Candle

    To estimate the single-cell MscL abundance via microscopy, we needed to determine a calibration factor that could translate arbitrary fluorescence units to protein copy number. To compute this calibration factor, we relied on a priori knowledge of the mean copy number of MscL-sfGFP for a particular bacterial strain in specific growth conditions. In Bialecka-Fornal et al. 2012 (1), the average MscL copy number for a population of cells expressing an MscL-sfGFP fusion (E. coli K-12 MG1655 \(\phi\)(mscL-sfGFP)) cells was measured using quantitative Western blotting and single-molecule photobleaching assays. By growing this strain in identical growth and imaging conditions, we can make an approximate measure of this calibration factor. In this section, we derive a statistical model for estimating the most-likely value of this calibration factor and its associated error.

Definition of a calibration factor

    We assume that all detected fluorescence signal from a particular cell is derived from the MscL-sfGFP protein, after background subtraction and correction for autofluorescence. The arbitrary units of fluorescence can be directly related to the protein copy number via a calibration factor \(\alpha\), \[ I_\text{tot} = \alpha N_\text{tot}, \qquad(1)\] where \(I_\text{tot}\) is the total cell fluorescence and \(N_\text{tot}\) is the total number of MscL proteins per cell. Bialecka-Fornal et al. report the average cell Mscl copy number for the population rather than the distribution. Knowing only the mean, we can rewrite Eq. 1 as \[ \langle I_\text{tot}\rangle = \alpha \langle N_\text{tot} \rangle, \qquad(2)\] assuming that \(\alpha\) is a constant value that does not change from cell to cell or fluorophore to fluorophore.

    The experiments presented in this work were performed using non-synchronously growing cultures. As there is a uniform distribution of growth phases in the culture, the cell size distribution is broad the the extremes being small, newborn cells and large cells in the process of division. As described in the main text, the cell size distribution of a population is broadened further by modulating the MscL copy number with low copy numbers resulting in aberrant cell morphology. To speak in the terms of an effective channel copy number, we relate the average areal intensity of the population to the average cell size, \[ \langle I_\text{tot} \rangle = \langle I_A \rangle \langle A \rangle = \alpha \langle N_\text{tot} \rangle, \qquad(3)\] where \(\langle I_A\rangle\) is the average areal intensity in arbitrary units per pixel of the population and \(\langle A \rangle\) is the average area of a segmented cell. As only one focal plane was imaged in these experiments, we could not compute an appropriate volume for each cell given the highly aberrant morphology. We therefore opted to use the projected two-dimensional area of each cell as a proxy for cell size. Given this set of measurements, the calibration factor can be computed as \[ \alpha = {\langle I_A \rangle\langle A \rangle \over \langle N_\text{tot} \rangle}. \qquad(4)\]

    While it is tempting to use Eq. 4 directly, there are multiple sources of error that are important to propagate through the final calculation. The most obvious error to include is the measurement error reported in Bialecka-Fornal et al. 2012 for the average MscL channel count (1). There are also slight variations in expression across biological replicates that arise from a myriad of day-to-day differences. Rather than abstracting all sources of error away into a systematic error budget, we used an inferential model derived from Bayes' theorem that allows for the computation of the probability distribution of \(\alpha\).

Estimation of \(\alpha\) for a single biological replicate

    A single data set consists of several hundred single-cell measurements of intensity, area of the segmentation mask, and other morphological quantities. The areal density \(I_A\) is computed by dividing the total cell fluorescence by the cell area \(A\). We are interested in computing the probability distributions for the calibration factor \(\alpha\), the average cell area \(\langle A \rangle\), and the mean number of channels per cell \(\langle N_\text{tot} \rangle\) for the data set as a whole given only \(I_A\) and \(A\). Using Bayes' theorem, the probability distribution for these parameters given a single cell measurement, hereafter called the posterior distribution, can be written as \[ g(\alpha, \langle A \rangle, \langle N_\text{tot} \rangle\,\vert\, A, I_A) = {f(A, I_A\,\vert \, \alpha,\langle A \rangle, \langle N_\text{tot} \rangle) g(\alpha,\langle A \rangle, \langle N_\text{tot} \rangle) \over f(\alpha, I_A)}, \qquad(5)\] where \(g\) and \(f\) represent probability density functions over parameters and data, respectively. The term \(f(A, I_A\,\vert\, \alpha, \langle A \rangle, \langle N_\text{tot} \rangle)\) in the numerator represents the likelihood of observing the areal intensity \(I_A\) and area \(A\) of a cell for a given values of \(\alpha\), \(\langle A \rangle\), and \(\langle N_\text{tot} \rangle\). The second term in the numerator \(g(\alpha,\langle A \rangle, \langle N_\text{tot} \rangle)\) captures all prior knowledge we have regarding the possible values of these parameters knowing nothing about the measured data. The denominator, \(f(I_A, A)\) captures the probability of observing the data knowing nothing about the parameter values. This term, in our case, serves simply as a normalization constant and is neglected for the remainder of this section.

    To determine the appropriate functional form for the likelihood and prior, we must make some assumptions regarding the biological processes that generate them. As there are many independent processes that regulate the timing of cell division and cell growth, such as DNA replication and peptidoglycan synthesis, it is reasonable to assume that for a given culture the distribution of cell size would be normally distributed with a mean of \(\langle A \rangle\) and a variance \(\sigma_{\langle A \rangle}\). Mathematically, we can write this as \[ f(A\,\vert\,\langle A \rangle, \sigma_{\langle A \rangle}) \propto {1 \over \sigma_{\langle A \rangle}}\exp\left[-{(A - \langle A \rangle)^2 \over 2\sigma_{\langle A \rangle}^2}\right], \qquad(6)\] where the proportionality results from dropping normalization constants for notational simplicity.

     While total cell intensity is intrinsically dependent on the cell area the areal intensity \(I_A\) is independent of cell size. The myriad processes leading to the detected fluorescence, such as translation and proper protein folding, are largely independent, allowing us to assume a normal distribution for \(I_A\) as well with a mean \(\langle I_A \rangle\) and a variance \(\sigma_{I_A}^2\). However, we do not have knowledge of the average areal intensity for the standard candle strain a priori. This can be calculated knowing the calibration factor, total MscL channel copy number, and the average cell area as \[ I_A = {\alpha\langle N_\text{tot} \rangle \over \langle A \rangle}. \qquad(7)\] Using Eq. 7 to calculate the expected areal intensity for the population, we can write the likelihood as a Gaussian distribution, \[ f(I_A\,\vert\,\alpha,\langle A \rangle,\langle N_\text{tot} \rangle, \sigma_{I_A}) \propto {1 \over \sigma_{I_A}}\exp\left[-{\left(I_A - {\alpha \langle N_\text{tot} \rangle\over \langle A \rangle}\right)^2 \over 2 \sigma_{I_A}^2}\right]. \qquad(8)\]

    With these two likelihoods in hand, we are tasked with determining the appropriate priors. As we have assumed normal distributions for the likelihoods of \(\langle A \rangle\) and \(I_A\), we have included two additional parameters, \(\sigma_{\langle A \rangle}\) and \(\sigma_{I_A}\), each requiring their own prior probability distribution. It is common practice to assume maximum ignorance for these variances and use a Jeffreys prior (2), \[ g(\sigma_{\langle A \rangle}, \sigma_{I_A}) = {1 \over \sigma_{\langle A \rangle}\sigma_{I_A}}. \qquad(9)\]

    The next obvious prior to consider is for the average channel copy number \(\langle N_\text{tot} \rangle\), which comes from Bialecka-Fornal et al. 2012. In this work, they report a mean \(\mu_N\) and variance \(\sigma_N^2\), allowing us to assume a normal distribution for the prior, \[ g(\langle N_\text{tot}\rangle\,\vert\, \mu_N,\sigma_N) \propto {1 \over \sigma_N}\exp\left[-{(\langle N_\text{tot} \rangle - \mu_N)^2 \over 2 \sigma_N^2}\right]. \qquad(10)\] For \(\alpha\) and \(\langle A \rangle\), we have some knowledge of what these parameters can and cannot be. For example, we know that neither of these parameters can be negative. As we have been careful to not overexpose the microscopy images, we can say that the maximum value of \(\alpha\) would be the bit-depth of our camera. Similarly, it is impossible to segment a single cell with an area larger than our camera's field of view (although there are biological limitations to size below this extreme). To remain maximally uninformative, we can assume that the parameter values are uniformly distributed between these bounds, allowing us to state \[ g(\alpha) = \begin{cases} {1 \over \alpha_\text{max} - \alpha_\text{min}} & \alpha_\text{min} \leq \alpha \leq \alpha_\text{max} \\ 0 & \text{otherwise} \end{cases}, \qquad(11)\] for \(\alpha\) and \[ g(\langle A \rangle) = \begin{cases} {1 \over \langle A \rangle_\text{max} - \langle A \rangle_\text{min}} & \langle A \rangle_\text{min} \leq \langle A \rangle \leq \langle A \rangle_\text{max}\\ 0 & \text{otherwise} \end{cases} \qquad(12)\] for \(\langle A \rangle\).

Piecing Eq. 6 through Eq. 12 together generates a complete posterior probability distribution for the parameters given a single cell measurement. This can be generalized to a set of \(k\) single cell measurements as \[ \begin{aligned} g(\alpha,\langle A \rangle, \langle N_\text{tot} \rangle, \sigma_{I_A}, \sigma_{\langle A \rangle}\,\vert\, [I_A, A], \mu_N, \sigma_N) &\propto {1 \over (\alpha_\text{max} - \alpha_\text{min})(\langle A \rangle_\text{max} - \langle A \rangle_\text{min})}{1 \over (\sigma_{I_A}\sigma_{\langle A \rangle})^{k+1}}\,\times\\ &{1 \over \sigma_N}\exp\left[- {(\langle N_\text{tot}\rangle - \mu_N)^2 \over 2\sigma_N^2}\right] \prod\limits_i^k\exp\left[-{(A^{(i)} - \langle A \rangle)^2 \over 2\sigma_{\langle A \rangle}^2} - {\left(I_A^{(i)} - {\alpha \langle N_\text{tot}\rangle \over \langle A \rangle}\right)^2 \over 2\sigma_{I_A}^2}\right] \end{aligned}, \qquad(13)\] where \([I_A, A]\) represents the set of \(k\) single-cell measurements.

    As small variations in the day-to-day details of cell growth and sample preparation can alter the final channel count of the standard candle strain, it is imperative to perform more than a single biological replicate. However, properly propagating the error across replicates is non trivial. One option would be to pool together all measurements of \(n\) biological replicates and evaluate the posterior given in Eq. 13. However, this by definition assumes that there is no difference between replicates. Another option would be to perform this analysis on each biological replicate individually and then compute a mean and standard deviation of the resulting most-likely parameter estimates for \(\alpha\) and \(\langle A \rangle\). While this is a better approach than simply pooling all data together, it suffers a bias from giving each replicate equal weight, skewing the estimate of the most-likely parameter value if one replicate is markedly brighter or dimmer than the others. Given this type of data and a limited number of biological replicates (\(n = 6\) in this work), we chose to extend the Bayesian analysis presented in this section to model the posterior probability distribution for \(\alpha\) and \(\langle A \rangle\) as a hierarchical process in which \(\alpha\) and \(\langle A \rangle\) for each replicate is drawn from the same distribution.

A hierarchical model for estimating \(\alpha\)

    In the previous section, we assumed maximally uninformative priors for the most-likely values of \(\alpha\) and \(\langle A \rangle\). While this is a fair approach to take, we are not completely ignorant with regard to how these values are distributed across biological replicates. A major assumption of our model is that the most-likely value of \(\alpha\) and \(\langle A \rangle\) for each biological replicate are comparable, so long as the experimental error between them is minimized. In other words, we are assuming that the most-likely value for each parameter for each replicate is drawn from the same distribution. While each replicate may have a unique value, they are all related to one another. Unfortunately, proper sampling of this distribution requires an extensive amount of experimental work, making inferential approaches more attractive.

This approach, often called a multi-level or hierarchical model, is schematized in Fig. 1. Here, we use an informative prior for \(\alpha\) and \(\langle A \rangle\) for each biological replicate. This informative prior probability distribution can be described by summary statistics, often called hyper-parameters, which are then treated as the "true" value and are used to calculate the channel copy number. As this approach allows us to get a picture of the probability distribution of the hyper-parameters, we are able to report a point estimate for the most-likely value along with an error estimate that captures all known sources of variation.

    The choice for the functional form for the informative prior is often not obvious and can require other experimental approaches or back-of-the-envelope estimates to approximate. Each experiment in this work was carefully constructed to minimize the day-to-day variation. This involved adhering to well-controlled growth temperatures and media composition, harvesting of cells at comparable optical densities, and ensuring identical imaging parameters. As the experimental variation is minimized, we can use our knowledge of the underlying biological processes to guess at the approximate functional form. For similar reasons presented in the previous section, cell size is controlled by a myriad of independent processes. As each replicate is independent of another, it is reasonable to assume a normal distribution for the average cell area for each replicate. This normal distribution is described by a mean \(\tilde{\langle A \rangle}\) and variance \(\tilde{\sigma}_{\langle A \rangle}\). Therefore, the prior for \(\langle A \rangle\) for \(n\) biological replicates can be written as \[ g(\langle A \rangle\, \vert\, \tilde{\langle A \rangle}, \tilde{\sigma}_{\langle A \rangle}) \propto {1 \over \tilde{\sigma}_{\langle A \rangle}^n}\prod\limits_{j=1}^{n}\exp\left[-{(\langle A \rangle_j - \tilde{\langle A \rangle})^2 \over 2 \tilde{\sigma}_{\langle A \rangle}^2}\right]. \qquad(14)\] In a similar manner, we can assume that the calibration factor for each replicate is normally distributed with a mean \(\tilde{\alpha}\) and variance \(\tilde{\sigma}_\alpha\), \[ g(\alpha\,\vert\,\tilde{\alpha}, \tilde{\sigma}_\alpha) \propto {1 \over \tilde{\sigma}_\alpha^n}\prod\limits_{j=1}^n \exp\left[-{(\alpha_j - \tilde{\alpha})^2 \over 2\tilde{\sigma}_\alpha^2}\right]. \qquad(15)\]

    With the inclusion of two more normal distributions, we have introduced four new parameters, each of which needing their own prior. However, our knowledge of the reasonable values for the hyper-parameters has not changed from those described for a single replicate. We can therefore use the same maximally uninformative Jeffreys priors given in Eq. 9 for the variances and the uniform distributions given in Eq. 11 and Eq. 12 for the means. Stitching all of this work together generates the full posterior probability distribution for the best-estimate of \(\tilde{\alpha}\) and \(\tilde{\langle A \rangle}\) shown in Eq. 2 given \(n\) replicates of \(k\) single cell measurements, \[ \begin{aligned} g(\tilde{\alpha}, \tilde{\sigma}_\alpha, \tilde{\langle A \rangle}, \tilde{\sigma}_{\langle A \rangle}, &\{\langle N_\text{tot} \rangle, \langle A \rangle, \alpha, \sigma_{I_A}\}\,\vert\, [I_A, A], \mu_N, \sigma_N) \propto\\ &{1 \over (\tilde{\alpha}_\text{max} - \tilde{\alpha}_\text{min})(\tilde{\langle A \rangle}_\text{max} - \tilde{\langle A \rangle}_\text{min})\sigma_N^n(\tilde{\sigma}_\alpha\tilde{\sigma}_{\langle A \rangle})^{n + 1}}\,\times\,\\ &\prod\limits_{j=1}^n\exp\left[-{(\langle N \rangle_j^{(i)} - \mu_N)^2 \over 2\sigma_N^2} - {(\alpha_j - \tilde{\alpha})^2 \over 2\tilde{\sigma}_\alpha^2} - {(\langle A \rangle_j - \tilde{\langle A \rangle})^2 \over 2\tilde{\sigma}_{\langle A \rangle}^2}\right]\,\times\,\\ &{1 \over (\sigma_{{I_A}_j}\sigma_{\langle A \rangle_j})^{k_j + 1}}\prod\limits_{i=1}^{k_j}\exp\left[-{(A_j^{(i)} - \langle A \rangle_j)^2 \over 2\sigma^{(i)2}_{\langle A \rangle_j}} - {\left({I_A}_{j}^{(i)} - {\alpha_j \langle N_\text{tot}\rangle_j \over \langle A \rangle_j}\right)\over 2\sigma_{{I_A}_j}^{(i)2}}\right] \end{aligned}, \qquad(16)\] where the braces \(\{\dots\}\) represent the set of parameters for biological replicates and the brackets \([\dots]\) correspond to the set of single-cell measurements for a given replicate.

    While Eq. 16 is not analytically solvable, it can be easily sampled using Markov chain Monte Carlo (MCMC). The results of the MCMC sampling for \(\tilde{\alpha}\) and \(\tilde{\langle A \rangle}\) can be seen in Fig. 2. From this approach, we found the most-likely parameter values of \(3300^{+700}_{-700}\) a.u. per MscL channel and \(5.4^{+0.4}_{-0.5}\) \(\mu\)m\(^2\) for \(\tilde{\alpha}\) and \(\tilde{\langle A \rangle}\), respectively. Here, we've reported the median value of the posterior distribution for each parameter with the upper and lower bound of the 95% credible region as superscript and subscript, respectively. These values and associated errors were used in the calculation of channel copy number.

Effect of correction

     The posterior distributions for \(\alpha\) and \(\langle A \rangle\) shown in Fig. 2 were used directly to compute the most-likely channel copy number for each measurement of the Shine-Dalgarno mutant strains, as is described in the coming section (Logistic Regression). The importance of this correction can be seen in Fig. 3. Cells with low abundance of MscL channels exhibit notable morphological defects, as illustrated in Fig. 3A. While these would all be considered single cells, the two-dimensional area of each may be comparable to two or three wild-type cells. For all of the Shine-Dalgarno mutants, the distribution of projected cell area has a long tail, with the extremes reaching 35 µm\(^2\) per cell (Fig. 3B). Calcuating the total number of channels per cell does nothing to decouple this correlation between cell area and measured cell intensity. Fig. 3C shows the correlation between cell area and the total number of channels without normalizing to an average cell size \(\langle A \rangle\) differentiated by their survival after an osmotic down-shock. This correlation is removed by calculating an effective channel copy number shown in Fig. 3D.