Conceived and designed the experiments: AAC TK AEM AS UA. Performed the experiments: AAC NGZ TD II NP. Analyzed the data: AAC TK AEM. Contributed reagents/materials/analysis tools: RBK RM AS. Wrote the paper: AAC TK AEM UA.
The authors have declared that no competing interests exist.
A current challenge in biology is to understand the dynamics of protein circuits in living human cells. Can one define and test equations for the dynamics and variability of a protein over time? Here, we address this experimentally and theoretically, by means of accurate time-resolved measurements of endogenously tagged proteins in individual human cells. As a model system, we choose three stable proteins displaying cell-cycle–dependant dynamics. We find that protein accumulation with time per cell is quadratic for proteins with long mRNA life times and approximately linear for a protein with short mRNA lifetime. Both behaviors correspond to a classical model of transcription and translation. A stochastic model, in which genes slowly switch between ON and OFF states, captures measured cell–cell variability. The data suggests, in accordance with the model, that switching to the gene ON state is exponentially distributed and that the cell–cell distribution of protein levels can be approximated by a Gamma distribution throughout the cell cycle. These results suggest that relatively simple models may describe protein dynamics in individual human cells.
A goal of systems biology is to understand the dynamics of protein levels, as proteins are produced and degraded. One aims for a mathematical description of these processes that captures the essentials and that can be understood intuitively.
Most current models of protein dynamics have been established and tested in micro-organisms. Proteins in bacterial cells, for example, are well described by production-degradation equations. These show that protein mass increases exponentially over time in growing cells, and that protein concentration approaches steady-state with exponential decays
Much less is known about the cell-cell variability of protein dynamics in human cells
To address this, we experimentally followed the dynamics of selected human proteins in human cancer cells and test simple models to describe these dynamics. The experiments were made possible by recent advances in obtaining dynamics of endogenously expressed proteins in individual human cells at very high resolution and accuracy
We find that the proteins are found to follow either an approximately linear or a quadratic dependence on time. They also differ in their onset time of production and in their variability across the cell cycle. We find that these features can be understood in terms of a single unified model. Approximately linear accumulation results from short mRNA half-life, and quadratic accumulation from long mRNA half-life, as suggested by transcription inhibition experiments. Cell-cell variability of all three proteins is consistent with a stochastic model with slow gene ON- and OFF-transitions suggested by Raser and O'shea
We used an approach for dynamic proteomics to measure the protein levels of PRC1 (protein regulator of cytokinesis 1, NM_003981), ANLN (anillin, NM_018685) and YB1 (Y-box binding protein-1, NM_004559) in individual living human cancer cells over several cell cycles. Cell clones were used in which these genes are tagged at their endogenous chromosomal locus with yellow fluorescent protein (EYFP) as an internal exon. The proteins are expressed as full length proteins tagged with fluorescent YFP (
The cells were followed using time-lapse fluorescence microscopy in incubated conditions controlled for temperature, humidity and CO2. This resulted in time-lapse movies over days of growth, showing about 10–20 cells per frame at a 20 minute temporal resolution. Movies were analyzed using automated image analysis software
The three proteins showed cell-cycle dependent dynamics (
(A) Each image is an overlay of the fluorescence of the tagged protein on the corresponding phase-contrast image. One cell for each tagged protein is shown. Images are ordered according to the fraction of the time elapsed between two division events. The cells are automatically centered and neighboring cells are removed for clarity. The percentage of elapsed cell cycle is indicated at the bottom. Bar denotes 10 µm. Note the asymmetric partition of fluorescently tagged PRC1 between the two daughter cells following cell division. (B,C) For each protein, PRC1 and YB1, tracks of 150 cells are plotted. Each line denotes the total cell fluorescence level as measured during the cell cycle of a single cell. Cells are
The second protein studied here, YB1, is a transcription and translational regulator related to cell cycle events and other processes
The third protein ANLN, is an actin bundling protein crucial for abscission at the final stage of the cytokinesis and is involved in coordinating the contractile activity of myosin
The degradation of these proteins in the early phase of the cell cycle is useful for the present study because it minimizes the effects of proteins carried over from previous generations. In practice, for PRC1 there was no correlation across cell cycles (R = 0.07±0.09 where R is the average Pearson correlation coefficient of accumulated protein levels at equal time points at two consecutive cell cycles of individual cells ), while YB1 showed weak correlation between daughter and mother cells (R = 0.38±0.18). See also
We calibrated the YFP fluorescence from the movies in terms of number of tagged protein concentrations and total number per cell. Calibration of concentrations was achieved using fluorescence correlation spectroscopy inside the cell's cytoplasm
Image analysis of the time-lapse movies allowed accurate quantitation of the total protein level in each cell, defined as the total YFP fluorescence within the cell boundaries. Cells were synchronized
We first consider the average protein accumulation over the cell population, and then, in the next sections, the variability between cells. Protein accumulation was considered for each cell as the increase of fluorescence above the basal fluorescence at the end of the degradation event that occurs after division (see
We find that the mean protein level increases over the cell cycle. PRC1 showed a quadratic increase with time,
(A) PRC1 total cell protein accumulation exhibits an approximately quadratic profile while (B) YB1 accumulation profile is approximately linear. Each grey line denotes accumulation dynamics of an individual cell. Protein accumulation onset time was defined for all cells as the time at which protein levels reached their minimum level. Black circles denote the mean total fluorescence. Red line is the fit calculated according to a simple model of transcription and translation as schematically shown in (C).
To understand the different accumulation profiles, we used a classical model of transcription and translation
In contrast to this quadratic accumulation, the approximately linear accumulation of YB1 is predicted to mainly result from a high degradation rate of its mRNA. In this case the mRNA concentration reaches a steady-state concentration, and as a result protein level increases linearly with time,
To test the model predictions, we inhibited transcription by adding α-amanitin, a commonly used transcription inhibitor
We found that after adding α-amanitin, PRC1 protein levels began to accumulate linearly with time, as compared to the quadratic accumulation without the drug (
(A) Mean of total fluorescence of tagged PRC1 with (dashed line) and without (solid line) transcription inhibitor added at time 0 hrs. Following addition of transcription inhibitor protein levels continue to rise, though linearly instead of quadratically. This suggests that PRC1 is translated from stable mRNA. (B) Mean of total fluorescence of tagged YB1 with (dashed line) and without (solid line) transcription inhibitor added at time 0 hrs. Following addition of transcription inhibitor protein levels cease to rise, suggesting YB1 is translated from unstable mRNA. Best fit of simple differential equations regarding mRNA and protein levels suggest that the inhibitor starts taking affect at time
Fitting the differential equations describing mRNA and protein levels to measured protein levels of YB1 following addition of α-amanitin (detailed in
(A) Cells are classed into three groups based on the time of their division relative to the time of administration of α-amanitin, a transcription inhibitor (time = 0 hrs). Top group – division
This indicates that mRNA is degraded after cell division. This agrees with previous micro array experiments on synchronized HeLa cell populations that show a sharp drop in PRC1 and ANLN mRNA levels after cell division
So far we have discussed the average behavior of the cell population observed in the movies. We now turn to analyze the
We quantified this variation between cells using the noise strength, defined by ratio of the variance to the mean. Thus, noise strength, is NS = variance/mean. We note that a second common measure, the coefficient of variation CV = std/mean was more susceptible to noise in the present data. We find that the NS increased over time for PRC1, and appeared to saturate with time for YB1 (
(A) Noise strength (std2/mean) of PRC1 rises linearly (blue circles) and of YB1 saturates (red circles). Both 2-step and 3-step stochastic models reproduce these profiles of noise strength (blue solid line for PRC1 and red solid line for YB1). (B) Time interval between end of degradation to beginning of accumulation for PRC1 in blue and YB1 in red. The time of accumulation has an exponential tail (decay rate of 0.68±0.24 hr−1 for PRC1 and of 1.88±0.44 hr−1 for YB1). (C) Schematic of three stage model, with slow transitions between open and closed DNA. (D) Parameter estimation based on the slow switching model. As explained in the text, several parameters are estimated directly from the data, for PRC1; kon = 0.68±0.24, αr = 0 and αp = 0. For YB1; kon = 1.88±0.44, αr = 0.77±0.2 and αp = 0. For the rest of the parameters, the model was fit to data (mean and noise strength) and kp and kr were determined for a range of koff. Red and blue circles denote YB1 and PRC1 respectively. Values of average mRNA levels over the cell cycle and koff are shown for several parameters sets (filled circles).
We also found that different cells began to significantly accumulate the proteins at different times after cell division. In particular, there appeared to be a large cell-cell variation of such ‘onset times’ for PRC1 (
To understand the present data on cell-cell variability, we employed models for stochasticity in protein expression. The observed exponential distribution of times for the onset of protein accumulation is consistent with a slow process that turns expression ON with a timescale of hours. This corresponds to a picture, suggested in previous studies
We also compared the data to a model with no gene switching. This model is a straightforward stochastic version of the model of
In summary, the slow-switching model in which genes switch from OFF to ON on the timescale of hours seems to generate the large observed variability naturally, because of the long times between opening and closing events, resulting in relatively rare bursts of mRNA production
Theoretical studies
We found that Gamma distributions describe the observed experimental data well at all times. Gamma distributions seem to describe the data better than other empirically suggested distributions, such as Gaussian or log-normal distributions (
(A) Example of cell tracks (each line denotes PRC1-tagged fluorescence measured in a single cell), Distributions of fluorescence levels at 3, 7 and 10 hours following accumulation onset are shown. Fit to log-normal (red) and Gamma (blue) distributions are overlaid. (B) Ratio between maximum likelihood estimation (MLE) of log-normal and gamma distributions was computed at a temporal resolution of 20 minutes for PRC1 (blue) and YB1 (red). Cell-cell distributions fit Gamma distribution better (ratio<1) across the cell cycle. (C) Variation of the effective Gamma distribution parameters
Tracking changes of protein levels at the individual cell level over time allows following the mixing of protein level ranking between cells over time.. Cellular mixing occurs when a cell lineage, given enough time, reaches the different states found in a snapshot of a cell population. Mixing was measured using two approaches (detailed by Sigal
We measured mixing across the entire cell cycle and during the shorter period of protein accumulation. While cells exhibit faster mixing for PRC1 at the beginning of the cell cycle compared to YB1 (
Mixing is measured by auto-correlation and an ergodicity measure. In both measurements, cells are ranked according to tagged protein fluorescence at the beginning of the movie. Ergodicity is defined as the measure of the fraction of the total ranks that each cell traversed as a function of time. (A) Mixing of cells synchronized to the
This study presented an experimental and theoretical analysis of the detailed dynamics and variability of selected proteins in human cells. This analysis was made possible by an experimental advance that allows highly accurate, time-resolved measurement of proteins expressed from their native chromosomal position, and under their natural regulation, in individual cells. We chose proteins that are degraded upon cell division and that accumulate throughout the cell cycle, and which do not significantly carry memory of their levels in the previous cell cycle.
We find either a linear accumulation with time or a quadratic accumulation with time. Linear accumulation is found to correspond to short mRNA life-time, and quadratic accumulation to long mRNA life-time, in accord with simple transcription-translation equations. The theoretical predictions were tested by using a transcription inhibitor, which showed how quadratic profiles turn to linear, and linear profiles turn to constant protein levels, as predicted.
Note that linear and quadratic accumulation of total protein/cell are quite different from the typical patterns found in bacteria, in which total protein levels typically rise exponentially (or nearly exponentially) across the cell cycle.
We also studied the cell-cell variability in the protein levels. A model with slow transitions between ON and OFF gene states captures the observed variability, including the exponential distribution of onset times and the Gamma-shaped cell-cell distribution of protein levels. The extent of the noise seems to be too large to be explained by stochastic models that lack slow ON-OFF transitions.
In addition to single-time point statistics, we also considered the rate at which protein levels mix so that cells higher or lower than average return to average. Except for a short period after cell division, in which memory of previous levels is rapidly lost, the protein levels seem to show a small extent of mixing, with accumulation trajectories that are similarly shaped and roughly parallel for different cells in the population. Thus, the proteins display distinct cell-cell individuality over the cell cycle.
One may ask whether the different accumulation profile shapes we observed for YB1 and PRC1 correspond to their biological function. PRC1 is essential for the final stage of the cell cycle
This study raises the hope that relatively simple models might accurately capture the behavior of protein dynamics. Future work can explore whether such models can be used as building blocks to understand more complex protein circuitry. It would be important to further test this approach by measurements of additional proteins. For this purpose, the present experimental approach may be of use, especially in conjunction with a comprehensive library of tagged human proteins
PRC1 and YB1 were chosen from a library of tagged proteins in the H1299 non-small cell lung carcinoma cell line. This library was constructed as described in Sigal
The present library has an additional feature which is crucial for accurate image analysis, and which goes beyond
Time-lapse movies at 20× magnification were obtained as described
α-amanitin (A2263 Sigma), was dissolved in DMSO (hybri-max, D2650 Sigma) for a stock solution of 1 mg/ml. Cells were grown and filmed as explained in previous sections. After ∼24 hours of growth in the microscope, growth medium (2 ml) was gently replaced by growth medium with α-amanitin (100 µg/ml in 2 ml) under the microscope.
Custom image analysis software performed cell tracking and segmentation, and background and bleaching corrections (only very low levels of bleaching ∼3% occur in this experiment). Segmentation was applied to images after flat-field correction and background subtraction
Protein accumulation onset was defined for each of the cells as the time at which protein level reached its minimum level. To compute the time delay between end of degradation to beginning of accumulation, the minimum level of each cell was subtracted from its profile. A threshold chosen to be 1 percent of the maximum mean protein levels was computed. For each cell the first and last time points in which the threshold was crossed were denoted as the end of degradation and beginning of accumulation phases, respectively. Results are robust to threshold levels between 0.03–3% of the maximum level.
Calibration between number of EYFP molecules and measured fluorescence levels was carried out using Fluorescence Correlation Spectroscopy (FCS)
A second method for calibration used purified protein standards in the following way: 5 µL of purified GFP solution containing ∼1012 molecules of GFP were placed under a slide cover slip with diameter of 13 mm. Images with linearly increasing exposure times were taken in order to compute the gain of the mean pixel level. Number of GFP molecules per pixel was computed (∼7700) and compared to the mean pixel fluorescence (∼3000 fluorescence A.U. per 1000 msec exposure), and summed up to ∼0.4 fluorescence counts per molecule.
Time-lapse movies were acquired for several hours in 2 ml RPMI medium. Then the medium was switched to 2 ml of medium containing 100 nM of α-amanitin and time –lapse movie acquisition was continued. Cells were manually chosen so that only cells that were in the accumulation stage and did not divide for 8 hours following drug adittion were used. To compare to theory, mean of data was fit to two functions. The first describing the mean behavior until the inhibitor becomes fully active (∼2 hrs) and the second from that stage allowing mRNA degradation rate to be obtained (see
The auto-correlation function was computed with a robust estimator
The ergodic metric
Immunoblots of YB1, PRC1, and ANLN with anti-GFP. Estimated molecular weight of each protein without yellow fluorescent tag is: YB1 ∼36 kDa, PRC1 ∼71 kDa and ANLN ∼124 kDa. YFP ∼27 kDa.
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Protein distribution within cellular compartments during cell cycle. (A,B) Mean of total fluorescent levels in different compartments of the cell (cyan - cytoplasm, magenta - nucleus and black- whole cell) for PRC1 and YB1. (C,D) Mean of mean fluorescent levels in different compartments of the cell (cyan - cytoplasm, magenta - nucleus) for PRC1 and YB1. (E) Mean of total fluorescent levels of 2 proteins tagged with mCherry in the PRC1 and YB1 clones. The mCherry fluorescence and distribution is used for cell segmentation and tracking.
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Correlation across consecutive cell cycles. (A,B) cells were tracked for two consecutive cell cycles. Shown is the correlation matrix of each time point in first cell cycle and all time points in the second cell cycle for PRC1 and YB1 respectively. Note that PRC1 doesn't retain any memory across cell cycles while YB1 shows relatively high correlation between fluorescent levels of a mother cell and fluorescent levels in daughter cell.
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Response of single cells expressing tagged PRC1 at different stages of their cell cycle to transcription inhibitor reveal mRNA degradation at beginning of cell cycle. (A) Cells were divided into three groups based on the time of their division relative to the time of administration of α-aminitin, a transcription inhibitor (time = 0 hrs). Top group - division before drug addition, middle group - division during drug addition and bottom group - division after drug addition. Left panel - 2 ml medium was exchanged with 2 ml fresh medium (at t = 0 hrs), right panel - 2 ml medium was exchanged with 2 ml fresh medium containing 100 µg/ml α-aminitin (at t = 0 hrs). Each grey line denotes measurements of a single cell, black line is a chosen cell for eye guidance. Note that as cell divide later in the experiment the slope of protein levels decreases. (B) all trajectories of right panel overlaid and color ordered based on time of cell division, early divisions in blue and late divisions in red.
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Single cell trajectories display linear profiles in the case of YB1 and quadratic profiles in the case of PRC1. Each single cell trajectory denoting fluorescent levels measured over the cell cycle was fit to a polynomial function of first and second degree. Plotted are the pearson correlation coefficient values (R2) of the experimental data and the fit for PRC1 (red) and YB1 (blue). All trajectories of YB1 showed R2>0.9 already in the linear fit, while PRC1 showed R2>0.9 for all cells only when using a polynomial of 2nd degree.
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Coefficient of Variance (CV) of PRC1 and YB1. CV of PRC1 (blue) and YB1 (red) of protein levls across cells synchronized to the beginning of protein accumulation. Inset denotes the CV of the first 4 hours.
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Cell cycle dependent behavior of ANLN. (A) Example of one cell automatically tracked through the cell cycle. Each image is an overlay of fluorescent tagged protein on phase contrast image. Images are ordered according to the fraction of the time elapsed between two division events. The cells are automatically centered. The percentage of elapsed cell cycle is indicated at the bottom, Bar, 10 µm. Note asymmetric division of fluorescently tagged PRC1 between the two daughter cells following cell division. (B,C) For each protein, PRC1 and YB1 tracks of 150 cells are plotted. Each line denotes the total fluorescence level as measured during the cell cycle of a single cell. Cells are synchronized to beginning of cell cycle. Each cell has different cell cycle length (PRC1: μ = 21.6±2 hrs, YB1: 17.2±2 hrs). Both proteins are degraded following cell division.
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Cell cycle dependent behavior of ANLN. (A) Mean of total fluorescent levels in different compartments of the cell (cyan - cytoplasm, magenta - nucleus and black- whole cell). (B) Mean of mean fluorescent levels in different compartments of the cell (cyan - cytoplasm, magenta - nucleus). (C) The same as Legend of
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Time-lapse movie of transmitted light images of the clone with YFP CD-tagged PRC1. Movie duration is 46 hours. (time-lapse: 1 frame per 20 minutes).
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Time-lapse movie of yellow fluorescence images of the clone with YFP CD-tagged PRC1. Movie duration is 46 hours. (time-lapse: 1 frame per 20 minutes).
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Time-lapse movie of yellow fluorescence images overlaid on transmitted light images of the clone with YFP CD-tagged PRC1. Movie duration is 46 hours. (time-lapse: 1 frame per 20 minutes).
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Time-lapse movie of transmitted light images of the clone with YFP CD-tagged YB1. Movie duration is 46 hours. (time-lapse: 1 frame per 20 minutes).
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Time-lapse movie of yellow fluorescence images of the clone with YFP CD-tagged YB1. Movie duration is 46 hours. (time-lapse: 1 frame per 20 minutes).
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Time-lapse movie of yellow fluorescence images overlaid on transmitted light images of the clone with YFP CD-tagged YB1. Movie duration is 46 hours. (time-lapse: 1 frame per 20 minutes).
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Time-lapse movie of transmitted light images of the clone with YFP CD-tagged ANLN. Movie duration is 46 hours. (time-lapse: 1 frame per 20 minutes).
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Time-lapse movie of yellow fluorescence images of the clone with YFP CD-tagged ANLN. Movie duration is 46 hours. (time-lapse: 1 frame per 20 minutes).
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Time-lapse movie of yellow fluorescence images overlaid on transmitted light images of the clone with YFP CD-tagged ANLN. Movie duration is 46 hours. (time-lapse: 1 frame per 20 minutes).
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We thank the Kahn Family Foundation and the Israel Science Foundation for support. We thank Michael Elbaum and his lab for discussions and assistance with the calibration of fluorescence levels.