Ribosome self-assembly leads to overlapping reproduction cycles and increases growth rate
(cid:105)(cid:105) (cid:105) “overlapping˙ribosome˙reproduction˙cycles˙via˙self˙assembly˙Arxiv˙Version” — 2018/7/25 — 1:39 — page 1 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105)
Ribosome self-assembly leads to overlappingreproduction cycles and increases growth rate
Rami Pugatch ∗ † , Yinon M. Bar-On ‡ ∗ Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel, † Quantitative Life Science Section, The Abdus SalamInternational Center for Theoretical Physics, Strada Costiera 11, 34014, Trieste, Italy, and ‡ department of Plant and Environmental Science, Weizmann Institute of Science,Rehovot 7610001, IsraelSubmitted to Proceedings of the National Academy of Sciences of the United States of America In permissive environments, E. coli can double its dry mass every ∼ minutes. During this time, ribosomes, RNA polymerases, andthe proteome are all doubled. Yet, the question of how to relatebacterial doubling time to other biologically relevant time scales inthe growth process remains illusive, due to the complex temporal or-ganization of these processes. In particular, the relation between thecell’s doubling time and the ribosome assembly time is not known.Here we develop a model that connects growth rate to ribosomeassembly time and show that the existence of a concurrent ribosomeself-assembly step increases the growth rate, because during ribo-some self-assembly existing ribosomes can start a new round of re-production, by making a new batch of ribosomal proteins prior to thecompletion of the previous round. This overlapping of ribosome re-production cycles increases growth rate beyond the serial-limit that istypically assumed to hold. Using recent data from ribosome profilingand established measurements of the average translation rate, rigidbounds on the in-vivo ribosome self-assembly time are set, which arerobust to the assumptions regarding the biological noises involved.Utilizing these physiological parameters, we find that at minutesdoubling time, the ribosome assembly time is ∼ minutes — threefold larger than the common estimate. We further use our modelto explain the detrimental effect of a recently discovered ribosomeassembly inhibitor drug, and predict the effect of limiting the expres-sion of ribosome assembly chaperons on the overall growth rate. self-assembly | ribosome | growth rate | branching processes A ll known single-cell organisms share the same basic archi-tecture first suggested by Von-Neumann [1, 2]. In par-ticular, all single cells have a membrane, metabolic machinerythat is responsible for supplying ample amounts of energy andsubstrates, a transcription-translation machinery, and DNA toinstruct it. The cell reproduces by allowing the transcription-translation machinery to read the DNA instructions, and makecopies of all the molecular machines in the cell, as well as copiesof itself. Concurrently to this process the preexisting molec-ular machines, produced in previous production rounds, keepsupplying energy and substrates, produce membrane boundvolume and replicate DNA.Complex production and assembly processes are comprisedof many indivisible tasks that are constrained to occur accord-ing to a given partial temporal order. This partial temporalorder forces some tasks to occur in series, allowing other tasksto occur concurrently. When all tasks are completed, a func-tional end product emerges. The duration of the longest set oftasks that are bound to occur in series defines a natural timescale, known as the critical path duration, which sets a lowerbound on the production time of a specific product.There are three generic methods to increase the productionrate of such complex processes. A straightforward method toincrease the rate is to decrease the critical path duration T c . Ifthis is impossible e.g. due to constraints such as accuracy [3],there are two quintessential alternatives that allow productionrate to become larger than the reciprocal of the critical pathduration: (i) parallel production — having multiple produc-tion lines that run in parallel; (ii) pipelining — on a single production line, starting a new round of production prior tothe completion of the previous round.To illustrate these three methods for increasing the produc-tion rate, consider a single ribosome translating mRNA. Thecritical path duration is the average translation time. To geta production rate that is twice as fast, we can double thespeed of translation — reducing by half the critical path du-ration. Alternatively, we can parallelize the production line,by doubling both the number of mRNA’s and the number oftranslating ribosomes per mRNA. As a result, the rate of pro-tein production will double but the critical path duration willnot change.Finally, we can use pipelining, by allowing ribosomes to starta new round of protein production prior to the completion ofthe previous round. For example, if on a single strand ofmRNA we allow a second ribosome to start a new round ofprotein translation while the previous ribosome is on aver-age midway in the process of making the previous round, atsteady state we will increase the production rate by a factor oftwo, without reducing the critical path duration. This factor-of-two increase will be valid as long as neighboring translat-ing ribosomes are far enough apart to be effectively decou-pled from each other in order to avoid extra delays caused byself-exclusion (“traffic jams”). Having multiple mRNAs eachpipelined with multiple ribosomes allows the methods of par-allelization and pipelining to be combined. Significance
The transcription-translation machinery has a dual role to syn-thesize the proteome, and to synthesize copies of itself. Twokey players in this process, RNA-polymerases and ribosomes,self-replicate by jointly producing their sub-components whichsubsequently self-assemble to new RNA-polymerases and ribo-somes. We show that a self-assembly step allows ribosomesto perform more tasks, including starting another round of self-reproduction, prior to the completion of the previous round. Thisoverlapping of self-reproduction cycles increases growth rate rel-ative to the serial case. We devise a model for concurrentself-replication with a self-assembly step and employ it to in-fer in-vivo duration of ribosome self-assembly in fast growing
E.coli and predict the effect of limiting assembly chaperons on thegrowth rate.
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Volume Issue Number – a r X i v : . [ q - b i o . S C ] J u l (cid:105) “overlapping˙ribosome˙reproduction˙cycles˙via˙self˙assembly˙Arxiv˙Version” — 2018/7/25 — 1:39 — page 2 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) Perhaps surprisingly, in the context of self-replication, onlytwo of the three methods help to decrease the doubling time.The argument is simple — let U be a self-replicating machinewhich can make a copy of itself in T c time units, where T c isthe critical path duration for making a single copy. Let’s in-vestigate how we can increase the doubling rate. One straightforward way to increase the doubling rate is to decrease thecritical path duration T c . What about parallelization? If westart with two machines instead of one, the production rate— how many U’s are made per unit time, will increase by afactor of two. However, the doubling time will remain T c sincenow, in order to double, we need to produce two U -machinesrather than one.More generally, if we start with n self-replicating machinesthat need to double to 2 n machines, increasing the numberof machines at the beginning of the process to 2 n will notincrease the doubling time, since the target will be to reach4 n machines. We conclude that parallelization does not helpincreasing the doubling rate.What about pipelining? If the machine U can start the pro-cess of making a new copy but can leave this process in themiddle in order to restart another round of U production whilethe immature U machine continues to mature independently,then the growth rate will increase without changing the criticalpath duration. We call this process pipelined self-replication([2], chapter 7). The mathematical structure of such a pro-cesses turns out to be identical to a branching model firststudied by Crump and Mode and by Jagers [4], in the con-text of population demography and also in the study of bud-ding yeast [5], and is known as a general branching process orCrump-Mode-Jagers branching process [4, 6].An important recent result in the field of bacterial physiol-ogy is the bacterial growth law developed and experimentallytested in [7]. This bacterial growth law connects the growthrate µ , the ribosome workload — the time it takes a ribosometo translate all ribosomal proteins τ RP , and the percentageof ribosomes allocated toward making ribosomal proteins α , µ = ατ RP (see also [8, 9, 10]).Here we present a novel quantitative relation (“growth law”)that connects the growth rate µ to the ribosome assembly time τ SA as well as other more standard cellular parameters, speci-fied below. The connection between growth rate and assemblytime is not trivial, since in the presence a non-zero assemblytime, the process of self-reproduction of ribosomes is pipelined,i.e. several generations are produced concurrently, in contrastto the tacitly assumed serial reproduction model, where theoverall ribosome doubling time is simply the assembly timeplus the time to produce all the ribosomal proteins.We develop a model that can robustly predict rigid boundson the in-vivo ribosome self-assembly duration, and find thatthe assembly time is roughly three times higher than the preva-lent estimate. We further employ our model to explain therelation between the newly discovered effect of the drug Lam-otrigie on ribosome self-assembly in live E. coli [11] and itsgrowth. Finally, we study dependence of the growth rate onthe reliability of the assembly process, which can also be ex-perimentally tuned by limiting assembly chaperons.
A new growth law that accounts for ribosome self-assem-bly and noise.
Our main result is an implicit functional rela-tion (“growth law”) between the fraction of ribosomes busytranslating ribosomal proteins — α , the single cell biomassgrowth rate — µ and the following Laplace transforms: (i)Laplace transform of the distributions of ribosome idling times— P ( s ); Laplace transform of the distribution of durations totranslate the i th ribosomal protein — P i ( s ) = P ( s ) assumed equal among all ribosomal proteins [12]; (iii) Laplace trans-form of the distribution of ribosome assembly times — P SA ( s ).The Laplace-transform naturally appears in this problem be-cause upon averaging, a complicated convolution of many ran-dom independent factors e.g. assembly and translation du-rations, factor under its operation thus leading to a simplerequation [4], (see SI, section III). The equation we obtain is, P ( s ) P ( s )( αn P SA ( s ) + 1) = 1 , [1] where n = 54 is the number of ribosomal protein subunits ina ribosome (there are 52 ribosomal protein species but two ofthem appears in tandem dimers and hence n = 54 [13]). Therate parameter s that solves Eq. 1 above is the asymptoticgrowth rate, which is the growth rate of a large asynchronouscollection of ribosomes, under the assumption that the supplyof material inputs and rRNA is not limiting. Our formalismallows us to derive other growth laws, when one of these as-sumptions breaks (SI, section III).To illustrate the use of Eq. 1 consider a deterministic settingwith all the process durations fixed. The resulting growth lawis then αn e − µ ( τ + τRPn + τ SA ) + e − µ ( τ + τRPn ) = 1 with τ RP is theribosome workload — duration to make all the ribosomal pro-teins by a single ribosome, τ SA is the ribosome self-assemblytime, and τ is the ribosome rest time between consecutivetranslations. Two models for ribosome self-assembly.
Before deriving Eq.1, we discuss how we model the self-assembly process — a cru-cial ingredient in the derivation. Our first assembly model isvery simple, assuming that whenever all 54 ribosomal proteinsare present in stoichiometry — one per type, one ribosome as-sembly process is initiated which will end after an assemblyduration τ SA . We summarize this by the following reactionequation RP + . . . + RP + rRNA → R ∗ , [2] where R ∗ is a new ribosome.Actual ribosome assembly is significantly more elaborate.As discovered by Nomura [14], Nierhaus [15], Williamson [20]and others [13] the process of ribosome assembly proceedaccording to a partial temporal order. Some proteins can-not bind before other proteins are docked and after the sub-assembled ribosome had properly conformed (also see SI, sec-tion II).To test the sensitivity of our growth law to the intricacies ofthe self-assembly process we also considered a second model,where we roughly arranged all the ribosomal proteins intothree groups. The first group is the primary binders — ribo-somal proteins that directly attach to rRNA, if it is present.After a duration τ SA , the sub-assembled ribosome we denoteby A is ready and can allow the secondary binders — thesecond group of ribosomal proteins, to attach to it. After aduration τ SA the sub-assembled structure we denote by B isformed, and the third group of ribosomal proteins can bindto it thus forming, after a duration τ SA a new ribosome. Wesummarize the second model as: RP + . . . + RP l + rRNA → A [3] RP l +1 + . . . + RP l + A → BRP l +1 + . . . + RP l + B → R ∗ , where l , l and l are the sizes of the three ribosomal proteingroups and l + l + l = 54.We find that as long as rRNA is not limiting, the twomodels lead to the same overall growth law as describedby Eq. [1], under the following conditions; (i) If the limit-ing ribosomal proteins are the primary binders and we set (cid:105) “overlapping˙ribosome˙reproduction˙cycles˙via˙self˙assembly˙Arxiv˙Version” — 2018/7/25 — 1:39 — page 3 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) τ SA = τ SA + τ SA + τ SA ; (ii) If the secondary binders arelimiting, and we set τ SA = τ SA + τ SA ; (iii) If the tertiarybinders are limiting and we set τ SA = τ SA . This model couldbe readily generalized to accommodate actual assembly maps.See SI, section III for derivation of this result, and for treat-ment of the case where rRNA is limiting, and section II foran example of the distribution of assembly times for the 16Ssmall ribosomal subunit, using Nomura’s assembly map. Pipelined ribosome assembly — derivation of the new growthlaw.
To derive Eq. 1, consider a collection of ribosomes thatare busy translating mRNA of various proteins and of ribo-somal proteins. Each ribosome, upon completing its currenttask, idle for a certain period τ after which with probability α it will synthesize a random ribosomal proteins, and withthe complementary probability β = 1 − α will make anotherprotein.The ribosome lifetime is an important parameter whose ma-nipulation affect the level of ribosomes in the cell relative toother proteins (and their lifetime), but its effect on the growthlaw is negligible as long as the life time is much larger thanthe doubling time [5]. Thus, with the purpose of deriving anequation for the growth rate in mind, we assume ribosomelifetime is infinite.In the first self-assembly model (Eq. 2), if all 54 ribosomalproteins exist in stoichiometry then self-assembly of a new ri-bosome is initiated. In the second self-assembly model, if theentire set of primary binding ribosomal proteins and their tar-get rRNA exist, self-assembly is initiated. However, if thesecondary binders and their target A do not exist in stoi-chiometry, the second step in the assembly will be delayed,and similarly for the third stage (see Eq. 3) .Let n ( t ) be the number of ribosome that enter the “rest”state at time t . Three processes contribute to n ( t ). The firstcontribution is from the ribosomes that finished translating ri-bosomal proteins. The second contribution is from ribosomesthat finished translating other proteins. The last contributionis from the ribosomes that were just finished being assembled.If all the durations are deterministic then the number ofribosomes that just finished translating ribosomal proteins ison average αn ( t − τ − τ RP n ) since α is the fraction of freeribosomes that upon exiting rest mode will be allocated formaking ribosomal proteins, and the average duration of mak-ing a single ribosomal protein is τ RP n . Similarly, the num-ber of ribosomes that finished translating other proteins is βn ( t − τ − τ P ), where β = 1 − α , and τ P is the average timeto make a non-ribosomal protein.The number of ribosomes that just finished being as-sembled at time t , R ∗ ( t ) depends on the assembly model.Using the first assembly model (Eq. 3) yields R ∗ ( t ) =min i ∈{ ,..., } ( n ( t − τ SA ) , ..., n ( t − τ SA )), where n i ( t ) is thenumber of ribosomal proteins of type i that completed beingsynthesized at time t .On the other hand, the number of ribosomal proteins oftype i that finished being synthesized at time t is given by n i ( t ) = αn n ( t − τ − τ RP n ), for all i . We thus obtain, n ( t ) = αn ( t − τ − τ RP n ) + βn ( t − τ − τ P ) + αn n ( t − τ − τ RP n − τ SA ) . [4] This formula is valid under the assumption that n ( t ) (cid:29) τ RP n and the fraction allocatedtowards any particular ribosomal protein is αn . This homo- geneity of translation rates and allocations is not essential butsimplifies the analysis.Equations 4 takes into account the pipelining in the repro-duction process, where ribosomes are free to restart anotherround of translation of ribosomal proteins with probability α or other proteins with probability β , before the self-assemblyprocess is finished. To obtain the growth law, we now as-sume that all the durations are random and independentlydistributed. Averaging over all possible durations, and takingthe Laplace transform, we obtain n ( s ) = n ( s ) P ( s ) (cid:16) αP ( s ) + βP p ( s ) + αn P ( s ) P SA ( s ) (cid:17) . [5] The average length of all non-ribosomal proteins in
E. coli ’sgenome is 2 . .
27 [7, 21] at doubling times of 21.5 min, onefinds that P p ( s ) = P ( s ), i.e. the actual load for making ageneric protein is the same as the load of making a ribosomalprotein. Inserting this to Eq. 5 and dividing both sides by n ( s ) we recover Eq. 1. Serial ribosome assembly.
Consider next serial ribosome repro-duction. In serial reproduction ribosomes cannot start a newround of reproduction before the current round is finished. Toaccommodate that, Eq. 4 has to be modified as follows: n ( t ) = αn ( t − τ − τ RP n − τ SA ) + βn ( t − τ − τ P ) + αn n ( t − τ − τ RP n − τ SA ) [6] where we emphasized the only change made in Eq. 4 — theaddition of a delay τ SA for all the ribosomes that are involvedin the process of making ribosomal protein subunits. Thisdelay ensures that all the ribosomes that are involved in ribo-somal protein synthesis will be able to continue to translateonly after the new ribosome they were involved in makingfinish the assembly process. Hence, no new generation of ri-bosome is started before the previous generation is completed,and so, no overlapping reproduction cycles occurs in the se-rial model. The typical, tacit assumption is a limiting caseof this serial model, whereas the assembly time is negligiblecompared to the synthesis of all the ribosomal proteins andhenceforth dropped.Averaging over all durations in Eq. 6, taking the Laplacetransform as before and assuming P p ( s ) ≈ P ( s ) we obtain n ( s ) = n ( s ) P ( s ) P ( s ) P SA ( s )( α + αn )+ βn ( s ) P ( s ) P ( s ). Di-viding by n ( s ) we obtain the serial growth law: P ( s ) P ( s ) P SA ( s )( αn + α ) + βP ( s ) P ( s ) = 1 . [7] Both models — pipelined self-replication (Eq. 1) and serialself-replication (Eq. 7), agree in the limit where the assemblytime tends to zero τ SA → On the serial limit with zero idling and assembly time.
An im-portant limiting case of Eq. 1 we now turn to consider is thecase with zero idling and assembly durations. To get this limitwe set P SA ( s ) = P ( s ) = 1 in Eq. 1. The resulting equationis then P ( s )(1 + αn ) = 1. Setting P SA ( s ) = P ( s ) = 1 in Eq. 7leads to the same equation (see Fig. 2).Lets assume that the duration for translating all ribosomalproteins is distributed exponentially with an average τ = τ RP n ,i.e. all ribosomal proteins are produced in parallel (as before, τ RP is the average duration to make all the ribosomal proteins,and n = 54). Then the Laplace transform of the exponentialdistribution is given by P ( s ) = sτ . This yields the growth Footline Author PNAS
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Volume Issue Number (cid:105) “overlapping˙ribosome˙reproduction˙cycles˙via˙self˙assembly˙Arxiv˙Version” — 2018/7/25 — 1:39 — page 4 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) Fig. 1.
Model for pipelined self-reproduction of ribosomes. (A) Ribosomes trans-late ribosomal proteins probabilistically (with probability α ). After each translationthey idle for a duration τ , after which they return to translate proteins (ribosomalwith probability α and other with the complementary probability). When ribosomalproteins accumulate in stoichiometry, the self-assembly process starts. While self-assembly proceeds, more ribosomal proteins are concurrently being synthesized. Uponcompletion of the self-assembly process a new ribosome is added to the pool of exist-ing ribosome, and joins the rest-work cycle. Inset (B) shows the temporal structure ofthis process, with time running from up to bottom. Note the overlapping reproductioncycles, caused because ribosomes keep making ribosomal proteins with probability α ,concurrently with the assembly process. Inset (C) shows the corresponding serial pro-cess, where the old and the newly added ribosome, initiate a new reproduction roundsimultaneously. law µ = ατ RP , which is also the growth law for a serial assem-bly, where each ribosome makes all the ribosomal proteins.Alternatively, we can assume that the time to make all n ri-bosomal proteins is Deterministic by setting P ( s ) = e − τRPn s .This yields the growth law µ = nτ RP ln (cid:0) αn (cid:1) ≈ ατ RP , since α < n = 54 (cid:29)
1. For an extension to second orderof this model, with interesting biological implication see [12].Hence we conclude that in the absence of an assembly step,to first order, both exponential and deterministic distributionseffectively yield the same growth rate, if the number of parallelprocesses is n is large. Comparing pipelined to serial self-replication.
In Fig. 1A weillustrate the pipelined self-reproduction model with determin-istic variables. Ribosomes in the rest-work cycle translate ri-bosomal proteins or other proteins, then rest, then translateagain. The ribosomal proteins enter into “pools” and when-ever a full set exists, the self-assembly process of another ri-bosome is initiated. Self-assembly happens concurrently tothe translation process, and, upon completion, the newly syn-thesized ribosome joins the rest-work cycle, starting from restmode. Setting τ SA = 0 yields a particular serial limit. Fig-ure 1B illustrates the growing tree of ribosomes as a functionof time — advancing from top to bottom, and assuming forsimplicity that β = 0.Also noticeable in Fig. 1B, is the overlap between differentgenerations, due to the fact that existing ribosomes start anew generation prior to the completion of the previous gen-eration. For comparison, in Fig. 1C presents the serial limit(also with β = 0), whereas newly formed ribosomes and ex-isting ones start a new round simultaneously. Evidently, for agiven critical path duration for the formation of a single givenribosome τ RP + nτ + τ SA the serial limit has slower growthrate, see also Fig. 2. Fig. 2.
Comparison between the overlapping reproduction cycle model (Eq.1) and the serial model (Eq. 7). Relative increase in growth rate for Markovian(red line) and deterministic (blue line) distributions, as a function of the percent ofoverlap between the self-assembly duration and the overall net duration of making anew ribosome T = τ RP + τ SA assumed constant. The parameter τ RP is theribosomal protein workload — the time to make all the ribosomal proteins by a singleribosome. The nominal growth rate, µ = αT + nτ represents the completely seriallimit where the doubling time of a single ribosome is solely composed of the timeto produce all its ribosomal proteins and the assembly time is set to zero. Using anallocation parameter α = 28% as measured in [16] we calculated the growth rateusing Eq. 1 as a function of the assembly time. A clear increase in growth rate isobserved for the overlapping reproduction cycle model (as described by Eq. 1), aswell as a difference between completely Markovian (red upper line) and completelydeterministic (blue lower line) limits. For comparison we plotted the relative growthrate for the serial growth law (as described by Eq. 7), for which the relative growthrate decrease with the overlap parameter, and the difference between the Markovianand the deterministic distributions is negligible. To aid understanding we also presentan inaccurate illustration of the structure of the temporal tree for overlap parameters τ SA T = 0 . and τ SA T = 0 . but with α = 1 . Note that for τ SA T = 50% the increase in growth rate of the overlapping reproduction cycles model is by . Self assembly step increase growth rate relative to the se-rial case for a given critical path duration.
To study the effectof overlapping reproduction cycles on the growth rate let usassume, hypothetically, that the parameters τ RP and τ SA aretunable, provided that their sum τ RP + τ SA is kept constant. Inaddition we also keep τ constant, so overall, the critical pathduration to make a single particular ribosome τ RP + nτ + τ SA is assumed constant throughout the following discussion.In Fig. 2 we plot the overall growth rate as a function ofthe ratio between the self-assembly time to the constant netproduction time, τ SA τ RP + τ SA .The monotonically increasing solid red line represent thecase where the duration to make ribosomal proteins, the as-sembly duration, and the idling duration are all exponentiallydistributed (with an average duration of τ RP n and τ SA and τ respectively). The monotonically increasing solid blue linerepresent the case where the durations to make ribosomal pro-teins, the assembly duration, and the idling duration are alldeterministically distributed (with an exact duration of τ RP n , τ SA and τ respectively). The growth rate in the Markoviancase is noticeably larger than in the deterministic case as seenin Fig. 2, in contrast to the serial limit.As seen in Fig. 2, as the overlap increases, the growth rateincreases monotonically and non-linearly, due to the overlap inthe reproduction cycles. Specifically, when the overlap param-eter is 50%, the increase in growth rate relative to the serial (cid:105) “overlapping˙ribosome˙reproduction˙cycles˙via˙self˙assembly˙Arxiv˙Version” — 2018/7/25 — 1:39 — page 5 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) limit, is 50%. To contrast, the monotonically decreasing blueand red dashed lines represent the Markovian (red) and de-terministic (blue) serial limits, which are essentially identical.The reason for the decrease in the growth rate as a functionof the overlap in the serial model is because as the assemblytime increases, ribosomes translating ribosomal proteins aredelayed for longer durations before they are allowed to starttranslating ribosomal proteins for the next generation.We emphasize the difference between the serial ribosomereproduction scenario with a zero assembly time and the over-lapping ribosome reproduction scenario with an assembly time τ SA that is equal to the synthesis time of all ribosomal pro-teins τ RP . When these two scenarios are compared withthe same critical path duration for making a single ribosome T c = τ RP + nτ + τ SA , the overlapping reproduction cyclescenario will have a growth rate that is 50% larger than theserial ribosome reproduction scenario. This is in-spite of thefact that in both these scenarios at steady-state a new ribo-some will be added whenever an existing ribosome completestranslation of a single ribosomal protein.This is because in the overlapping reproduction scenariowith τ SA = τ RP the existing ribosomes will finish on average anew set of ribosomal protein subunits for the next generation,just when the previous generation completes the assembly ofan earlier ribosome. In contrast, in the serial reproductionscenario with τ SA = 0, all the ribosomes will finish on aver-age one round, and only then will start another round. As weargue below, the latter scenario is what we predict for E. coli growing in rich defined medium.
Calculating ribosome in-vivo assembly time.
We now turn toapply Eq. 1 for estimating the in-vio ribosome assembly timein
E. coli growing at 21 min doubling time. Using Eq. 1
Fig. 3.
Relation between the ribosome assembly time τ SA and the ribosomeprotein workload τ RP for a fixed doubling time of . [min] as measured in [16].Upper bound is set by a Markovian (Exponential) distribution. Lower bound is set bya deterministic distribution. Green area is calculated assuming the fraction of idlingribosome is zero φ = 0 . Transparent purple area is calculated assuming that thefraction of idling ribosome is φ = 15% . Dashed lines represent bounds on thein-vivo ribosome assembly time when the ribosome workload is τ RP = 6 . [min]which we deduce from an experimental measurements of average translation rate in arich defined medium. The ribosome assembly time is at least . min. If the fractionof idling ribosome is φ = 10% , the minimal ribosome assembly time is ∼ min. Fig. 4.
Growth rate as a function of Lamotrigine concentration. We calculatedthe growth rate as a function of Lamotrigine concentration assuming its biologicalmechanism halts assembly irreversibly. Blue circles are data points taken from [11].Red solid line is a fit using Eq. 9 and 8. The two fitting parameters were the saturationparameter K and Hill’s factor h . Inset A shows the effect of limiting the expressionof a ribosome assembly chaperon on the growth rate, assuming lack of chaperonsmay cause the traversed assembly pathway duration to be longer, e.g. because of theneed to escape from kinetic traps that would have been avoided if chaperons wereabundant. When chaperons are abundant ( cK (cid:29) ) the probability to reversibly fallto a kinetic trap is negligible and chaperons do not limit growth rate. On the otherhand, when chaperons are deficient ( cK (cid:28) ) the probability to fall into a kinetictrap increases, which in turn cause an increased delays in the ribosome assembly time.We present three such delays; (i) Four times longer than the nominal (solid line); (ii)Eight times longer than nominal (dotted line); (iii) Sixteen times longer than nominal(dashed line). In the limit of a deep trap the delay tend to infinity and we recover theirreversible model. along with experimental measurements of ribosome cellularallocation fractions and the average translation speed, we candeduce upper and lower bounds on the in-vivo ribosome as-sembly time. This is done as follows. First, we use ribosomeprofiling data from [16] which reports that for growth in arich defined (MOPS) medium at 37 o C the fraction of ribo-somes translating ribosomal protein genes is α A = 28% (thedoubling time in these conditions was 21 . ± . − τ thatyields φ = 15% idling ribosomes to be τ = 0 .
02 minutes.Next using measurements of in-vivo ribosome translationrates we calculated the ribosome workload — the average du-ration for a single ribosome to translate all 54 ribosomal pro-tein subunits. The total number of amino-acids in all theribosomal proteins (including the proteins that are presentin duplicate) is 7249 amino-acids. The average translationrate of a ribosome in rich defined medium is measured to be19 . B in [8],see also references therein. Thus, the ribosome workload is τ RP = . = 377 sec = 6 . ln 2 µ = 21 . α = 0 .
28 we numerically found all possible pairsof parameters ( τ SA , τ RP ) that yield the same doubling timeof 21 . Footline Author PNAS
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Volume Issue Number (cid:105) “overlapping˙ribosome˙reproduction˙cycles˙via˙self˙assembly˙Arxiv˙Version” — 2018/7/25 — 1:39 — page 6 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) sis and ribosome assembly. The first extreme distribution isa deterministic distribution, i.e. a distribution without anynoise.The second extreme distribution is the distribution that hasmaximal noise given the average duration, which we derive bymaximizing the entropy given the average. The maximal en-tropy distribution with a positive support (as durations cannotbe negative) and a given average is the exponential (Marko-vian) distribution [18]. In the absence of contrary evidenceregarding the coefficient of variation being larger than one,any actual distribution of durations would have to lie some-where in between these two extremes. We thus utilize thesetwo distributions to set upper and lower bounds on the assem-bly durations without worrying about the accuracy of thesedistributions as models for biological noises.For ribosome workload of τ RP = 6 . φ = 0, the assembly time in rich definedmedium is found to lie between 10 . . φ = 15% the assembly time is estimated to bebetween τ SA = 5 . τ SA = 6 min, for exponential distributions. This is ∼ −
21 aa/sec,with the best fitted value for rich defined (MOPS) mediumand doubling time of 21 . Modeling the effect of a drug that inhibits ribosome assembly.
In this section and the next we demonstrate that our model-ing approach is versatile by applying it to quantitatively studyphenotypes with faulty assembly of ribosomes.Recently, the drug Lamotrigine was reported to have a di-rect detrimental effect on ribosome assembly in live E. coli [11].In the presence of Lamotrigine, E. coli cells were observed toaccumulate non-functional partially assembled ribosome com-plexes, which subsequently led to slower overall growth [11].We can model the effect of such a drug, by expanding oursimplified model, adding a probability that the ribosome self-assembly process will fall into an irreversible trap, with a prob-ability that depends on the Lamotrigine concentration. We usetwo parameters to characterize the probability of an assemblyfailure p AF as a function of the Lamotrigine concentration —the saturation parameter K and Hill’s cooperativity factor h : p AF ( c ) = K h K h + c h . [8] The probability or an assembly failure goes to one when c (cid:29) K , and tends to zero when c (cid:28) K . The functionalequation for the growth rate Eq. 1 is modified to read P ( s ) P ( s )( αn P SA ( s ) p AF ( c ) + 1) = 1 . [9] Using Eq. 9 to calculate the growth rate as a function ofthe Lamotrigine concentration, and the inferred assembly time from the previous section, we fitted the data from [11] to ob-tain an estimate for K and h . We find K = 52 [nM/L] and h = 0 .
75 indicating non-cooperative effect on the assembly.It would be of interest to biochemically measure K and h to ascertain our prediction and to elucidate the biochemicalmechanism underlying the non-cooperative Hill factor. Predicting the effect of limiting assembly chaperons.
In theprevious section we assumed that the effect of Lamotrigineon the ribosome assembly process is irreversible. We also as-sumed that all the assembly chaperons are abundant. Whatif we want to model a situation in which the expression of aspecified assembly chaperon is externally controlled? Clearly,if lack of certain assembly chaperons cause an effectively irre-versible failure in the assembly process, we can utilize Eq. 9by adapting the K and h parameters to model the chaperonunder consideration. However, the role of certain chaperonsis to reduce the probability of falling intro a kinetic trap [13].If the kinetic trap is “shallow” falling to it only slows downthe assembly process. This cause two detrimental complemen-tary effects; the average assembly duration increases and thecoefficient of variation of the assembly duration increases.To model shortage of chaperons that cause such combineddetrimental effect we introduce the Laplace transform for themodified distribution of assembly times ˆ P SA ( s ) as a functionof p ( c ) — the probability to choose a longer assembly pathwayas a function of the concentration of the chaperon — c . Whenthe concentration of the chaperon protein far exceeds its satu-ration parameter c (cid:29) K the assembly time distribution is thepreviously used assembly time distribution ˆ P SA ( s ) = P SA ( s ).When c (cid:28) K the distribution shifts to a longer distributionwhich we assume for the sake of simplicity to be exponential,hence with Laplace transform that equals to λ c s + λ c . We requirethat λ c > dP SA ( s ) ds | s =0 so that the alternative assembly routewhich the chaperon helps avoiding is longer than the nominalone on average. Hence,ˆ P SA = c h K h + c h P SA ( s ) + K h K h + c h λ c s + λ c . [10] It follows that the relation between the growth rate and theconcentration of the chaperon will be given by, P ( s ) P ( s )( αn ˆ P SA ( s ) + 1) = 1 . [11] In Fig. 4A we plot the relation between growth rate and theconcentration of a chaperon protein that assist in assembly asdescribed above, by solving Eq. 11 . We show three types ofkinetic traps; shallow (solid upper line); medium (dotted mid-dle line) and deep (dashed lower line). From an evolutionaryperspective we can expect the level of expression of ribosomeassembly chaperon to be correlated with the “severity” of thekinetic trap it assist to mitigate. This can be tested by corre-lating their expression levels and detailed biophysical knowl-edge, which steadily accumulates, regarding the free energylandscape of ribosome assembly and the role of the chaperonsin the assembly process.
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