Supplemental Information. Exploiting Natural Fluctuations. to Identify Kinetic Mechanisms. in Sparsely Characterized Systems

Transcription

1 Cell Systems, Volume Supplemental Information Exploiting Natural Fluctuations to Identify Kinetic Mechanisms in Sparsely Characterized Systems Andreas Hilfinger, Thomas M. Norman, and Johan Paulsson

2 Exploiting natural fluctuations to identify kinetic mechanisms in sparsely characterized systems (Supplementary Information) Andreas Hilfinger Thomas M Norman Johan Paulsson Dept of Systems Biology, Harvard University, 00 Longwood Ave, Boston, MA 05, USA Corresponding authors: andreas_hilfinger@hms.harvard.edu and johan_paulsson@hms.harvard.edu Contents S. General fluctuation constraints description of Eq. () S. Illustrating network invariants for classes of systems S.3 Protein fluctuations due to transcriptional noise derivation of Eq. (4) S.4 Fluctuating translation rates derivation of Eq. (6) S.5 The effect of fluorescent maturation on correlations derivation of Eq. (7) S.6 The effect of undercounting molecules on mrna-protein correlations derivation of Eq. (9) 6 S. General fluctuation constraints description of Eq. () While Eq. () as presented in the main text is formally proven in reference [], we here provide an intuitive explanation. Eq. () can be intuitively understood as a general fluctuation constraint for any two components of interest within a complex network. For simplicity we denote those components X and X, while allowing their dynamics to be influenced by an arbitrary network of fluctuating components X 3,X 4,X 5,... whose dynamics in turn are allowed to be affected by X and X in arbitrary ways. The chemical reactions that change the number of X or X molecules are then listed as (x, x ) r k(x,x,x 3,x 4,x 5,...)! (x + d k, x + d k ) k =,..., m (S.) where each reaction changes the state by adding or subtracting integer numbers of molecules according to the jump sizes d k, d k, and the rate functions r k (x, x, x 3, x 4, x 5,...) quantify the probability per unit time with which reaction k occurs when the system is in state X = x,x = x,x 3 = x 3,X 4 = x 4,X 5 = x 5,... The general fluctuation constraint follows from considering the instantaneous dynamics of two components X,X which obey a master equation with conditional rates d dt P(x, x ; t) =Âhr k x d k, x d k ; tip(x d k, x d k ; t) Âhr k x, x ; tip(x, x ; t), (S.) k k where we have introduced conditional averages defined as h f (x) x, x ; ti := Â f (x)p(x 3, x 4,... x, x ). x 3,x 4,... While it is generally impossible to determine the values of the conditional rates from just specifying the reactions involving X,X, Eq. (S.) can be used to derive general moment equations that describe systems that reach a stationary probability distribution. The first order moment equation is given by hr i i = hr + i i

3 for (i =, ) and the normalized second order moment equation is given by (for i =, and j =, ) Cov(R j t j hr ± j ihx i i + Rj, x i ) with birth and death fluxes defined as + t i Cov(R i R + i, x j ) hr ± i ihx j i = t i hs ij i hx j i + t j hs ji i hx i i, (S.3) R + i := Â d ik r k R i := Â d ik r k, k:d ik >0 k:d ik <0 for reactions with rates r k = r k (x, x, x 3, x 4, x 5,...) that can depend in arbitrary ways on the state of the entire system, and event-sizes d k, d k as defined in Eq. (S.). In the fluctuation constraint Eq. (S.3) hx i i denotes the average abundance of component X i, t i is the average lifetime of X i, and hs ij i are average jump sizes, denoting the average change in the number of X j molecules as an X i molecule is made or degraded, formally defined as hs ij i = Â p ik d jk sgn(d ik d jk ), (S.4) k where p ik denotes the relative flux of component X i going through reaction reaction k. In this supplement we will discuss hs ij i as they becomes relevant for the processes analyzed in this paper. See reference [] for an illustration of how to determine the step-sizes hs ij i in general using various example systems. Note that Eq. (S.3) neither requires that the dynamics of the entire system is Markovian, nor that the waiting time distributions between events that change X and X -levels are exponentially distributed. The only assumption is that X,X numbers change in discrete steps, and the probability of those numbers to change is described by instantaneous propensities r k. For large Markovian systems at stationarity Eq. (S.3) follows directly from summing the usual master equation over all unspecified components, while for non- Markovian systems with formally unspecified X 3 = x 3,X 4 = x 4,X 5 = x 5,... dynamics it can be derived from first principles []. This latter class of systems includes explicitly time-dependent oscillatory systems, in which case the terms in Eq. (S.3) describe the long-term time-averaged behavior of the system appropriately sampling over the period of the oscillation. Eq. (S.3) thus not only applies to stationary systems but also periodically driven systems as illustrated by the example in Fig. S.. However, the equations do not apply to systems that are exhibiting transient dynamics from one state into another. To analyze such systems, a similar approach can be used that involves corrective terms that depend on the difference in system properties at the beginning and the end of the experiment (not shown). S. Illustrating network invariants for classes of systems We next illustrate how Eq. (S.3) translates local assumptions into relations that apply to all possible systems that satisfy those assumptions regardless of all other (indirect) network effects. Consider one molecular component, say X, within a large network of interacting molecules. Then Eq. (S.3) implies the following simple relation Cov(x, R ) hx ihr i = hs i hx i + Cov(x, R + ) hx ihr + i. (S.5) Making assumptions about the rate of production and degradation of that component then allows us to substitute those assumptions for R + and R into Eq. (S.5) which in turn leads to a simple relationship that must be satisfied by all systems with those R ±. For example, in Fig. a-d of the main text we consider all possible systems in which the degradation of component X is first order, i.e. R µ x, the X -production is proportional to X, i.e. R + µ x, and that X

4 Abundance Time Figure S.: Eq. (S.3) applies to time-averages of non-stationary systems. This supplemental figure relates to Fig. of the main text: The time-averaged statistics of periodically oscillating systems must satisfy Eq. (S.3) as long as the system is sampled equally over periods. Intuitively, Eq. (S.3) applies even to non-stationary systems because in the long run time-averaged fluxes and other dynamical properties must balance as long as the systems are non-diverging. See reference [] for a rigorous derivation. molecules are degraded and produced one by one. Nothing else about the systems is specified, 3 arbitrary dynamics of 4 X,X 3,X 4, x ax! x + bx. (S.6) including feedback loops x! x {z } {z } unspecified network dynamics specified X -dynamics Here, all degradation and production events of X lead to changes of size one, which implies that hs i =. For all such systems Eq. (S.5) then simply becomes AbCov(x, x ) Abhx ihx i = hx i + AaCov(x, x ) Aahx ihx i =) CV = hx i + Cov(x, x ), (S.7) hx ihx i where we have introduced the coefficient of variation CV i := s i /hx i i. This simple relation must be satisfied by any system within the above class of possible networks. Of course we cannot determine any of the terms without specifying the dynamics of the entire system because the unspecified details of the rest of the network will affect those terms. However, none of the unspecified details can change the relation between those terms, as illustrated by numerical examples in Fig. d of the main text. Eq. (S.7) thus characterizes the specified reactions in the class of networks defined by Eq. (S.6) and is strictly invariant with respect to changes in the unspecified parts. Similarly, in Fig. e-g we consider a different class of systems in which the specified production and degradation events take the form 3 arbitrary dynamics of 4 X,X 3,X 4, x ax 3! x + bx (x ) including feedback loops x! x {z } {z } unspecified dynamics specified X -dynamics. (S.8) Substituting the birth and death fluxes R + = ax3 and R = bx (x ) into Eq. (S.3) for i =, j = then 3

5 yields bcov(x, x x bhx i(hx i hx i) AaCov(x, x 3 ) Aahx ihx 3 i =.5 hx i (S.9) hs i = + =.5. (S.0) Eq. (S.9) is the network invariant illustrated in Fig. g in the main text. Note that the average step-sizes are defined as an average taken per molecule that appears and disappears and not per reaction that occurs. In Eq. (S.0) we thus use relative weights of / because at stationarity the birth and death fluxes must balance, even though for the above system on average we must have twice as many birth reactions than death reactions at stationarity. See reference [] for a detailed discussion of how to determine hs ij i in more complicated example systems. The above results imply that all systems within the class defined by Eq. (S.6) must satisfy Eq. (S.7) while all systems within the class defined by Eq. (S.8) must satisfy Eq. (S.9). Strictly speaking these results say nothing about the discriminatory power of such invariant relations. However, numerical simulations indicate that the invariant relations are specific, in the sense that systems of the wrong type do not generally satisfy a network invariant corresponding to another class of systems, as illustrated in Fig. S.. a 5 b Figure S.: Invariant relations can discriminate between different classes. This supplemental figure relates to Fig. of the main text: a, Systems within the class defined by Eq. (S.8) must satisfy the network invariant Eq. (S.9) (dashed red line). This is illustrated here with numerical simulations of various realization of such systems for different parameter values and network topologies (blue dots). b, Numerical simulation data (blue dots) for systems within the class defined by Eq. (S.8) but now compared against the wrong network invariant Eq. (S.7) (dashed red line). The many systems that deviate significantly from Eq. (S.7) illustrate that network invariants in principle have the power to discriminate between systems of different types. S.3 Protein fluctuations due to transcriptional noise derivation of Eq. (4) To analyze the fluctuation data reported in reference [] we first consider all gene expression models that include the common assumption that proteins (X ) are made at a rate proportional to cognate mrna levels (X ) and are subject to first order degradation, i.e., 3 4 arbitrary dynamics of X,X 3,X 4,... 5 including feedback loops {z } unspecified dynamics + x ax! x + x x /t! x {z } specified dynamics, (S.) 4

6 which is equivalent to the class of systems defined above and illustrated in Fig. a-d of the main text. The same invariant relation Eq. (S.7) thus applies, which we rewrite as h = hx i + h, (S.) where h ij := Cov(x i, x j )/(hx i ihx j i) denotes the normalized (co)variances. Here the specified dynamics assumes that X is made and degraded one molecule at a time implying hs i =. Note that this stepsize corresponds to the size of individual chemical reactions, which do not change even when considering the regime in which X -levels are very low and extremely short-lived leading apparent bursts of protein production. While this regime is commonly referred as the bursting regime it does not corresponds to chemical events of larger size. Its larger step-sizes correspond to a non-physical abstraction in which the dynamics of two components is mathematically condensed into just one step. Introducing the correlation coefficient r ij := h ij / p h ii h jj we can rewrite Eq. (S.) as CV = hx i + r CV CV. (S.3) which is Eq. (4) of the main text. For high copy number proteins with /hx i CV we then obtain CV CV r. (S.4) No high copy system that includes the above protein production and degradation steps can violate this constraint as Eq. (S.4) holds for any type of mrna fluctuations, including arbitrarily nonlinear feedback and oscillations. While we cannot predict the value of correlations without knowing the underlying dynamics of mrna fluctuations, r must equal CV /CV in any such system, which is depicted as the red line in Fig. of the main text. Quantifying how much measurements deviate from Eq. (S.4) Because not all data points are easily visible in Fig. of the main text we here explicitly report the deviation of each gene from the predicted relation Eq. (S.4): For each data point we calculate the Euclidean distance to the predicted straight line, with deviations in y-direction rescaled in units of the reported r - error bar, and deviations in x-direction rescaled in terms of the reported CV /CV -error bar. These relative deviations in terms of the respective error bar lengths are listed in the table of Fig. S.3. Of 37 measured genes only 8 exhibit fluctuations that deviate less than one error-bar unit from the prediction. The distribution of deviations is plotted in Fig. S.3b, with some of the biggest outliers depicted in Fig. S.3c. We here do not give a probabilistic interpretation of the observed deviations, but simply determine the magnitude of each gene s deviation in terms of its reported error bars to establish whether the disagreement between theory and data is meaningful for that gene. We are not making any statements based on a collective interpretation of the data or interpret those data points that approximately satisfy relation Eq. (S.4). In absolute terms the biggest outliers are the genes b555, cspe, gade. These and the biggest relative outliers hupa, rpsv, acee, rpoz are thus clear candidates to be experimentally investigated in greater detail. They are either dramatic measurement errors or their dynamics strongly deviate from the simple class defined in Eq. (S.). For completeness, we next show explicitly that for the experimental data reported in [] the /hx i term in Eq. (S.3) is negligible, and thus that the above deviations are not due to the finite abundances. Instead of assuming /hx i 0 in Eq. (S.3) we can analyze the exact relation between the correlation coefficient 5

7 a Distance 0<d< <d< <d<3 3<d<4 4<d<5 5<d<6 6<d<7 7<d<8 8<d<9 9<d<0 0<d< <d< <d<3 3<d<4 5<d<6 6<d<7 7<d<8 <d< Genes purb, yebj, malk, rpse, yrda, pyka, yhgi, many, yadr, lpd, tnaa, ybex, lon, ndk, glf, nrda, acnb, pepd b530, ydfg, deob, trps, bola, fabz, yajq, trxa, fba, fabi, vals, ybbn, dead atph, sspa, deod, ynaf, ylij, pps, pepn, glya, b3836, fusa, atpa, ybdb, yaeh, cspa, gyrb, galf, yccj, yaie, yhby, aspc, sers, eno, metj, ybed, hslu, lpxd, ydhd, yrfi kate, map, lyss, wrba, yhbh, thrs, yain, tufb, yebc, cyda, vacb, mreb, pstc, acef, cyod, nusa, ppib, nuoa, b59, gltb, hslt, adk, prsa cspe, atpd, b555, pssa, yfia, infa, pyrh, clpa, serc gade, pura, alda, flda, tkta, yajc, gros, yggx, rfad, thdf, ribb elab, pflb, ppa zwf, dnak, rpob, tig, yjgd, rcsb, pstb, fur yqjd, yiiu, yjiy, slyd, tufa, pnp, tpia Distance illustration for clpb clpb, secb gltd, hupb, metk fklb, clpp, gatz, gada, pgk hfq yjgf csra hupa, rpsv acee rpoz b Number of genes closer than d to line Relative distance d to line prediction 0 c rpoz gatz Ratio of noises acee Figure S.3: Quantifying how much the data deviate from the theoretical prediction. This supplemental figure relates to Fig. of the main text: a, Using the error bars of the data as reported in reference [] as yardsticks we quantify how much each gene deviates from the predicted relation Eq. (S.4). The distance d is the Euclidean distance of each data points from the straight line prediction, with deviations in y-direction rescaled in units of its r -error bar, and deviations in x-direction rescaled in terms of its CV /CV -error bar. This definition is illustrated graphically for the clpb data point, which lies a distance of d = 9.7 away from the theoretical prediction. Of all measured genes only the data for eighteen genes satisfy Eq. (S.4) within error bars. The majority of reported genes lie many times further away from the predicted relation. The reported fluctuations of such genes are inconsistent with the hypothesis that their protein production rate is proportional to their mrna levels and that proteins are subject to first order degradation. b, Illustrating the cumulative distribution of how far the individual genes deviate from the prediction of Eq. (S.4). The dotted line indicates d =. Inset: histogram of deviations. Same data as listed in the table, i.e. each gene s deviation is measured in terms of the error bars reported for that measurement. c, Graphically depicting the scale of the deviations for some of the extreme violators. 6

8 and the CVs r = CV CV hx i CV!. (S.5) For the observed data the correction term in Eq. (S.5) is small and leads to negligible changes in the analysis as illustrated in Fig. S.4. a Averages b Correction terms c Figure S.4: The effect of /hx i terms is negligible for the data analysis in the main text. a, The average protein abundances hx i of the data reported in [] are very large. Depicted here is a histogram of the protein averages corresponding to the 37 reported genes. b, The analysis in the main text used the assumption /hx i 0. In this panel we explicitly present the magnitude of the correction term in Eq. (S.5) for the data reported in []. The majority of genes have a relative error of less than.5%. c, For completeness we compare the data against Eq. (S.5) without setting the /hx i term equal to zero. We see that the resulting picture is virtually unchanged compared Fig. of the main text. For simplicity we thus present the approximate analysis in the main text rather than the above exact one. Eq. (S.) applies to all possible systems of the type defined in Eq. (S.). While not necessary for our analysis we can consider the special case of the simple gene expression model discussed in Taniguchi et al.[], in which the production rate of mrnas is constant, excluding the possibility of feedback or upstream fluctuations. Because this standard model of stochastic gene expression is completely linear and completely specified it can be solved using existing standard methods [3]. In the limit of small /hx i this model exhibits s s CV = CV + t /t and r =, (S.6) + t /t where t and t are the average life-times of mrna and protein, respectively. For the range of parameters reported in [] (t between and 0 minutes, and t = 80 minutes), the predicted correlation coefficients should then fall along the blue part of the line indicated in Fig. in the main text, corresponding to 0. apple r apple 0.3. S.4 Fluctuating translation rates derivation of Eq. (6) In the supplement of Taniguchi et al. [] it was suggested that fluctuations in the translation rate per transcript could explain the low correlations between mrna and protein levels. To investigate the effect of such translation noise in general we specify the mrna dynamics as in reference [], but allow for an arbitrarily fluctuating translation rate per mrna f (u) where u is a vector of fluctuating variables. This analysis differs from the previous analysis in two ways: First, we do not consider static distribution for rate constants but allow for any temporal behavior of the translation machinery, on arbitrary time scales. Second, we derive a relationship between correlation coefficients and fluctuations rather than focus on the magnitude of the correlations. The class of systems we are analyzing is given by specifying the dynamics of mrna (X ) and protein 7

9 (X ) while allowing for an unspecified vector of variables that randomize the translation rate per mrna apple arbitrary dynamics of u := x 0, x 3, x 4,... {z } unspecified dynamics + x l f (u)x! x + x! x + x x /t x! x x /t! x {z } specified dynamics (S.7) where l is a constant and t and t are the average life-times of mrna and protein, respectively. Applying Eq. (S.3) to Eq. (S.7) for i =, j = yields 0 = Cov(x, x /t f (u)x ) + Cov(x, x /t l ) t hx ihx f (u)i t hx ihx /t i () + Cov(x, x ) = Cov(x, f (u)x ) t t hx ihx i t hx ihx f (u)i Cov(x () h =, x f (u)) + t /t hx ihx f (u)i (S.8) In the first step above we exploited that hs i = 0 since no reactions change abundances of mrna and protein simultaneously, and in the second step we used Little s law hx i/t = hx f (u)i to rewrite one of the denominators. Though we normally suppress time dependence in variables, we note that we allow for arbitrary time-dependent fluctuations in the translation rate f (u) = f (u(t)). This is in contrast to modeling approaches in which parameters such as the translation rate are assumed to take fixed values drawn from a static probability distribution. Independent translation rate fluctuations When the fluctuations in the translation rate f are independent of the mrna levels, we have as a direct consequence of such independence that Cov(x, x f (u)) hx ihx f (u)i Combining Eqs. (S.8) and (S.9) then yields = Cov(x, x )h f (u)i hx ihx ih f (u)i = h. (S.9) r = + t /t CV CV, (S.0) which is Eq. (6) presented in the main text. Because this invariant relations involves the mrna lifetimes t we check whether it holds for those genes for which t was reported in reference []. For 87 of the genes mrna life-times were reported, allowing us to directly compare the RHS of Eq. (S.0) versus its LHS for that subset of the data, see Fig. 3 of the main text. Counteracting transcriptional and translational noise The above analysis shows that translation noise that is uncorrelated with the fluctuations in mrna-levels cannot explain the data. We thus next analyze systems in which translation rate fluctuations are allowed to correlate with mrna levels, as described by the following system x g(u)! x + x f (u)x! x + x x /t! x x x /t! x (S.) 8

10 where the unspecified functions f, g are randomizing the mrna production and protein translation rates. Applying Eq. (S.3) to X and X then gives rise to 0 = Cov(x, x /t f (u)x ) + Cov(x, x /t g(u)) t hx ihx f (u)i t hx ihg(u)i () + Cov(x, x ) = Cov(x, f (u)x ) + Cov(x, g(u)) t t hx ihx i t hx ihx f (u)i t hx ihg(u)i () + t h t = Cov(x, x f (u)) + t Cov(x, g(u)), hx ihx f (u)i t hx ihg(u)i where in the second line the balance of average fluxes was used to substitute hx i/t = hg(u)i and hx i/t = hx f (u)i in the denominator on the LHS. We can rewrite this relationship in terms of the correlation coefficients as + t r t CV CV = r x,x f CV CV x f + t r x,gcv CV g. (S.) t For many of the genes mrna protein correlations are so small that the LHS becomes negligible. For those data Eq. (S.) shows that one of the correlation coefficients on the RHS must become negative. In fact in the most extreme cases of the data in which r is large in magnitude but negative we expect both correlations to be negative. This raises the question of what kind of mechanism could give rise to such negative correlations even in the absence of strong feedback, which is not expected for the fluorescent reporter library used in reference []. We next show that such correlations can in fact be generated by a common upstream factor that inversely affects the rates g and f, i.e., when f is up g is down and vice versa. To test whether such systems can exhibit behavior that is consistent with the reported data (rather than just not ruled out by the constraints Eqs. (4) and (6) of the main text), we simulated systems in which a common upstream factor u(t) affected the transcription and translation rate such that g /u and f u. Picking parameter values such that the mrna and protein abundances, CVs and lifetimes are comparable to the data we found systems that matched the majority of the reported correlations. A subset of those simulations is presented in Fig. 5 of the main text (blue data points). Specifically, the simulated systems were of the type defined in Eq. (S.) a with the following choice of rate functions f (x 0 )=l x 0 and g(x 0 )=l a+x 0, i.e., X 0 is a component that randomizes the translation rate per mrna, while antagonistically affecting the mrna production rate. The dynamics of X 0 itself followed x 0 l 0 z! x0 + z x 0 x 0 /t 0! x 0 z l z +fz/t z! z + z/t z! z. (S.3) Varying the parameters of the above X 0 and Z-dynamics allowed us to simulate systems with varying upstream noise strength and time-scales affecting the transcription and translation rates f, g. We simulated, 000 systems, of which we picked a subset to illustrate the accessible range of such systems (blue dots in Fig. 5 of the main text). The parameters used in the simulations spanned the following ranges : t 0 = t z : 0.007!.4, l 0 : /t 0! /t 0, f : 0! 0.9, l z : 0.4( f)/t z! 50( f)/t z, l : 0!, 000, l : 500!, 500, t :! 0, t =80, a : 0.00! 0. (all rate constants and life-times are reported in units of minutes). Most of the resulting systems had averages, correlations, and CVs comparable to that of the data reported in reference []. The representative ranges for the simulations were hx i : 0.000! 6.9, hx i : 0.5! 7, r : 0.67! 0.7, CV : 0.6! 0, CV : 0.! 0 and mrna Fano-Factor:! 7. The randomizing upstream variable exhibited a variability CV 0 : 0.8!. The specific model choice for randomizing the simple transcription and translation steps is arbitrary. We are not advocating this particular model but merely illustrate that simple gene expression models can exhibit fluctuations that are consistent with the data as as long as significant upstream randomization antagonistically affects the transcription and translation rates. 9

11 Stochastic interactions of the noise cancellation type Eq. (S.) subject to strong upstream variability are clearly very different from the simple mrna-protein model in which both g = const and f = const. However, although the underlying molecular explanations are markedly different, the resulting protein distributions are virtually indistinguishable as illustrated in Fig. S.5 where we present 4 distributions corresponding to systems that deviated the most from the prediction of Eq. (S.0). The majority of those distributions fit almost perfectly to a negative binomial distribution just like the experimentally observed distributions reported in reference []. This confirms that vastly different mechanisms can give rise to very similar distributions, and highlights the increased discriminatory power the correlation analyses presented in the main text afford. Similar conclusions apply to systems in which the mrna degradation is affected by an upstream factor that randomizes the translation rate. We also note that the choice of inverse rate dependence for g and f that we have chosen here is to some extent arbitrary, and other pairs will yield similar effects. 0

12 Figure S.5: Complicated noise cancellation mechanisms can generate negative binomials distributions. This supplemental figure relates to Fig. 5 of the main text: Here we plot the protein distributions P(x ) for numerical simulations in which transcription and translation rates are affected by a common upstream factor u(t) such that g /u and f u as defined in Eq. (S.). We picked the 4 simulations for which the deviations of the mrna-protein correlations deviated the most from the network invariant Eq. (S.0) that applies to all systems in which the translation rate fluctuations are independent of mrna-levels. These simulations thus correspond to systems with strongly antagonistic noise effects in transcription and translation. Depicted here are the corresponding distributions obtained from the numerical simulation data (blue dots) and a fitted negative binomial distribution (dashed red lines). Strikingly, in the majority of cases the fits are near perfect even though the negative binomial distribution is more typically associated with a simple mrna-protein model in which g = const and f = const. This highlights that protein distributions are not unique and can often not distinguish between mechanisms.

13 S.5 The effect of fluorescent maturation on correlations derivation of Eq. (7) Protein levels in Taniguchi et al. [] were inferred from fluorescent measurements, so even if proteins are made at a rate proportional to mrna levels, the apparent production rate will depend on maturation of the fluorophore. As the maturation time becomes faster and faster, the difference becomes negligible, but in reality the maturation times of fluorescent proteins are comparable to the lifetime of mrna molecules. Next we address, whether the data could potentially be explained by simple gene expression models if we account for finite fluorescent maturation times. Maturation and transcriptional noise models Previous work [4] analyzed how a delay between observed protein levels and translation events affect correlations between observed protein and mrna levels. By considering a simple mrna-protein model in which proteins were subject to fixed maturation delays the authors concluded that reporter maturation delay explains the lack of correlation between mrna and protein observed in Taniguchi et al. []. In this section we first revisit the special case of this previously discussed model to illustrate that it is not so much the overall lack of correlation, but the value of correlations for a given value of CV /CV that makes the data so difficult to explain. We then use our general approach to consider classes of systems subject to an additional maturation step. Special case of previously motivated model Previous work [4] analyzed the correlations between mrna and proteins for the simplest type of gene expression model corresponding to the special case of Eq. (S.7) in the absence of any translation rate fluctuations but subject to a delayed read out in which actually observed protein levels x at time t simply correspond to the immature protein levels x 0 at t t mat x l! x + x 0 ax! x 0 + x x /t! x x 0 x 0 /t! x 0 + x (t) =x 0 (t t mat ). (S.4) As reported in Ref. [4] for such systems in the high copy number limit hx 0 i! the correlations between mrna and (observed) protein levels for such a system are given by r t r = exp t + t tmat t. (S.5) Next we consider the above special mrna-protein model with a more realistic description of the maturation step, assuming that the maturation of fluorescent proteins X 0 follows an exponentially distributed times [5] such that immature proteins convert into (visible) mature ones X x l! x + x x /t! x x 0 ax! x 0 + (x 0, x ) x 0 /t mat! (x 0, x + ) x x /t! x. (S.6) In this specific model the mrna dynamics is a simple birth and death process and no feedback is allowed. We here analyze this special case to illustrate how sharp delays differ from distributed delays and how CVs and correlations are connected even in systems subject to maturation. Because the system defined in Eq. (S.6) is completely linear and completely specified we can explicitly solve for the covariances h ij using standard methods [6]. Specifically looking at the large copy number limit (hx 0 i!, hx i! ), we find that r s t t r = (t mat + t )(t mat + t ), (S.7) t + t t mat + t t mat t + t mat t + t t which differs significantly from Eq. (S.5). For example, assuming the same range of parameters as in [, 4] the expected range of correlations after accounting for exponentially-distributed delays is roughly two-fold

14 higher than reported in [4] for sharp delays. This aside establishes that a sharp delay significantly overestimates the reduction in correlations due to a stochastic maturation step. However, focusing on the reduction in the magnitude of the correlation coefficient is misleading without considering the CVs. To illustrate this point we next derive the relation between correlations and CVs for this specific case of a simple mrna-protein model with finite maturation times. The simple linear model of Eq. (S.6) can be solved for the ratio of CVs using standard methods [6] CV CV = s t (t t mat + t (t mat + t )) (t + t )(t mat + t )(t mat + t ). (S.8) Combining Eqs. (S.8) and (S.7) defines the accessible range of correlations and ratio of CVs. For any given value of t mat, the accessible range is a one-dimensional curve that is traced out as the mrna life-time t varies. See red lines in Fig. S.6 for t mat = 7.5 min and t mat = 5 min respectively. Assuming a realistic range min apple t apple 0 min for mrna life-times leads to a subset of allowed values as highlighted by the blue part of the curves in Fig. S.6. In Fig. 4a of the main text the blue line highlights this accessible range for t mat = 7.5 min, t = 80 min. X 0 mrna X 0 X immature protein mature protein Ratio of noises Ratio of noises Figure S.6: The experimental data is inconsistent with the simple mrna-protein model even when accounting for finite maturation times. This supplemental figure relates to Fig. 4 of the main text: Here we plot the accessible range of r and CV /CV -values for the simple mrna-protein model subject to finite maturation as defined in Eq. (S.6). For any given value of t mat the accessible range corresponds to a simple one-dimensional curve. We here plot these curves for t mat = 7.5 min and t mat = 5 min respectively (red lines) allowing for arbitrarily fast or slow mrna-lifetimes. Note that these curves are not trend lines: for this model to be correct all the data would have to fall onto the curve (within experimental error). Many of the data points clearly do not fall onto this curve regardless of the value of t mat. For a realistic range of mrna life-times min apple t apple 0 min only a subregion is accessible as highlighted by the blue sections under the assumption that t = 80 min. This accessible range is highlighted in Fig. 4a of the main text for t mat = 7.5 min. General classes of systems subject to maturation The above analysis of a specific model showed that focusing on the correlation coefficient is misleading: it is not so much the overall lack of correlation, but the value of correlations for a given value of CV /CV that makes the data so difficult to explain. Furthermore, sharp delays as assumed in reference [4] overestimate the reduction in correlations due to maturation. Next we show how the fluctuation relation Eq. (S.3) generally constrains the correlation between mrna and mature proteins even when we allow for largely unspecified networks. This analysis again considering arbitrary feedback, mrna-dynamics, or extrinsic influences suggests that many of the data are inconsistent with any simple gene expression model in which protein production is proportional to mrnalevels, even when allowing for an unobserved intermediary component with a finite maturation time. The 3

15 fundamental starting point is specifying the reactions characterizing translation and maturation 3 4 arbitrary dynamics of X,X 3,X 4,... 5 including feedback loops {z } unspecified dynamics + ax x 0! x 0 + x (x 0, x ) 0 /t mat! (x 0, x + ) x x /t! x {z } specified dynamics, (S.9) where X still refers to mrnas, whose dynamics are left completely unspecified and which are translated at a rate proportional to its abundance, to yield immature (non-fluorescent) proteins denoted by X 0. These proteins mature at an exponential rate with time constant t mat to yield detectable proteins X, which in turn are diluted by cell division at rate t. Because fluorescence maturation times are much faster than the reported degradation time (effectively set by the cell cycle time) we here ignore the degradation reaction of the immature proteins X 0. In the ensuing analysis we exploit the fact that the data corresponds to the high copy number limit (see Fig. S.4a). For example, applying the fluctuation balance Eq. (S.3) with i = j = 0 yields Cov(x 0, x 0 /t mat ax ) hx 0 ihx 0 /t mat i = () h 00 = h hx 0 it 0 + mat hx 0 it mat {z } 0 where we used the flux balance hx 0 i/ht mat i = hax i to identify h 0. Thus in this limit we must have h 00 = h 0. This result together with those equations corresponding to i, j =, and i, j = 0, yield three simple equations h 00 = h 0 h = h 0 h 0 = + t /t mat h 00 + t /t mat + t /t mat h. (S.30) Substituting the second relation into the third and applying (h 0 ) apple h h 00 to the first equation (which implies that h 00 apple h ) allows us to eliminate the X 0 -dynamics and obtain the following inequality h apple + t /t mat h + t /t mat + t /t mat h. (S.3) Substituting the definition of the correlation coefficient and simple rearrangement then give rise to + t mat CV t CV t mat t CV CV apple r. (S.3) Similarly, applying the Cauchy-Schwarz inequality (h 0 ) apple h h 00 to the second relation implies that h apple h 00. Note, that fluctuations in X must be less than those of X 0 makes intuitive sense since the fluctuations in X -levels are time-averaged linear responses to changes in X 0 -levels, regardless of the nature of upstream fluctuations or feedback loops affecting the X 0 -dynamics. Applying this inequality to the third relation leads to h h + + t /t mat h t /t mat + t /t. mat Substituting the definition of the correlation coefficient r and simple rearrangement then gives rise to r apple CV CV. (S.33) Combining these two inequalities thus constrains the correlation coefficient to + t mat CV t CV t mat t CV CV apple r apple CV CV. (S.34) 4

16 arbitrary network mature protein 0.4 X 0 X mrna 0 X 0 immature protein Ratio of noises Figure S.7: The effect of slower maturation times on the class of systems defined in Eq. (S.). This supplemental figure relates to Fig. 4 of the main text: Here we plot the allowed region defined by the inequality Eq. (S.34) for the same parameters used in the main text (red region) as well as the extended region when assuming t mat = 5 min (additional blue region). Note that unlike the correlations themselves, this inequality is strictly independent of mrna lifetimes because this bound on correlations and CV-ratios constrains all possible systems regardless of the details of the mrna dynamics, e.g. mrna lifetimes could be as fast as mere seconds or as slow as one hour and the system still has to satisfy Eq. (S.34). The relationship only depends on the protein maturation and degradation times. In Fig. 4a of the main text we plot Eq. (S.34) for t mat = 7.5 min and t = 80 min. These values correspond to estimates for the fast maturing YFP-variant Venus [7] used in Taniguchi et al. [], and the effective protein lifetime set by the reported cell-cycle time []. While accounting for finite protein maturation times makes a lot more data points consistent with theoretical predictions (colored area), many of the data points remain inconsistent with the theoretical prediction. Changing the exact parameters used for t mat and t changes the conclusions only little as illustrated in Fig. S.7. While the proven bound of Eq. (S.34) is an inequality that rigorously constrains all possible systems of the type defined in Eq. (S.9) it is not the tightest possible bound. It can be made tighter by utilizing the semi positive-definiteness constraint of covariance matrices applied to the three components X 0,X,X. However, in the range for CV /CV corresponding to the experimental data the difference is negligible. In the main text (Fig. 4a) we thus plot the straightforward bound of Eq. (S.34). Maturation and translational noise models For translational noise we analyze the effect of a finite maturation time by considering the class of Eq. (S.7) subject to an additional maturation step, i.e., f (u)x apple arbitrary dynamics of + x l x 0! x 0 +! x + x u:=x 3,X 4,X 5,... x {z } x /t (x 0, x ) 0 /t mat! (x 0, x + )! x x unspecified dynamics x /t! x {z } specified dynamics, (S.35) describing the dynamics of mrna (X ), immature protein (X 0 ) and mature protein (X ). Here we again allow for arbitrary time-dependence of the translation rate per mrna molecule f (u) = f (u(t)) rather than assuming that translation rates take some constant value drawn from a fixed probability distribution. 5

17 Applying Eq. (S.3) to i =, j = 0 and i =, j = then yields h 0 = Cov (x, x R (u)) + t mat /t hx ihx f (u)i h = + t /t h 0. Assuming again that the fluctuations of f (u) are independent of the mrna levels implies that Cov(x, x f (u)) hx ihx f (u)i = h and we thus find r = CV. (S.36) + t /t + t mat /t CV Compared to Eq. (S.0) the correlation coefficient is multiplied by the additional time-averaging factor t /(t + t mat ). In Fig. 4b of the main text we plot the data reported in Taniguchi et al. [] for t mat = 7.5 min and t = 80 min. S.6 The effect of undercounting molecules on mrna-protein correlations derivation of Eq. (9) Next we address, whether the data could potentially be explained by simple gene expression models if we account for the possibility of stochastic undercounting of fluorescent molecules. Although the experimental methods in reference [] were checked as rigorously as possible, with fluorescent hybridization methods it is hard to prove that in fact all molecules were detected. Such undercounting that the experimentally determined numbers only represent a certain percentage of the true numbers is a potential candidate explanation for the low correlations between mrna and protein levels [8]. It is straightforward to include undercounting within our framework by introducing an additional variable describing components that are detected with some fixed probability. This introduces a simple mathematical mock process that allows us to determine the (co)variances when observed populations are a binomial cut of the true population. Undercounting of mrna in transcriptional noise models For the class of systems with arbitrary transcriptional noise as defined in Eq. (S.) mrna undercounting can be analyzed in the following system in which proteins X are made at a rate proportional to mrna levels X 0 (whose dynamics is completely unspecified), but where mrna levels are measured via a mock process modeling the binomial read-out species X as follows 3 4 arbitrary dynamics of X 0,X 3,X 4,... 5 including feedback loops {z } unspecified dynamics + x ax 0! x + x x /t! x {z } specified dynamics & x l(x 0 x )! x + x bx! x {z } binomial read out. (S.37) For large values of l and b, the component X corresponds to a binomial read-out of the mrna X 0 with a counting success rate of q = l/(l + b), i.e., each mrna molecule has a probability of q to be detected experimentally. This approach again allows for arbitrary mrna dynamics and can be solved for relations between (co)-variances in exactly the same way as in the previous sections. Applying Eq. (S.3) to compo- 6

18 nents X and X yields + t b h = hx i + h 0 h = hx i + Cov(x, x 0 x ) hx i(hx 0 i hx ) h = Cov(x, x 0 x ) hx i(hx 0 i hx ) + t b h 0 Making use of b /t (which is not an approximation but satisfied by construction of the mathematical mock system to define X as a binomial cut of X 0 ) and q = l/(l + b) this simplifies to h = hx i + h 0 h = q hx i + h 0 h = h 0, which we can combine to solve for the correlation between observed protein levels X and undercounted mrna levels X! h p =: r h h = CV. CV hx i This is exactly the same relation as Eq. (S.5). We have thus shown that whether or not mrna molecules are undercounted, the predicted relation between correlations and the ratio of CVs must be satisfied by any system in which protein production is proportional to mrna levels. While the actual values of observed CVs and corrections will strongly depend on the undercounting factor q, these artifacts cancel out as far as Eq. (S.5) is concerned. If the undercounting of mrna molecules is not binomial (because e.g. different cells have differently permeable membranes to the FISH probes) the above analysis does not apply. However, in a scenario in which measurement errors cannot be restricted at least in some ways, experimental data cannot be reliably analyzed regardless of mathematical approach. Undercounting of mrna in translational noise models Similarly, we can analyze the effect of mrna undercounting on the predicted relations for the class of translational noise models defined by Eq. (S.7) by introducing a mock variable X that generates a binomial read-out of mrnas (X 0 ) apple arbitrary dynamics of u := x 3, x 4, x 5... {z } unspecified dynamics CV + x l 0 f (u)x 0! x0 + 0 x! x + x 0 /t 0 x x 0! x 0 x /t! x {z } specified dynamics & x l(x 0 x )! x + x bx! x {z } binomial read out (S.38) where the mrna molecules (X 0 ) are produced at a constant rate and the translation rate f (u) = f (u(t)) can vary arbitrarily in time in response to an arbitrary network of other variables as long as those fluctuations are independent of mrna levels. We can proceed as before in Section S.4 where we derived (for a differently enumerated set of components) r 0 = Applying Eq. (S.3) to components X and X again yields + t /t 0 CV 0 CV. (S.39) h = q hx i + h 0 h = h 0, where the second equation implies that the sought correlation between proteins and detected mrnas r simply follows r = r 0 CV 0 /CV. To relate this to the observed protein CV we apply Eq. (S.3) to X 0 and. 7

19 X to obtain + t 0 b h 00 = hx 0 i h 0 = Cov(x 0, x 0 x ) hx 0 i(hx 0 i hx i) Since b /t 0 (by construction of the mock system), the pre-factor multiplying h 0 is essentially equal to. Making use of qhx 0 i = hx i and thus hx 0 i hx i =( q)hx 0 i), the equation above simply reduces to h 0 = h 00. Substituting back into the equation for h shows that h = q hx i + hx 0 i = hx i = qhx 0 i = h 00 q. Thus CV 0 = CV p q and the correlation now follows from Eq. (S.39), which implies r = CV 0 CV 0 = p p q CV q = CV + t /t 0 CV + t /t 0 CV q + t /t 0 CV CV, (S.40) where r is the observed correlation between (undercounted) mrna levels and proteins, t 0 is the mrna lifetime, CV is the observed noise in (undercounted) mrna levels, and CV is the observed noise in protein levels. The probability q with which each mrna molecule was detected thus lowers the slope of the straight line prediction depicted in Fig. 3 of the main text. However, the experimental data does not fall on a straight line regardless of the value of the slope q, showing that much of the data set is not consistent with the above assumptions characterizing mrna-protein dynamics. Undercounting proteins If proteins were undercounted the above analysis remains unchanged because the observed protein abundances in the data set are large (see Fig. S.4a). For such large populations binomial thinning is wellapproximated by a simple re-scaling. Because all correlations and (co)-variances discussed here are normalized, their value is unaffected by this simple rescaling. Artifacts due to protein undercounting thus cannot explain the data either. In fact, if the proteins were undercounted the only effect on the analysis here would be that the already small correction in Fig. S.4 would be even smaller. 8

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