Non-genetic inheritance restraint of cell-to-cell variation

Heterogeneity in physical and functional characteristics of cells (e.g. size, cycle time, growth rate, protein concentration) proliferates within an isogenic population due to stochasticity in intracellular biochemical processes and in the distribution of resources during divisions. Conversely, it is limited in part by the inheritance of cellular components between consecutive generations. Here we introduce a new experimental method for measuring proliferation of heterogeneity in bacterial cell characteristics, based on measuring how two sister cells become different from each other over time. Our measurements provide the inheritance dynamics of different cellular properties, and the ‘inertia’ of cells to maintain these properties along time. We find that inheritance dynamics are property specific and can exhibit long-term memory (∼10 generations) that works to restrain variation among cells. Our results can reveal mechanisms of non-genetic inheritance in bacteria and help understand how cells control their properties and heterogeneity within isogenic cell populations.

One of the main challenges in biological physics today is to quantitatively predict the change in cells' physical and functional characteristics over time. Cellular characteristics are regulated by genetic and non-genetic (proteins, RNA, chemicals, etc.) factors that interact in order to determine the state of the cell at all times. While genetic information passed from generation to the next is the main scheme, by which cells conserve their characteristics, non-genetic cellular components are also transferred between consecutive generations and thus influence the state of the cell in future generations 1,2 . The mechanism of genetic information transfer between generations, as well as how this information is expressed, are mostly understood [3][4][5] . This information can be altered by rare occurring processes such as mutations, lateral gene transfer, or gene loss 6,7 . Therefore, changes resulting from genetic alterations emerge over very long timescales (several 10s of generations).
On the other hand, inheritance of non-genetic cellular components, which are subject to a considerable level of fluctuations, can influence cellular characteristics at shorter timescales [8][9][10][11] .
Here we focus on understanding how robust cellular characteristics are to intrinsic sources (stochastic gene expression and division noise) and extrinsic sources (environmental fluctuations) of variation, and how cells that emerge from a single mother develop distinct features and over what time scale. This requires understanding and quantitatively characterizing 1) how intrinsic and extrinsic sources of variation contribute to the proliferation of heterogeneity in a population, and 2) how non-genetic inheritance contributes to the maintenance of cellular properties along time.
While our understanding of variation sources have increased significantly over the past two decades [12][13][14] , progress in understanding non-genetic inheritance and its contribution to restraining the proliferation of heterogeneity has been extremely limited. Extensive studies have been dedicated to revealing the different non-genetic mechanisms that influence specific cellular processes and how they are inherited over time [15][16][17][18][19] . However, since the state of the cell (or its phenotype) is determined by the integration of multiple processes, the inheritance dynamics of most cellular phenotypes cannot be predicted by characterizing the effect of individual inheritance mechanisms separately. Instead, there is a need to measure and characterize the inheritance dynamics of the phenotype directly. Progress in this research has been drastically hindered by the limited experimental techniques that can provide reliable quantitative measurements.
The recent development of the microfluidic device known as the "mother machine", has allowed us to trap single bacterial cells and follow their growth and division, as well as their protein expression dynamics for hundreds of generations 20,21 . These measurements have been used by several groups to gain insight into non-genetic inheritance and cellular memory of different cellular properties. The results obtained have consistently showed that non-genetic memory in bacteria is almost completely erased within one generation 20,22,23 . This consensus is founded on the calculation of the autocorrelation function (ACF) for the different measurable cellular properties, such as cell size, growth rate, cell cycle time, and protein content. It is important to note that in calculating the autocorrelation function, measurements of cells from different traps of the mother machine are averaged together. These cells might experience slightly different environments at different times resulting from thermal fluctuations and their dynamic interaction with their surroundings. As a result of the individuality of the cell-environment interaction, different microniches can be created in different traps 23,24 . Thus, averaging over many traps erases the dynamics of cellular memory.
To overcome this hurdle, we have developed a new measurement technique, which enables us to separate environmental effects from cellular ones. The new technique is based on a new microfluidic device that allows trapping two cells immediately after they divide from a single mother simultaneously, and sustain them right next to each other for tens of generations. This enables us to measure how two cells that originate from the same mother become different over time, while experiencing exactly the same environment. Thus, we are able to measure the nongenetic memory of bacterial cells for several different traits. Our results reveal important features of cellular memory. We find that different traits of the cell exhibit different memory patterns with distinct timescales. While the cell cycle time and cell size exhibit slow exponential decay of their memory that extends over several generations, other cellular features exhibit complex memory dynamics over time. The growth rates of two sister cells, for example, diverge immediately after division, but re-converge towards the end of the first cell cycle and subsequently persist together for several generations. In comparison, the mean fluorescence intensities, reporting gene expression, are identical in both cells immediately after they separate but diverge within two cell cycles.
Our new microfluidic device, dubbed the "sisters machine" (Fig. 1a), is similar to the mother machine used in earlier studies 14,15,21 . It consists of 30µm long narrow trapping channels (1µm × 1µm) open at one end to a wide channel (30 µm × 30µm), through which fresh medium is continuously pumped to supply nutrients to cells in the traps and wash away cells that are pushed out of them. Here however, every two neighboring trapping channels are joined on the closed end through a v-shaped connection of the same width and height. The tip of the v-shaped connection is made 0.5µm narrower than the rest of the channel to reduce the likelihood of cells passing from one side to the other (Fig. 1b). Therefore, once it happens, the cells at the tip will remain there, while we track their growth and division events, and measure their size and protein expression ( Fig. 1c and d), until the next cell passage occurs, which can take many 10s of generations (see Supplementary movie). The environment in this setup is identical for both cells at the tip of the vshaped connection, as they are kept in close proximity to each other. This ensures that differences observed between the two cells are due to internal cellular factors only.
Using this setup, we successfully trapped pairs of cells next to each other for 20 -160 generations. Images of the cells in both DIC and fluorescence modes were acquired every 3 minutes and used to measure various cellular characteristics as a function of time, including cell size, protein concentration, growth rate, and generation time. To measure cellular memory, we replace the ACF, used in previous studies, with the Pearson correlation function (PCF) between pairs of cells: where is the cellular property of interest, t is the measurement time, n is the number of cell pairs measured, is the population standard deviation of , and (1) and (2)  and their lineages are aligned artificially even though they can be measured at different times. In this case, t is measured relative to the alignment point, which is chosen to be at the start of the cell cycle for both cells. Since NCs and RPs do not originate from the same mother at time 0, the PCF is measured from the first generation only, and we set it to be 1 at time 0. Comparing the correlation of NCs, which experience the same environmental conditions at the same time, with that of RPs allows us to determine the effect of the environment on the correlation. On the other hand, the comparison of SCs with NCs provides the effect of cellular factors (i.e. epigenetics) that are shared between SCs, on the correlation function. This in turn allows us to determine the cellular memory of a specific property resulting from shared information passed on from the mother to the two sisters.
We measured the correlations between the different pair types for cell cycle time (T) ( Fig.   S1 and S2). We find that T of SCs remain strongly correlated for up to 8 successive cell divisions ( Fig. 2b) regardless of the environmental conditions ( Fig. S3), while the NCs correlation decays to zero within 3 generation (Fig 2c). These results clearly reveal the effects of epigenetics and environmental conditions on cellular memory when compared to the RPs correlation, which as expected decays to zero within one generation similar to the ACF (Fig. 2b and c).
Next, we applied our method to cell size. Also here, our measurements show that SCs correlation decays slowly over ~7 generations (Fig. 2d), while the correlation of NCs exhibit fast decay to zero within 2 generations similar to the ACF (Fig. 2e). Note that RPs exhibit no correlation from the start of the measurement (Fig.2d and e). These results further demonstrate the existence of strong non-genetic memory that restrains the variability of cell size between SCs for a long time. Unlike the cell cycle time however, the effect of both epigenetic factors and environmental conditions on the cellular memory, appears to extend for a slightly shorter time.
To quantify the increase in variability among cells along time differently, we measured the change in the variance of a cellular property as time advances, which is expected to reach an equilibrium saturation value at long timescales. Measuring how the variance reaches saturation, provides information about cellular memory and the nature of forces acting to restrain variation.
The cellular memories of cell cycle time and length, measured using this method, agree well with our previous PCF results ( Fig. S5 and S6). Thus, we have measured the relative fluctuations in the exponential elongation rate of the cell pairs , defined as: where ( ) = ln ( ) is the exponential elongation rate of the cell, ( ) is the cell length at time t, and (1) and (2) distinguish the cell pair ( Fig. S7a-c). As expected, for all pairs of lineages is randomly distributed with < >=0 (Fig. S7d), as the elongation rate of all cells fluctuate about a fixed value, identical for all cells in the population and depends on the experimental conditions.
The variance of for both RPs ( 2 ( )) and NCs ( 2 ( )), was found to be constant over time and is similar for both types of cell pairs (Fig. 3a). However, the variance of for SCs ( 2 ( )) exhibits a complex pattern (Fig. 3b), which eventually converges to the same value as 2 ( ) and 2 ( ). The time it takes for 2 ( ) to reach saturation extends over almost 10 generations, which again reflects a long memory resulting from epigenetic factors. These results show that, unlike cell cycle time and cell length, elongation rates of SCs immediately after their division from a single mother exhibit the largest variation. This variation decreases to its minimum value within a single cell cycle time (~30 min.). To understand the source of this large variation immediately following separation, we have measured the growth rate over a moving time window of 6 minutes throughout the cell cycle, and compared the results between SCs. Our comparison clearly shows that a SC that receives a smaller size-fraction from its mother exhibits a larger growth rate immediately after division. The growth rate difference between the small and large sisters, decreases to almost zero by the end of the first cell cycle after separation (Fig. 3b inset).
This result reveals that the exponential growth rate of a cell immediately after division inversely scales with the size-fraction the cell receives from its mother. It also demonstrates that the difference in the growth rates between SCs changes during the cell cycle indicating that they are not constant throughout the whole cycle as has been accepted so far 20,25,26 .
We have also examined how the protein concentration varies over time between the two cells by measuring the concentration of GFP (green fluorescent protein), via its fluorescence intensity, expressed from a constitutive promoter in a medium copy-number plasmid. The variance of fluorescence intensity difference between cell pairs , was calculated as for the growth rate (see Fig. S8 for details). Upon division, soluble proteins are partitioned symmetrically with both daughters receiving almost the same protein concentration. As expected, 2 starts from zero initially, and diverges to reach saturation within 2 generations (Fig. 3c). On the other hand, NCs and RPs exhibit constant variance throughout the whole time. 2 is twice as large as 2 , which reflects the influence of the shared environment resulting in additional correlations between NCs. The relatively short-term memory in protein concentration, may be protein specific (Fig. S8), or it could reflect the fact that in this case the protein is expressed from a plasmid. Nevertheless, this result indicates that cellular properties are controlled differently and can exhibit distinct memory patterns. It is important therefore to distinguish between different cellular characteristics and to examine their inheritance patterns individually.
There has been a rising interest over the past two decades in understanding the contribution of epigenetic factors to cellular properties and their evolution over time. Here, we introduce a new measurement technique that can separate environmental fluctuations from cellular processes. This allows for quantitative measurement of non-genetic memory in bacteria, and reveals its contribution to restraining the variability of cellular properties. Our results show that the restraining force dynamics vary significantly among different cellular properties, and its effects can extend up to ~10 generations. In addition, the growth rate variation emphasizes the effect of division asymmetry, which can help in understanding the mechanism that controls cellular growth rate. The slow increase in the growth rate variance that follows, reflects the effect of inheritance.
Since both cells inherit similar content, which ultimately determines the rate of all biochemical activities in the cell and thus its growth rate, it is expected that both cells would exhibit similar growth rates once they make up for the uneven partitioning of size acquired during division. The Finally, in order to understand and characterize the evolution of population growth rate as it reflects its fitness, there is a need to incorporate inheritance effects, which has been thus far assumed to be short lived. This study confirms that cellular memory can persist for several generations, and therefore limits the variation in certain cellular characteristics, including growth rate. Such memory should be considered in future studies and has the potential of changing our perception of population growth and fitness.

METHODS SUMMARY
Device fabrication. The master mold of the microfluidic device was fabricated in two layers.
Initially, the growth channels for the cells were printed on a 1mm x 1 mm fused silica substrate using Nanoscribe Photonic professional (GT). The second layer, containing the main flow channels that supply nutrients and wash out excess cells, was formed using standard soft lithography techniques 27,28 . SU8 2015 photoresist (MicroChem, Newton, MA) was spin coated onto the substrate to achieve a layer thickness of 30 μm and cured using maskless aligner MLA100 (Heidelberg Instruments). Following a wash step with SU8 developer, the master mold was baked and salinized. The experimental setup described in the main text was then prepared using this master mold, from PDMS prepolymer and its curing agent (Sylgard 184, Dow Corning) as described in previous studies. Image acquisition, and data analysis. Images of the channels were acquired every 3 minutes (in LB medium) or 7 minutes (in M9CL medium) in DIC and fluorescence modes using a Nikon eclipse Ti2 microscope with a 100x objective. The size and protein content of the sister cells were measured from these images using the image analysis software Oufti 30 . The data were then used to generate traces such as in Fig. 1d, and for further analysis as detailed in the main text.
Single-cell measurements were analyzed using MATLAB. Sample autocorrelation functions, Pearson correlation coefficients, sample distributions and curve fitting were all calculated by their implementations in MATLAB toolboxes.  of SCs, on the other hand, exhibits large variance immediately after separation (~50% higher than NCs and RPs) and rapidly drops to its minimum value within one generation time (~30 minutes), and increases thereafter for 4 hours (~8 generations) until saturating at a fixed value equivalent to that observed for NCs and RPs. Each point in a and b is the average over 3 frames moving window (6 minutes), and the error bars represent the standard deviation of that average. The inset depicts the growth rate difference between a cell and its larger sister calculated over a 3 frame moving window during the first generation after separation. The values are normalized by the average growth rate of two cells. Also here, we see that the difference between the two cells is largest immediately after separation but it degreases to almost zero by the end of the cell cycle. (c) Unlike , of SCs increases to its saturation value within ~2 generation (see Fig. S8 for the details of the calculation). Here, each point represents the average of three different experiments, and the error bars are their standard deviation.

PCF and error calculation
The PCF was calculated using following equation: and the standard deviation 1 : Where n is the number of cell pairs considered in the calculation  (Fig. S2).

Fig. S1. Distributions of different cell parameters.
In order to avoid artifacts arising in calculations due to differences between experiments carried out on different days, raw data from these experiments was normalized by subtracting the mean (μ) and dividing by the standard deviation (σ) for each experiment separately. Later, this normalized data was combined and used for calculating the PCF and variances for different parameters.       Fig. S5. 2 for SCs increase much slower than that of NCs and RPs, and reaches saturation at a fixed value after ~7 generations (mean lifetime ~3.5 generations) similar to the time scale observed in the PCF. Each point in the graph represent an average over a 6 frames window (15 minutes), and the error bars represent the window size for the x value and the standard deviation of the window average for the y value. The lines depict the best fits to the data described in the graphs.