Intercellular Variability in Protein Levels from Stochastic Expression and Noisy Cell Cycle Processes
[['The level of a protein can deviate considerably from cell-to-cell , in spite of the fact that cells are genetically-identical and are in the same extracellular environment [1–3] .', 'This intercellular variation or noise in protein counts has been implicated in diverse processes such as corrupting functioning of gene networks [4–6] , driving probabilistic cell-fate decisions [7–12] , buffering cell populations from hostile changes in the environment [13–16] , and causing clonal cells to respond differently to the same stimulus [17–19] .', 'An important source of noise driving random fluctuations in protein levels is stochastic gene expression due to the inherent probabilistic nature of biochemical processes [20–23] .', 'Recent experimental studies have uncovered additional noise sources that affect protein copy numbers .', 'For example , the time take to complete cell cycle ( i . e . , time between two successive cell-division events ) has been observed to be stochastic across organisms [24–32] .', 'Moreover , given that many proteins/mRNAs are present inside cells at low-copy numbers , errors incurred in partitioning of molecules between two daughter cells are significant [33–35] .', 'Finally , the time at which a particular gene of interest is duplicated can also vary between cells [36 , 37] .', 'We investigate how such noise sources in the cell-cycle process combine with stochastic gene expression to generate intercellular variability in protein copy numbers ( Fig 1 ) .', 'Prior studies that quantify the effects of cell division on the protein noise level have been restricted to specific cases .', 'For example , noise computations have been done in stochastic gene expression models , where cell divisions occur at deterministic time intervals [33 , 38 , 39] .', 'Recently , we have analyzed a deterministic model of gene expression with random cell-division events [40] .', 'Building up on this work , we formulate a mathematical model that couples stochastic expression of a stable protein with random cell-division events that follow a general class of probability distributions .', 'Moreover , at the time of cell division , proteins are randomly partitioned between two daughter cells based on a framework that allows the partitioning errors to be higher or lower than as predicted by binomial partitioning .', 'For this class of models , we derive an exact analytical formula for the protein noise level as quantified by the steady-state squared Coefficient of Variation ( CV2 ) .', 'This formula is further decomposed into individual components representing contributions from different noise sources .', 'A systematic investigation of this formula leads to novel insights , such as identification of regimes where increasing randomness in the timing of cell-division events decreases the protein noise level .', 'Next , we extend the above model to include genome-duplication events that increase the gene’s transcription rate [36 , 41] .', 'To our knowledge , this is the first study integrating randomness in the genome-duplication process with stochastic gene expression .', 'An exact formula for the protein noise level is derived for this extended model and used to investigate how the timing of duplication affects different noise components .', 'Counter intuitively , results show that doubling of the transcription rate within the cell cycle can lead to smaller fluctuations in protein levels as compared to a constant transcription rate through out the cell cycle .', 'Finally , we discuss how formulas obtained in this study can be used to infer parameters and characterize the gene expression process from single-cell studies .'], ['We consider the standard model of stochastic gene expression [42 , 43] , where mRNAs are transcribed at exponentially distributed time intervals from a constitutive gene with rate kx .', 'For the time being , we exclude genome duplication and the transcription rate is fixed throughout the cell cycle .', 'Assuming short-lived mRNAs , each transcription event results in a burst of proteins [43–45] .', 'The corresponding jump in protein levels is shown as', 'x ( t ) ↦ x ( t ) + B , ( 1 )', 'where x ( t ) is the protein population count in the mother cell at time t , B is a random burst size drawn from a positively-valued distribution and represents the number of protein molecules synthesized in a single-mRNA lifetime .', 'Motivated by observations in E . coli and mammalian cells , where many proteins have half-lives considerably longer than the cell-doubling time , we assume a stable protein with no active degradation [46–48] .', 'Thus , proteins accumulate within the cell till the time of cell division , at which point they are randomly partitioned between two daughter cells .', 'Let cell division events occur at times ts , s ∈ {1 , 2 , …} .', 'The cell-cycle time', 'T ≔ t s - t s - 1 , ( 2 )', 'follows an arbitrary positively-valued probability distribution with the following mean and squared coefficient of variation ( CV2 )', '〈 T 〉 = 〈 t s - t s - 1 〉 , C V T 2 = 〈 T 2 〉 - 〈 T 〉 2 〈 T 〉 2 , ( 3 )', 'where 〈 .', '〉 denotes expected value through out this paper .', 'The random change in x ( t ) during cell division is given by', 'x ( t s ) ↦ x + ( t s ) , ( 4 )', 'where x ( ts ) denotes the protein levels in the mother cell just before division and x+ ( ts ) denotes the protein levels in one of the daughter cells just after division .', 'Conditioned on x ( ts ) , x+ ( ts ) is assumed to have the following statistics', '〈 x + ( t s ) | x ( t s ) 〉 = x ( t s ) 2 , x + 2 ( t s ) - 〈 x + ( t s ) 〉 2 | x ( t s ) = α x ( t s ) 4 .', '( 5 )', 'The first equation implies symmetric partitioning , i . e . , on average each of the daughter cells inherits half the number protein molecules just before division .', 'The second equation in Eq ( 5 ) describes the variance of x+ ( ts ) and quantifies the error in partitioning of molecules through the non-negative parameter α .', 'For example , α = 0 represents deterministic partitioning where x+ ( ts ) = x ( ts ) /2 with probability equal to one .', 'A more realistic model for partitioning is each molecule having an equal probability of being in the each daughter cell [49–51] .', 'This results in a binomial distribution for x+ ( ts )', 'Probability { x + ( t s ) = j | x ( t s ) } = x ( t s ) ! j !', '( x ( t s ) - j ) !', '1 2 x ( t s ) , j ∈ { 0 , 1 , … , x ( t s ) } , ( 6 )', 'and corresponds to α = 1 in Eq ( 5 ) .', 'Interestingly , recent studies have shown that partitioning of proteins that form clusters or multimers can result in α > 1 in Eq ( 5 ) , i . e . , partitioning errors are much higher than as predicted by the binomial distribution [33 , 39] .', 'In contrast , if molecules push each other to opposite poles of the cell , then the partitioning errors will be smaller than as predicted by Eq ( 6 ) and α < 1 .', 'The model with all the different noise mechanisms ( stochastic expression; random cell-division events and partitioning errors ) is illustrated in Fig 2A and referred to as the full model .', 'We also introduce two additional hybrid models [52 , 53] , where protein production and partitioning are considered in their deterministic limit ( Fig 2B and 2C ) .', 'Note that unlike the full model , where x ( t ) takes non-negative integer values , x ( t ) is continuous in the hybrid models .', 'We will use these hybrid models for decomposing the protein noise level obtained from the full model into individual components representing contributions from different noise sources .', 'In order to quantify the steady-state protein mean and noise , we need to define the stochastic process that governs the timing of cell division .', 'Variations in the duration of cell cycle can result from a variety of factors , such as cell physiology , growth rate , cell size and expression of genes that affect cell-cycle time such as FtsZ [24–32] .', 'Given these complexities , we take a phenomenological approach to modeling cell-cycle time , and assume it to be an independent and identically distributed random variable that is drawn from a mixture of Erlang distributions ( also known as phase-type distribution ) .', 'The motivation for choosing this distribution is two fold: Consider a mixture of n Erlang distributions with mixing probabilities pi , i = {1 , … , n} .', 'Recall that an Erlang distribution of order i is the distribution of the sum of i independent and identical exponential random variables .', 'The cell-cycle time is assumed to have an Erlang distribution of order i with probability pi and can be represented by a continuous-time Markov chain with states Gij , j = {1 , … , i} , i = {1 , … , n} ( Fig 3 ) .', 'Let Bernoulli random variables gij = 1 if the system resides in state Gij and 0 otherwise .', 'The probability of transition Gij → Gi ( j+1 ) in the next infinitesimal time interval [t , t + dt ) is given by ikgij dt , implying that the time spent in each state Gij is exponentially distributed with mean 1/ik .', 'To summarize , at the start of cell cycle , a state Gi1 , i = {1 , … , n} is chosen with probability pi and cell division occurs after transitioning through i exponentially distributed steps .', 'Based on this formulation , the probability of a cell-division event occurring and a new cell cycle obtained from an Erlang distribution of size i starting in the next time interval [t , t + dt ) is given by k p i ∑ j = 1 n ( j g j j ) d t , and whenever the event occurs , the protein level changes as per Eq ( 4 ) .', 'Finally , the mean , the squared coefficient of variation , and the skewness of the cell-cycle time in terms of the Markov chain parameters are given by', '〈 T 〉 = 1 k , C V T 2 = ∑ i = 1 n p i i , Skewness = 〈 T 3 〉 - 3 〈 T 〉 ( 〈 T 2 〉 - 〈 T 〉 2 ) - 〈 T 〉 3 ( 〈 T 2 〉 - 〈 T 〉 2 ) 3 / 2 = 2 ∑ i = 1 n p i i 2 ( 7 )', '[55] , where 〈T3〉 is the third-order moment of the cell-cycle time .', 'An important property of this class of distributions is that increasing C V T 2 also makes the distribution highly skewed , because from Eq ( 7 ) both the CV and skewness are linear combinations of pi , albeit with different linear coefficients that decrease with i .', 'Considering that ∑ i = 1 n p i = 1 , the only way to increase C V T 2 is by increasing smaller-index components and decreasing larger-index components of the distribution ( i . e . increasing pi and decreasing pj , where i < j ) .', 'Since higher values of i are more penalized in the skewness equation , this would correspond to making the distribution more positively skewed .', 'Hence high values of C V T 2 also means high values of skewness , thus occurrences of longer cell cycles are more probable .', 'As we will shortly see , this property leads to mean protein levels being dependent on C V T 2 .'], ['All the models shown in Fig 2 are identical in terms of finding 〈x ( t ) 〉 and in principle any one of them could have been used .', 'We choose to analyze the full model illustrated in Fig 2A .', 'Time evolution of the statistical moments of x ( t ) can be obtained from the Kolmogorov forward equations corresponding to the full model in Fig 2A combined with the cell-division process described in Fig 3 .', 'We refer the reader to [52 , 56 , 57] for an introduction to moment dynamics for stochastic and hybrid systems .', 'Analysis in Appendix A in S1 Text shows', 'd 〈 x 〉 d t = k x 〈 B 〉 - k 2 ∑ j = 1 n j 〈 x g j j 〉 .', '( 8 )', 'Note that the time-derivative of the mean protein level ( first-order moment ) is unclosed , in the sense that , it depends on the second-order moment 〈xgij〉 .', 'Typically , approximate closure methods are used to solve moments in such cases [52 , 57–62] .', 'However , the fact that gij is binary can be exploited to automatically close moment dynamics .', 'In particular , since gij ∈ {0 , 1}', '〈 g i j n x m 〉 = 〈 g i j x m 〉 , n ∈ { 1 , 2 , … } ( 9 )', 'for any non-negative integer m .', 'Moreover , as only a single state gij can be 1 at any time', '〈 g i j g r q x m 〉 = 0 , if i ≠ r or j ≠ q .', '( 10 )', 'Using Eqs ( 9 ) and ( 10 ) , the time evolutions of 〈gij〉 and 〈xgij〉 are obtained as', 'd 〈 g i 1 〉 d t = k p i ∑ j = 1 n ( j 〈 g j j 〉 ) − i k 〈 g i 1 〉 , ( 11 ) d 〈 g i j 〉 d t = i k 〈 g i ( j − 1 ) 〉 − i k 〈 g i j 〉 , j = { 2 , … , i } , ( 12 ) d 〈 x g i 1 〉 d t = k x 〈 B 〉 〈 g i 1 〉 + k 2 p i ∑ j = 1 n ( j 〈 x g j j 〉 ) − i k 〈 x g i 1 〉 , ( 13 ) d 〈 x g i j 〉 d t = k x 〈 B 〉 〈 g i j 〉 − i k 〈 x g i j 〉 + i k 〈 x g i ( j − 1 ) 〉 , j = { 2 , … , i } ( 14 )', 'and only depend on 〈gij〉 and 〈xgij〉 ( see Appendix A in S1 Text ) .', 'Thus , Eqs ( 8 ) and ( 11 ) – ( 14 ) constitute a closed system of linear differential equations from which moments can be computed exactly .', 'To obtain an analytical formula for the average number of proteins , we start by performing a steady-state analysis of Eq ( 8 ) that yields', '∑ j = 1 n j 〈 x g j j 〉 ¯ = 2 k x 〈 B 〉 k , ( 15 )', 'where 〈 .', '〉 ¯ denotes the expected value in the limit t → ∞ .', 'Using Eq ( 15 ) , 〈 x g i 1 〉 ¯ is determined from Eq ( 13 ) , and then all moments 〈 x g i j 〉 ¯ are obtained recursively by performing a steady-state analysis of Eq ( 14 ) for j = {2 , … , i} .', 'This analysis results in', '〈 x g i j 〉 ¯ = k x 〈 B 〉 i k p i 1 + j i .', '( 16 )', 'Using Eqs ( 7 ) , ( 16 ) and the fact that ∑ i = 1 n ∑ j = 1 i g i j = 1 we obtain the following expression for the mean protein level', '〈 x 〉 ¯ = x ∑ i = 1 n ∑ j = 1 i g i j ¯ = ∑ i = 1 n ∑ j = 1 i 〈 x g i j 〉 ¯ = k x 〈 B 〉 〈 T 〉 3 + C V T 2 2 .', '( 17 )', 'It is important to point that Eq ( 17 ) holds irrespective of the complexity , i . e . , the number of states Gij used in the phase-type distribution to approximate the cell-cycle time distribution .', 'As expected , 〈 x 〉 ¯ increases linearly with the average cell-cycle time duration 〈T〉 with longer cell cycles resulting in more accumulation of proteins .', 'Consistent with previous findings , Eq ( 17 ) shows that the mean protein level is also affected by the randomness in the cell-cycle times ( C V T 2 ) [40 , 63] .', 'For example , 〈 x 〉 ¯ reduces by 25% as T changes from being exponentially distributed ( C V T 2 = 1 ) to periodic ( C V T 2 = 0 ) for fixed 〈T〉 .', 'Next , we determine the noise in protein copy numbers , as quantified by the squared coefficient of variation .', 'Recall that the full model introduced in Fig 2A has three distinct noise mechanisms .', 'Our strategy for computing the protein noise level is to first analyze the model with a single noise source , and then consider models with two and three sources .', 'As shown below , this approach provides a systematic dissection of the protein noise level into components representing contributions from different mechanisms .', 'The full model introduced in Fig 2 assumes that the transcription rate ( i . e . , the protein burst arrival rate ) is constant throughout the cell cycle .', 'This model is now extended to incorporate gene duplication during cell cycle , which increases the burst arrival ( transcription ) rate by f times ( Fig 5 ) .', 'Note that due to gene dosage compensation , doubling the genome does not always correspond to f = 2 [72–74] .', 'If f > 1 , then accumulation of proteins will be bilinear as illustrated in Fig 1 . As before , we again take a phenomenological approach to model the timing of gene duplication .', 'The cell-cycle time T is divided into two intervals: time from the start of cell cycle to gene duplication ( T1 ) , and time from gene duplication to cell division ( T2 ) .', 'T1 and T2 are independent random variables , each drawn from a mixture of Erlang distributions ( see Fig . B in S1 Text ) .', 'The mean cell-cycle duration and its noise can be expressed as', '〈 T 〉 = 〈 T 1 〉 + 〈 T 2 〉 , β = 〈 T 1 〉 〈 T 〉 , C V T 2 = β 2 C V T 1 2 + ( 1 - β ) 2 C V T 2 2 , ( 42 )', 'where C V X 2 denotes the squared coefficient of variation of the random variable X . An important variable in this formulation is β , which represents the average time of gene duplication normalized by the mean cell-cycle time .', 'Thus , β values close to 0 ( 1 ) imply that the gene is duplicated early ( late ) in the cell-cycle process .', 'Moreover , the noise in the gene-duplication time is controlled via C V T 1 2 . We refer the reader to Appendix F in S1 Text for a detailed analysis of the model in Fig 5 and only present the main results on the protein mean and noise levels .', 'The steady-state mean protein count is given by', '〈 x 〉 ¯ = k x 〈 B 〉 〈 T 1 〉 2 f ( 1 - β ) + 3 β + β C V T 1 2 2 + k x 〈 B 〉 〈 T 2 〉 3 f ( 1 - β ) + 4 β + f ( 1 - β ) C V T 2 2 2 , ( 43 )', 'and decreases with β , i . e . , a gene that duplicates early has on average , more number of proteins .', 'When β = 1 , then the transcription rate is kx throughout the cell cycle and we recover the mean protein level obtained in Eq ( 17 ) .', 'Similarly , when β = 0 the transcription rate is fkx and we obtain f times of the amount as in Eq ( 17 ) .', 'As per our earlier observation , more randomness in the timing of genome duplication and cell division ( i . e . , higher C V T 1 2 and C V T 2 2 values ) increases 〈 x 〉 ¯ .', 'Our analysis shows that the total protein noise level can be decomposed into three components', 'C V 2 = C V E 2 + C V R 2 + C V P 2 ( 44 )', 'where C V E 2 is the extrinsic noise from random genome-duplication/cell-division events , and the sum of the contributions from partitioning errors ( C V R 2 ) and stochastic expression ( C V P 2 ) is the intrinsic noise .', 'We refer the reader to Appendix F in S1 Text for noise formulas for any f , and only present formulas for f = 2 here .', 'In this case , the intrinsic noise is obtained as', 'C V R 2 + C V P 2 = 4 α ( 2 - β ) 3 ( β 2 - 4 β + 6 ) + β 2 C V T 1 2 + 2 ( 1 - β ) 2 C V T 2 2 1 〈 x 〉 ¯ ︷ C V R 2 + ( 10 - 8 β + 3 β 2 ) + 6 ( 1 - β ) 2 C V T 2 2 + 3 β 2 C V T 1 2 3 ( β 2 - 4 β + 6 ) + β 2 C V T 1 2 + 2 ( 1 - β ) 2 C V T 2 2 〈 B 2 〉 〈 B 〉 1 〈 x 〉 ¯ ︷ C V P 2 .', '( 45 )', 'Note that for β = 0 and 1 , we recover the intrinsic noise level in Eq ( 39 ) from Eq ( 45 ) .', 'Interestingly , for B = 1 with probability 1 and α = 1 , the intrinsic noise is always 1/Mean irrespective of the values chosen for C V T 1 2 , C V T 2 2 and β .', 'For high precision in the timing of cell-cycle events ( CVT1 → 0 , CVT2 → 0 )', 'CV2≈\u20094−3 ( β−2 ) 2β2︷CVE23 ( β2−4β+6 ) 2︸Extrinsic noise\u2009+\u20094α ( 2−β ) 3 ( β2−4β+6 ) 1〈x〉¯︷CVR2\u2009+ ( 10−8β+3β2 ) 3 ( β2−4β+6 ) 〈B2〉〈B〉1〈x〉¯︷CVP2︸Intrinsic noise , ( 46 )', 'where mean protein level is given by', '〈 x 〉 ¯ ≈ k x 〈 B 〉 〈 T 1 〉 4 - β 2 + k x 〈 B 〉 〈 T 2 〉 3 - β .', '( 47 )', 'We investigate how different noise components in Eq ( 46 ) vary with β as the mean protein level is held fixed by changing kx .', 'Fig 6 shows that C V P 2 follows a U-shaped profile with the optima occurring at β = 2 - 2 ≈ 0 .', '6 and the corresponding minimum value being ≈ 5% lower than its value at β = 0 .', 'An implication of this result is that if stochastic expression is the dominant noise source , then gene duplication can result in slightly lower protein noise levels .', 'In contrast to C V P 2 , C V R 2 has a maxima at β = 2 - 2 which is ≈ 6% higher than its value at β = 0 ( Fig 6 ) .', 'Analysis in Appendix F5 in S1 Text reveals that C V R 2 and C V P 2 follow the same qualitative shapes as in Fig 6 for any C V T 1 2 and C V T 2 2 . Interestingly , when C V T 1 2 = C V T 2 2 , the maximum and minimum values of C V R 2 and C V P 2 always occur at β = 2 - 2 albeit with different optimal values than Fig 6 ( see Fig . C in S1 Text ) .', 'For example , if C V T 1 2 = C V T 2 2 = 1 ( i . e . , exponentially distributed T1 and T2 ) , then the maximum value of C V R 2 is 20% higher and the minimum value of C V P 2 is 10% lower than their respective value for β = 0 .', 'Given that the effect of changing β on C V P 2 and C V R 2 is small and antagonistic , the overall affect of genome duplication on intrinsic noise may be minimal and hard to detect experimentally .', 'While the above analysis is for a stable protein , a natural question to ask is how would these results change for an unstable protein ?', 'Consider an unstable protein with half-life considerably shorter than the cell-cycle duration .', 'This rapid turnover ensures that the protein level equilibrates instantaneously after cell-division and gene-duplication events .', 'Let γx denote the protein decay rate .', 'Then , the mean protein level before and after genome duplication is 〈 x 〉 ¯ = k x 〈 B 〉 / γ x and 〈 x 〉 ¯ = 2 k x 〈 B 〉 / γ x , respectively .', 'Note that in the limit of large γx there is no noise contribution from partitioning errors since errors incurred at the time of cell division would be instantaneously corrected .', 'The extrinsic noise , which can be interpreted as the protein noise level for deterministic protein production and decay is obtained as ( for analysis on general f see Appendix G in S1 Text )', 'C V E 2 = ( 1 - β ) β ( 2 - β ) 2 , ( 48 )', 'which is similar to noise level reported in [75] .', 'When β = 0 or 1 , the transcription rate and the protein level are constant within the cell cycle and C V E 2 = 0 .', 'Moreover , C V E 2 is maximized at β = 2/3 with a value of 1/8 .', 'Thus , in contrast to a stable protein , extrinsic noise in an unstable protein is strongly dependent on the timing of gene duplication .', 'Next , consider the intrinsic noise component .', 'Analysis in Appendix G in S1 Text shows that the noise contribution from random protein production and decay is', 'C V P 2 = 1 2 〈 B 2 〉 〈 B 〉 + 1 1 〈 x 〉 ¯ , 〈 x 〉 ¯ = k x 〈 B 〉 ( 2 - β ) γ x .', '( 49 )', 'While the mean protein level is strongly dependent on β , the intrinsic noise Fano factor = C V P 2 × 〈 x 〉 is independent of it .', 'Thus , similar to what was observed for a stable protein , the intrinsic noise in an unstable protein is invariant of β for a fixed 〈 x 〉 ¯ .'], ['In this first-of-its-kind study , we have investigated how discrete f-fold changes in the transcription rate due to gene duplication affect the intercellular variability in protein levels .', 'Not surprisingly , the timing of genome duplication strongly affects the mean protein level—〈 x 〉 ¯ can change up to f folds depending on whether the gene duplicates early ( β = 0 ) or late ( β = 1 ) in the cell cycle .', 'Results show that genome duplication has counter intuitive effects on the protein noise level ( Fig 6 ) .', 'For example , if stochastic expression is the dominant source of noise , then doubling of transcription due to duplication results in lower noise , as compared to constant transcription throughout the cell cycle .', 'This is because for the same mean protein level , there are more burst ( transcription ) events in the case of genome duplication ( f = 2 ) than constant transcription ( f = 1 ) .', 'For example , consider deterministic timing ( C V T 1 2 = C V T 2 2 = 0 ) and gene duplication in the middle of the cell cycle ( β = 0 . 5 ) .', 'Then , for the case β = 1 , there are on average kx〈T〉 burst events per cell cycle .', 'For the same 〈 x 〉 ¯ , there are 1 . 05kx〈T〉 production events in the case of gene duplication ( β = 0 . 5 ) .', 'This slight increase in the number of transcription events leads to better averaging of bursty protein synthesis and lower noise levels .', 'Overall , the effect of β on different noise component is quite modest: as β varies , noise components deviate at maximum ≈ 20% from their values at β = 0 ( Fig 6 ) .', 'These results are in contrast to the case of an unstable protein , where noise from the cell-cycle process is strongly dependent on β as shown in Eq ( 48 ) .', 'The mathematical framework introduced for modeling timing of cell division can be easily used to compute noise in synchronized cells .', 'For example , let the cell-cycle duration be an Erlang distribution with shape parameter n and rate parameter nk ( i . e . , pn = 1 in Fig 3 ) , which can be biologically interpreted as cells moving through n cell-cycle stages Gn1 , Gn2 , … , Gnn .', 'Statistical moments conditioned on the cell-cycle stage Gnj can be obtained using', '〈 x m ∣ g n j = 1 〉 ¯ = 〈 g n j x m 〉 ¯ 〈 g n j 〉 ¯ , m ∈ { 1 , 2 } .', '( 50 )', 'Using Eq ( 50 ) and moments 〈 g n j x m 〉 ¯ obtained from Eqs ( 16 ) and ( 35 ) – ( 37 ) , yields the following conditional mean', '〈 x | g n j = 1 〉 ¯ = k x 〈 B 〉 〈 T 〉 1 + j n , ( 51 )', 'which increases with cell-cycle stage ( i . e . , higher values of j ) .', 'The protein noise level given that cells are in stage Gnj is given by', 'CV2|gnj=1≔〈x2|gnj=1〉¯−〈x|gnj=1〉¯2〈x|gnj=1〉¯2=n+3j3 ( n+j ) 2︷CVE2︸Extrinsic\xa0noise+2nα3 ( n+j ) 1〈x|gnj〉¯︷CVR2+n+3j3 ( n+j ) 〈B2〉〈B〉1〈x|gnj〉¯︷CVP2︸Intrinsic\xa0noise .', '( 52 )', 'Note that if n is large then the first term , which represents the noise contribution from the cell-cycle process , is negligible and can be dropped .', 'Interesting , while the noise contribution from partitioning errors C V R 2 decreases with cell-cycle stage , the noise contribution from stochastic expression C V P 2 increases with j .', 'Moreover , for B = 1 with probability 1 and α = 1 , the intrinsic noise is always 1/Mean irrespective of j .', 'Assuming high n , the noise at cell birth ( j = 1 ) and division ( j = n ) are obtained as', 'CV2|gn1=1=2α31〈x|gn1〉¯︷CVR2+13〈B2〉〈B〉1〈x|gn1〉¯︷CVP2︸Intrinsic\xa0noise ( 53 ) CV2|gnn=1=α31〈x|gnn〉¯︷CVR2+23〈B2〉〈B〉1〈x|gnn〉¯︷CVP2︸Intrinsic\xa0noise , ( 54 )', 'respectively .', 'Thus , measurements of Eqs ( 53 ) and ( 54 ) by synchronizing cells ( or by using cell size as a proxy for cell-cycle stage ) can be used to quantify α and 〈B2〉/〈B〉 , providing a novel way to separate these noise contributions .', 'Next , we discuss how noise in asynchronous cell can be used to quantify these parameters .', 'Simple models of bursty expression and decay predict the distribution of protein levels to be negative binomial ( or gamma distributed in the continuous framework ) [80 , 81] .', 'These distributions are characterized by two parameter—the burst arrival rate kx and the average burst size 〈B〉 , which can be estimated from measured protein mean and noise levels .', 'This method has been used for estimating kx and 〈B〉 across different genes in E . coli [47 , 82] .', 'Our detailed model that takes into account partitioning errors predicts ( ignoring gene-duplication effects )', 'Intrinsic noise = 4 α 3 ( 3 + C V T 2 ) 1 〈 x 〉 ¯ + 3 C V T 2 + 5 3 ( 3 + C V T 2 ) 〈 B 2 〉 〈 B 〉 1 〈 x 〉 ¯ .', '( 55 )', 'Using C V T 2 ⪡ 1 and a geometrically distributed B [50 , 83–85] , Eq ( 55 ) reduces to', 'Intrinsic noise = 4 α 9 1 〈 x 〉 ¯ + 5 9 1 + 2 〈 B 〉 〈 x 〉 ¯ .', '( 56 )', 'Given measurements of intrinsic noise and the mean protein level , 〈B〉 can be estimated from Eq ( 56 ) assuming α = 1 ( i . e . , binomial partitioning ) .', 'Once 〈B〉 is known , kx is obtained from the mean protein level given by Eq ( 17 ) .', 'Since for many genes 〈B〉 ≈ 0 . 5–5 [47] , the contribution of the first term in Eq ( 56 ) is significant , and ignoring it could lead to overestimation of 〈B〉 .', 'Overestimation would be even more severe if α happen to be much higher than 1 , as would be the case for proteins that form aggregates or multimers [33] .', 'One approach to estimate both 〈B〉 and α is to measure intrinsic noise changes in response to perturbing 〈B〉 by , for example , changing the mRNA translation rate through mutations in the ribosomal-binding sites ( RBS ) .', 'Consider a hypothetical scenario where the Fano Factor ( intrinsic noise times the mean level ) is 6 .', 'Let mutations in the RBS reduces 〈 x 〉 ¯ by 50% , implying a 50% reduction in 〈B〉 .', 'If the Fano factor changes from 6 to 4 due to this mutation , then 〈B〉 = 3 . 6 and 〈α〉 = 3 . 25 .', 'Our recent work has shown that higher-order statistics of protein levels ( i . e . , skewness and kurtosis ) or transient changes in protein noise levels in response to blocking transcription provide additional information for discriminating between noise mechanisms [86 , 87] .', 'Up till now these studies have ignored noise sources in the cell-cycle process .', 'It remains to be seen if such methods can be used for separating the noise contributions of partitioning errors and stochastic expression to reliably estimate 〈B〉 and α .', 'An important limitation of our modeling approach is that it does not take into account the size of growing cells .', 'Recent experimental studies have provided important insights into the regulatory mechanisms controlling cell size [88–91] .', 'More specifically , studies in E . coli and yeast argue for an “adder” model , where cell-cycle timing is controlled so as to add a constant volume between cell birth and division [78 , 91–93] .', 'Assuming exponential growth , this implies that the time taken to complete cell cycle is negatively correlated with cell size at birth .', 'More importantly , cell size also affects gene expression—in mammalian cells transcription rates linearly increase with the cell size [94] .', 'Thus , as cells become bigger they also produce more mRNAs to ensure gene product concentrations remains more or less constant .', 'A main direction of future work would be to explicitly include cell size with size-dependent expression and timing of cell division determined by the adder model .', 'This formulation will for the first time , allow simultaneous investigation of stochasticity in cell size , protein molecular count and concentration .', 'Our study ignores genetic promoter switching between active and inactive states , which has been shown to be a major source of noise in the expression of genes across organisms [95–104] .', 'Taking into account promote switching is particularly important for genome-duplication studies , where doubling the number of gene copies could lead to more efficient averaging of promoter fluctuations .', 'Another direction of future work will be to incorporate this additional noise source into the modeling framework and investigate its contribution as a function of gene-duplication timing .']]
['Introduction', 'Methods', 'Results', 'Discussion']
['Inside individual cells , expression of genes is inherently stochastic and manifests as cell-to-cell variability or noise in protein copy numbers .', 'Since proteins half-lives can be comparable to the cell-cycle length , randomness in cell-division times generates additional intercellular variability in protein levels .', 'Moreover , as many mRNA/protein species are expressed at low-copy numbers , errors incurred in partitioning of molecules between two daughter cells are significant .', 'We derive analytical formulas for the total noise in protein levels when the cell-cycle duration follows a general class of probability distributions .', 'Using a novel hybrid approach the total noise is decomposed into components arising from', 'i ) stochastic expression;', 'ii ) partitioning errors at the time of cell division and', 'iii ) random cell-division events .', 'These formulas reveal that random cell-division times not only generate additional extrinsic noise , but also critically affect the mean protein copy numbers and intrinsic noise components .', 'Counter intuitively , in some parameter regimes , noise in protein levels can decrease as cell-division times become more stochastic .', 'Computations are extended to consider genome duplication , where transcription rate is increased at a random point in the cell cycle .', 'We systematically investigate how the timing of genome duplication influences different protein noise components .', 'Intriguingly , results show that noise contribution from stochastic expression is minimized at an optimal genome-duplication time .', 'Our theoretical results motivate new experimental methods for decomposing protein noise levels from synchronized and asynchronized single-cell expression data .', 'Characterizing the contributions of individual noise mechanisms will lead to precise estimates of gene expression parameters and techniques for altering stochasticity to change phenotype of individual cells .']
['Inside individual cells , gene products often occur at low molecular counts and are subject to considerable stochastic fluctuations ( noise ) in copy numbers over time .', 'An important consequence of noisy expression is that the level of a protein can vary considerably even among genetically identical cells exposed to the same environment .', 'Such non-genetic phenotypic heterogeneity is physiologically relevant and critically influences diverse cellular processes .', 'In addition to noise sources inherent in gene product synthesis , recent experimental studies have uncovered additional noise mechanisms that critically effect expression .', 'For example , the time within the cell cycle when a gene duplicates , and the time taken to complete cell cycle are governed by random processes .', 'The key contribution of this work is development of novel mathematical results quantifying how cell cycle-related noise sources combine with stochastic expression to drive intercellular variability in protein molecular counts .', 'Derived formulas lead to many counterintuitive results , such as increasing randomness in the timing of cell division can lower noise in the level of a protein .', 'Finally , these results inform experimental strategies to systematically dissect the contributions of different noise sources in the expression of a gene of interest .']
['engineering and technology', 'cell cycle and cell division', 'signal processing', 'cell processes', 'binomials', 'random variables', 'noise reduction', 'dna transcription', 'probability distribution', 'mathematics', 'algebra', 'polynomials', 'gene expression', 'molecular evolution', 'probability theory', 'cell biology', 'genetics', 'biology and life sciences', 'gene duplication', 'physical sciences', 'evolutionary biology']