the extrinsic noise, as it is an unbiased estimator estimator for the covariance:

We note that the ELSS estimator (2) uses the scalar 1/n, which unlike the case of the intrinsic noise estimator (1) leads to a biased estimator in this case.

In order to find the estimator that minimizes the MSE, we consider the following general estimator:

We assume that M_i is normal and that μ = 0 and ϵ = 0. The MSE of S_ext is

which is minimized when

The last step in (12) is due to Equations (9), (10) and (11):

It is interesting to note that (12) comprises two parts: the first, (n−1)(1+1n) converges to n − 1 as n → ∞, while the second, (σ2+σμ2σμ2)2 is equal to 1ρ2 where ρ is the correlation between the two reporter expression vectors C and Y. See Appendices A and B for more details.

4 Intrinsic noise

Also starting with the law of total variance, the within-cell variability E[Var[Ci|Zi]] for cell i can be written as:

This leads to the following unbiased estimator for the intrinsic noise:

To find the estimator that minimizes the MSE, we consider estimators of the following general form

Assuming normality of the distribution G (i.e. cell-specific means M_i follow a normal distribution), as well as μ = 0 and ϵ = 0, the MSE is given by

The value of a that minimizes this expression is

See Appendices A and C for the complete derivation.

The analysis above can be simplified with an additional assumption, namely that C¯ = Ȳ. In some experiments this may be a natural assumption to make, whereas in others the condition is likely to be violated; we comment on this in more detail in the discussion. Here we proceed to note that assuming that C¯ = Ȳ, the estimator (15) simplifies to

The unbiased estimator with this form is easily derived by observing that

Thus, in order for S~int to be unbiased the parameter a must be equal to n. The resulting formula is the ELSS formula in (1). This makes clear that the assumption C¯ = Ȳ underlies the derivation of the ELSS intrinsic noise estimator.

In order to study the mean squared error and derive an estimator that minimizes it, we again assume normality of G. The MSE of Sint is then given by

Assuming again that μ = 0 and ϵ = 0, the MSE simplifies to

which is minimized when a = n + 2 (see Appendices A and D for the complete derivation).

5 Geometric interpretation

Figure 3A of Elowitz et al. (2002) shows a scatterplot of data (c_i, y_i) for an experiment and suggests thinking of intrinsic and extrinsic noise geometrically in terms of projection of the points onto a pair of orthogonal lines. While this geometric interpretation of noise agrees exactly with the ELSS intrinsic noise formula, the interpretation of extrinsic noise is more subtle. Here we complete the picture.

Figure 1

Geometric interpretation of intrinsic and extrinsic noise. The intrinsic noise, or the within-cell variability, is the variance of the points projected to the line y = −c, which is perpendicular to y = c. In other words, it is the average of the squared lengths 12(yi−ci)2. The red point is the projection of point (c_i, y_i) onto the line y = c. The green point is the centroid (c¯, ȳ) (or ((c¯+y¯)/2, 0) after projection) under the assumption that the two means are equal. See the main text for detail. The extrinsic noise, or the between-cell variability, is the sample covariance between c_i and y_i. The colored triangles around the blue point ( a randomly selected data point) illustrate the geometric interpretation of the sample covariance: it is the average (signed) area of triangles formed by pairs of data points: green triangles in Q1 and Q3 (some not shown) represent a positive contribution to the covariance, whereas the magenta triangles in Q2 and Q4 a negative contribution. Since most data points lie in the 1st (Q1) and 3rd (Q3) quadrants relative to the blue point, most of the contribution involving the blue point is positive. Similarly, since most pairs of data points can be connected by a positively signed line, their positive contribution will result in a positive covariance. In Elowitz et al. (2002) the direction along the line y = c is labeled extrinsic, which makes sense in terms of the intuition for positive sample covariance. However we have placed that label “extrinsic” in quotes because the extrinsic noise estimator corresponding directly to the sample variance for points projected onto the line y = c (in analogy with intrinsic noise) is heavily biased and not usable in practice.

To understand the intuition behind Figure 3A in Elowitz et al. (2002), we have redrawn it in a format that highlights the math (Figure 1). The projection of a point (c_i, y_i) onto the line y = c is the point ((yi+ci)/2,(yi−ci)/2), shown as the red point in Figure 1. Assuming equal means (c¯ = ȳ), the intrinsic noise, as estimated by the unbiased estimator (1), is then the mean squared distance of the points from the line y = c.

The ELSS estimate for the extrinsic noise is the sample covariance. Intuitively, it indicates how the measurements of one reporter track that of the other across cells. The geometric meaning of the sample covariance in Figure 1 is based on an alternative formulation of sample covariance (Hayes, 2011):

This formulation of the sample covariance has the interpretation of being an average of the signed area of triangles associated to pairs of points. Figure 1 illustrates these signed triangles using a randomly selected point (the blue point). This formulation is very different from what might be considered at first glance an appropriate analogy to intrinsic noise, namely the sample variance along the line y = c.

An alternative estimate for the extrinsic noise based on the sample variance of the projected points along the line y = c (using the projected centroid as the mean, which is shown as the green point in Figure 1) turns out to be biased by an amount equal to the total noise. This sample variance averages the squared distances of the data points from the centroid (green point) after projection onto the line y = c; see the distance between the red and green points in Figure 1. Since

the bias is

which is the true total noise.

The above calculation also shows that if the intrinsic and extrinsic noise are both estimated as variances along the projections to the lines y = −c and y = c respectively, then the total noise will be overestimated by a factor of two.

In summary, the caption to Figure 3A in Elowitz et al. (2002) is completely accurate in stating that “Spread of points perpendicular to the diagonal line on which CFP and YFP intensities are equal corresponds to intrinsic noise, whereas spread parallel to this line is increased by extrinsic noise.” However the geometric interpretation of covariance makes it precise how an increase in extrinsic noise relates to the spread of points in the direction of the line y = c.

6 Practical considerations

7 Re-analysis of published two-reporter experiment data

Michael Elowitz and Peter Swain have kindly shared with us their data published in Elowitz et al. (2002). Here we focus on the data in Figure 3A of their paper, which contain the unnormalized fluorescence intensities of CFP and YFP in the E. coli. strain D22 and in strain M22. We normalized the data as follows such that the resulting scatterplots are close to Figure 3A:

where ci* and yi* are the unnormalized intensity of the CFP and YFP, respectively, in the ith cell, c*¯ the sample mean, and sy* the sample standard deviation. The normalized intensities are close to normal distributions, and all four distributions have mean 1. At a sample size of over 200, the different estimators in Table 4 give essentially the same result. Additionally, the ratio of the estimated extrinsic and total noise is close to the sample correlation, verifying our theoretical result.

Nam Ki Lee and Sora Yang have also kindly shared with us their data published in Yang et al. (2014). Here we analyze the data in Figure 3A of their paper, which are the expression levels (intensities) of two reporters, CFP and mCherry (also see Sec. 6.2). The shared, unnormalized intensities have very different sample means (Table 4). Application of the estimators in Table 1 to these data gives two different estimates of the intrinsic noise, with the ELSS estimate being nearly three times the estimates under the equal mean assumption. To normalize the data, we removed the few negative values, log₂ transformed the data, and quantile normalized between the two reporters (see summary statistics in Table 4). Applying our estimators to the normalized data, all estimates are consistent with one another. This analysis illustrates the importance of the equal mean assumption: when this assumption is not satisfied, the ELSS estimator leads to overestimation of the intrinsic noise.

Additionally, we subsampled from these data sets and assessed the performance of the estimators as the sample size decreased. At each sample size, we repeated the subsampling 1000 times and computed the mean and standard deviation of the noise estimates (Table 5). Whereas the means of the estimates do not differ from those obtained using the entire data sets, the variation (measured by the standard deviation) increases quickly with decreasing sample sizes. For the Elowitz et al. data, the standard deviation in the estimates roughly doubles for both types of noise as the sample size halves. Comparing the standard deviation to the mean suggests that 200 is indeed a reasonable sample size for estimates with small variation (compare with their actual sample sizes of 284 and 250 for the two strains). For the Yang et al. data, the increase in the standard deviation is much less drastic, and 200 also appears a decent sample size for reasonably small variation in the estimates.

8 Conclusions and discussion

Our hierarchical model for Elowitz et al. (2002) provides statistically interpretable parameters representing intrinsic and extrinsic noise, and allows for the derivation of estimators with optimality guarantees. Furthermore, the model highlights experimental assumptions that need to be satisfied for the estimators to be valid, specifically that the two reporters need to have the same distribution (within a cell) and hence normalization may be necessary. Whereas similar hierarchical models have been proposed before to study heterogeneity among single cells (see, e.g. Finkenstädt et al., 2013, and Koeppl et al., 2012), our hierarchical model explicitly parameterize the two types of noise, and reveals their equivalence to other quantities, as indicated by (11) and (14), which enable derivation of closed-form estimators of these parameters (summarized in Table 1). We use bias and MSE to explicitly evaluate the performance of different estimators, and recognize the asymptotic equivalence of multiple estimators.

Other experiments have been set up to explore and assess intrinsic and extrinsic noise, and some of our results may be useful in those settings. For example, Volfson et al. (2006) used a single reporter but two Saccharomyces cerevisiae strains, with one strain containing only one copy of the reporter, and the other strain two copies. Assuming no strain effect, which may be thought of as batch effect, the authors applied the following estimators for (unscaled) intrinsic and extrinsic noise (consistent with their notation, and without the denominator of C¯Ȳ as used in the ELSS estimators in Table 1):

where V₁ and V₂ are the variance in the 1-copy and 2-copy strains, respectively, and V_i and V_e are intrinsic and extrinsic noise, respectively. These estimators are in fact consistent with (11) and (14) under our hierarchical model:

Together, (19) and (20) give rise to (17) and (18). Note that (19) and (20) imply that the extrinsic noise is also the covariance here, except that the covariance is between the 1-copy and 2-copy strains with the same reporter; this is also pointed out by Sherman et al. (2015). Additionally, the total (marginal) noise of the reporter is the sum of intrinsic and extrinsic noise (19). However, consistent with our analysis of the assumptions of the hierarchical model, these estimators hold only when the variance for each single copy in the 2-copy strain is identical to that in the 1-copy strain. This is equivalent to assuming no strain (batch) effect, which can be a rather strong assumption.

We note that during the preparation of this manuscript, Erik van Nimwegen independently examined the Elowitz et al. (2002) paper form a Bayesian point of view (van Nimwegen, 2016).