Biomolecular resource utilization in elementary cell-free gene circuits

Biomolecular resource utilization in elementary cell-free gene circuits
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  Biomolecular resource utilization in elementary cell-free gene circuits Dan Siegal-Gaskins 1 , Vincent Noireaux 3 , and Richard M. Murray 1 , 2  Abstract —We present a detailed dynamical model of thebehavior of transcription-translation circuits  in vitro  that makesexplicit the roles played by essential molecular resources. Aset of simple two-gene test circuits operating in a cell-freebiochemical ‘breadboard’ validate this model and highlight theconsequences of limited resource availability. In particular, weare able to confirm the existence of biomolecular ‘crosstalk’and isolate its individual sources. The implications of crosstalkfor biomolecular circuit design and function are discussed. I. INTRODUCTIONThe last several decades have witnessed significant ad-vances in the biological sciences, in part through the ap-plication of techniques from historically distinct areas suchas mathematics, computer science, physics, and engineering.Among the many insights that have come from this inter-disciplinary approach is the existence of indirect couplingor  crosstalk   between biological circuit (‘biocircuit’) compo-nents [1]–[5]. While evidence suggests that crosstalk comesabout through a shared molecular resource pool, neither thisfact nor the identity of the specific resources are in generalevident in the mathematical representations of biocircuits thatare commonly used.To illustrate, consider a simple biocircuit consisting of twoconstitutively-expressing genes,  x  and  y , which code for ageneric protein  X  and fluorescent reporter  Y , respectively(Fig. 1). One ‘naive’ description of the system is as follows: x  is transcribed into mRNA  x m  and translated into  X , and y  is transcribed into mRNA  y m  and translated into a dark reporter protein  Y d  that matures into the visible  Y . Allprotein and mRNA species are also degraded (and/or dilutedthrough cell division if in growing cells). If we assumethat reactions take place in sufficiently large volumes (sothat stochasticity in molecule concentrations does not affectthe overall dynamics), and that circuit dynamics can beapproximated using mass-action kinetics, then the systemmay be described by the following set of ordinary differentialequations (ODEs): d[ x m ] / d t  =  k x,TX [ x ] − k xm,deg [ x m ]  (1a) d[X] / d t  =  k x,TL [ x m ] − k X,deg [X]  (1b) For correspondence, please contact , , or . Affiliations:(1) Department of Bioengineering and (2) Department of Control andDynamical Systems, California Institute of Technology, 1200 E. CaliforniaBlvd., Pasadena, CA 91125, USA; (3) School of Physics and Astronomy,University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455,USA.  x x m X  y y m YY d Fig. 1. Schematic of a simple constitutive two-gene circuit. Noninteractinggenes  x  and  y  are transcribed into mRNAs  x m  and  y m , which are thentranslated into a generic protein  X  and immature fluorescent reporter  Y d ,respectively.  Y d  then matures into the visible  Y . mRNAs and proteins mayalso be degraded and/or diluted (not shown). d[ y m ] / d t  =  k y,TX [ y ] − k ym,deg [ y m ]  (1c) d[Y d ] / d t  =  k y,TL [ y m ] − ( k mat  +  k Y,deg )[Y d ]  (1d) d[Y] / d t  =  k mat [Y d ] − k Y,deg [Y]  (1e)where the  k i  are the various reaction rates of the circuit.While this type of model is common for the analysis of general biocircuit dynamics [6], it shows no crosstalk ormolecular competition effects—the model circuit output  [Y] is completely unaffected by the presence of   x .This particular model formalism assumes that the essentialtranscription and translation (TX-TL) machinery, includingtranscription initiation factors, RNA polymerase (RNAP),and ribosomes, exist in sufficiently high concentrations andthat their utilization by one component has no noticeableeffect on others. Clearly, for the study of crosstalk, anapproach that does not rely on these assumptions is required.To this end, various theoretical frameworks have recentlybeen developed (e.g., [7], [8 however, models that (i) may be used to explore resource utilization effects, (ii)distinguish between different sources, and (iii) are supportedby experimental data are still lacking.We developed a detailed mathematical model of   in vitro TX-TL circuits that consist of only two genes, with a levelof complexity sufficient to capture effects that may arisevia the sharing of fixed-concentration molecular resources.An important design criterion was that the model have ageneral form that could be easily expanded to more complexcircuits, and that the individual sources of crosstalk and theirrelative contributions to the total could be identified with asmall number of simple experiments. The resultant modelis shown in Section II, with the experimental ‘breadboard’used to validate it described in Section III. The simulated andexperimental results identifying the sources of biomolecularcrosstalk are presented in Section IV, along with predictionsas to how the level of crosstalk may be affected by resourceand DNA concentrations and binding affinities. In SectionV we show additional results for the special case when the  second gene is a constitutively-expressing alternative sigmafactor. A discussion of all these results is given in SectionVI.II. DETAILED MODEL FOR CONSTITUTIVEEXPRESSION OF TWO GENES IN VITROFor the two-gene system described above, a more detailedmodel for  in vitro  operation that makes explicit the role of TX-TL machinery is E + S1  ES1ES1 +  x  x :ES1ES1 +  y  y :ES1 x :ES1 → x  + ES1 +  x m y :ES1 → y  + ES1 +  y m x m  → ∅ y m  → ∅ R +  x m  x m :RR +  y m  y m :R x m :R → x m  + R + X y m :R → y m  + R + Y d Y d  → YX → ∅ Y d  → ∅ Y → ∅  . where  E ,  R ,  S1 , and  ES1  represent free core RNAP, free ri-bosome, the primary sigma factor (necessary for transcriptioninitiation), and the primary sigma factor-RNAP holoenzyme,respectively. RNAP holoenzymes bound to DNA and ribo-somes bound to mRNA transcripts are represented by ( : ).The ODEs for this expanded model are: d[ x m ] / d t  =  k x,TX [ x :ES1] − k xm,deg [ x m ] − k X+ [R][ x m ]+ ( k X − +  k x,TL )[ x m :R]  (2a) d[ y m ] / d t  =  k y,TX [ y :ES1] − k ym,deg [ y m ] − k Y+ [R][ y m ]+ ( k Y −  +  k y,TL )[ y m :R]  (2b) d[ x m :R] / d t  =  k X+ [R][ x m ] − ( k X − +  k x,TL )[ x m :R]  (2c) d[ y m :R] / d t  =  k Y+ [R][ y m ] − ( k Y −  + k y,TL )[ y m :R]  (2d) d[X] / d t  =  k x,TL [ x m :R] − k X,deg [X]  (2e) d[Y d ] / d t  =  k y,TL [ y m :R] − ( k mat  + k Y,deg )[Y d ]  (2f) d[Y] / d t  =  k mat [Y d ] − k Y,deg [Y]  (2g) d[ES1] / d t  =  k ES1+ [E]  [S1] tot − [ES1]  − k ES1 − [ES1] − k x + [ES1]  [ x ] tot − [ x :ES1]  + ( k x −  + k x,TX )[ x :ES1] − k y + [ES1]  [ y ] tot − [ y :ES1]  + ( k y − +  k y,TX )[ y :ES1]  (2h) d[ x :ES1] / d t  =  k x + [ES1]  [ x ] tot − [ x :ES1]  − k x − [ x :ES1]+  k x,TX [ x :ES1]  (2i) d[ y :ES1] / d t  =  k y + [ES1]  [ y ] tot − [ y :ES1]  − k y − [ y :ES1]+  k y,TX [ y :ES1]  (2j) with conservation relations [E] = [E] tot − [ x :ES1] f  ( x ) − [ y :ES1] f  ( y ) − [ES1]  (3a) and [R] = [R] tot − [ x m :R] g ( x ) − [ y m :R] g ( y )  .  (3b) [E] tot ,  [R] tot ,  [S1] tot ,  [ y ] tot , and  [ x ] tot  represent the fixedtotal concentrations of species in the reaction volume, andfactors of the form  f  ( a ) = 1 +  k a,TX ( L a /V  TX )  and g ( a ) = 1 +  k a,TL ( L a /V  TL )  account for the loading of multiple holoenzymes and ribosomes on the gene and mRNAtemplates [9].  L a  is the length (in bp) of gene  a  and  V  TX and  V  TL  represent the rates of progression (in nucleotidesper second) of RNAP along the DNA and ribosome alongthe mRNA, respectively.It is worth noting that, in contrast with the more com-mon type of biocircuit model that assumes the validity of the Michaelis-Menten kinetics approximation [10], we havemade no assumptions as to the timescales of various reactionsor the relative concentrations of reacting species.III. A CELL-FREE ‘BREADBOARD’ FORBIOCIRCUIT TESTINGExperimental verification of biocircuit models such as thisis often challenging, due in part to the complexity of thesystems and the context-dependence of their components(see, e.g., [11]–[13]). As a result there has been considerableinterest in using simple  in vitro  platforms for circuit develop-ment, characterization, and model validation [14]. Importantsteps have been made in recent years with the developmentof a TX-TL ‘breadboard’: an  in vitro  system that allows TX-TLprocesses to take place using molecular machinery extractedfrom  E. coli  [15], [16]. Endogenous DNA and mRNA fromthe cells is eliminated so that biocircuits of interest may bestudied in isolation with no other genetic material present.The breadboard also allows for tight control over reactionconditions and the concentrations of circuit components—control which is difficult to achieve  in vivo . It is thus an idealenvironment for establishing the validity of biocircuit modelsin general and for confirming the existence of crosstalk insimple genetic circuits.IV. RESULTSIn our cell-free system, protein species are stable againstdegradation and there is no dilution through cell division.We thus set  k X,deg  =  k Y,deg  = 0 . Under these conditions,the model (2) predicts the existence of a time  T   after whichthe fluorescent protein concentration increases linearly; i.e., [Y] ∝ t  for  t > T  . Time (mins) 206010014001020    [   Y   ]   (      μ    M   ) Reporter only Fig. 2. Expression of a fluorescent reporter  y  in the TX-TL breadboardsystem, with  [ y ] tot  = 2  nM. Solid line is the result of simulation, andshaded area indicates the standard deviation (n=2) of reporter concentrationas determined by fluorescence. When only the fluorescent reporter gene is present ( [ x ] tot  =0  and  [ y ] tot  = 2  nM), both simulation and experimentshow the expected linearly-increasing fluorescent protein  concentration (for  t > T  ) and are in good agreement (Fig. 2).The behavior of the system when two genes are presentis less easily predictable. It is reasonable to suspect thatRNAP ( E ) and ribosomes ( R ) contribute to total crosstalk,since an increase in the concentration of   x  could result ina sequestration of   ES1  away from  y , and an increase inthe amount of   x m  could decrease the amount of free  R available to translate  y m . We investigated—computationallyand experimentally—two different circuits designed to testfor RNAP and ribosome utilization effects and to distinguishbetween them: in one case,  x  encodes a small untranslatedRNA to which there is no ribosomal binding (Fig. 3A), and inthe other, it encodes a protein that has no direct interactionswith  y  (Fig. 3B). +  x y y m  x m Y (A) Untranslated RNA circuit  x y y m  x m Y (B) Noninteracting protein circuit XRS1EES1 + S1EES1R Fig. 3. Schematics of tested circuits. Genes  x  and  y , driven by constitutivepromoters (filled rectangles), are transcribed into RNAs  x m  and  y m .  x m  iseither (A) untranslated or (B) translated into a generic noninteracting protein X .  y m is translated into an immature fluorescent reporter  Y d  which maturesinto the visible  Y . Additional arrows represent complex formation and theregulatory roles of various molecular species as described in the text.  A. Untranslated RNA circuit: simulated and experimentalresults The use of an untranslated RNA molecule allows usto determine the contribution of RNAP alone to crosstalk (Fig. 3A). In simulations, the rate of association of   R  to x m ,  k X+ , is set to zero. At two low but biologically-relevantconcentrations,  [ x ] tot  = 0 . 1  nM and  [ x ] tot  = 1  nM, wefind no discernible effect on the rate of production of   Y ;both simulated functions are linear (for  t > T  ) and overlaidand consistent with experiment (Fig. 4A). Clearly, when theadditional DNA is present at low concentrations, and forthis particular set of rate constants and binding affinities,the crosstalk introduced by RNAP holoenzyme alone isnegligible.  B. Noninteracting protein circuit: simulated and experimen-tal results We now consider the effect of ribosome sequestration oncircuit output when the second gene codes for a noninteract-ing protein (Fig. 3B). We use the full model of Section IIwith all rates and concentrations positive. As with the single-gene control and untranslated RNA circuit, simulations showa linear increase in output for  t > T  ; however, the modelpredicts a slope for  d[Y] / d t  that is different for  [ x ] tot  = 0 . 1 nM and  [ x ] tot  = 1  nM (Fig. 4B), and the experimentaldata is consistent with this prediction. Thus, unlike RNAP,ribosomes appear to be a limiting resource and that even lowlevels of auxiliary ribosome targets can lead to a reductionin the circuit output. 20601001402060100140(B) Noninteracting protein01020 Time (mins)    [   Y   ]   (      μ    M   ) (A) Untranslated RNA 0.1 nM1 nM0.1 nM1 nM Fig. 4. Simulation and experimental results for implementations of thecircuits schematized in Fig. 3, in which the second gene encodes (A) asmall untranslated RNA, and (B) a noninteracting protein. Solid lines aresimulation results, and shaded areas indicate the standard deviation (n=2)of reporter concentration as determined by fluorescence for  [ x ] tot  = 0 . 1  nMand  [ x ] tot  = 1  nM. C. Sensitivity of output to changes in RNAP and ribosomeconcentrations Limited resources can have a significant effect on therobustness of even simple circuits [17], a fact that is trueboth  in vivo  and  in vitro . However, given the hard limitson resource concentrations in cell-free environments (ascompared with  in vivo  systems, in which the levels of RNAPmolecules and ribosomes are regulated to some degree by thecell [18], [19]), the potential for adverse limit-related effects is amplified. We thus used our full noninteracting proteinmodel to determine the sensitivity of the output to changesin the concentration of total core RNAPs  [E] tot  and ribosomes [R] tot . We find that  d[Y] / d t  is completely insensitive tochanges in RNAP concentration when RNAP levels are high(Fig. 5, left). Interestingly, the system naturally operates inthis regime with the nominal value of   [E] tot . On the otherhand, the total concentration of ribosomes has a significanteffect on  d[Y] / d t , one that increases dramatically withincreasing  [R] tot  (Fig. 5, right).  D. Gene concentrations, binding affinities, and resourcelimits We may also use our model to answer questions aboutthe system that would be difficult to address experimentally.These include determining the level of additional genesabove which RNAP becomes a limiting resource, and belowwhich the ribosomal loading does not lead to any signif-icant crosstalk. In Fig. 6 we see how  d[Y] / d t  is affected  200 500 2000 5000 [E] tot  (nM) [R] tot  (nM) 20 50 100 200 5002050100200500  0.1 nM1 nM0.1 nM1 nM     (   n    M    /   m    i   n    )    d    [    Y    ]    /    d    t    t   >   T Fig. 5. Fluorescent protein production rate for  t > T   as a function of total core RNAP (left) and ribosome (right) concentrations, for  [ x ] tot  = 0 . 1 nM and  [ x ] tot  = 1  nM. Dashed lines indicate nominal values of   [E] tot  and [R] tot  in breadboard environment. 0.01 0.1 1 10 100 1000501001500.01 0.1 1 10 100 1000 Second plasmid concentration (nM) Untranslated RNA Noninteractingprotein     (   n    M    /   m    i   n    )    d    [    Y    ]    /    d    t    t   >   T Fig. 6. Fluorescent protein production rate for  t > T   as a function of theconcentrations of the second gene when it codes for an untranslated RNA(left panel) or a typical noninteracting protein (right panel). Dashed linesindicate the concentrations  [ x ] tot  used elsewhere in this work. by the concentration of a second gene over 6 orders of magnitude. As before, we use the simulated untranslated-RNA gene (with  k X+  = 0 ) to isolate and predict the effectof holoenzyme utilization. We find that it is only at anadditional gene concentration of   ∼ 25 nM (Fig. 6, left), or12.5X the 2 nM total reporter concentration, that crosstalk arising from limited holoenzyme appears as a  > 1% changein the output. On the other hand, ribosome-related crosstalk begins to manifest itself (as a  > 1% change in the output) atconcentrations as low as 0.75% of   [ y ] tot , or ∼ 15 pM (Fig. 6,right).While crosstalk may be commonplace in natural circuits,we are not limited to naturally-occurring parts when con-structing new biocircuits; synthetic biological tools allow usto adjust many properties of circuit components, includingdegradation rates and resource binding affinities. Our modelmay thus be used as a circuit design aid, to predict, forexample, how much the ribosomal binding off-rate mustbe modified in order to ameliorate the effects of ribosomalcrosstalk. With  [ x ] tot  = 1  nM and all other parameters heldfixed, our model predicts that a greater than 50-fold increaseor decrease in  k X −  is needed to eliminate the ribosomeloading effects (Fig. 7). As might be expected, a significantincrease in  k X −  produces an output that is little different 0.01 0.1 1 10 100501001500 Relative off-rate  k X − / ( k X − ) 0      (   n     M     /   m     i   n     )     d     [     Y     ]     /     d    t    t   >   T Fig. 7. Fluorescent protein production rate for  t > T   as a function of thestrength of the second gene’s ribosomal binding site relative to that of thefluorescent reporter.  [ x ] tot  = 1  nM. from that of the single reporter control, while a significantdecrease reduces the output to near zero as all ribosomes aresequestered by  x m . Interestingly, we find that the sensitivityof the output to the off-rate parameter (as determined by theslope of   d[Y] / d t  vs.  k X − ) is highest at the natural valueof   k X − , but that this sensitivity is relatively unchanged overtwo orders of magnitude. The effects of variation in othercircuit parameters may be similarly tested.V. SPECIAL CASE: ALTERNATIVE SIGMAFACTORSCertain classes of proteins may contribute crosstalk effectsin addition to those introduced by ribosome sequestration; forexample, alternative sigma factors that compete for accessto free core RNAP and thus lead to a reduction in activityfrom orthogonal sigma factor-specific promoters (Fig. 8).Experimental evidence supporting sigma factor sequestrationhas been found  in vivo  [20] and using purified sigma factorsubunits [21]. The question of the effect of alternative sigmafactors on circuit performance is an important one giventheir relevance to complex biocircuit design: the promoterselectivity that sigma factors confer to RNAP [22] can leadto an increase in the number of available transcriptionalcontrol elements, beyond the standard library of repressorsand activators now commonly used. s2 y S2S1EES2ES1  y m s2 m Y Secondary sigma factor circuit R ++ Fig. 8. Schematic showing molecular interactions when a secondary sigmafactor is present. Symbols are as in Fig. 3. In order to determine if our model formalism predictsadditional resource-loading-type effects when a secondaryconstitutively-expressed sigma factor is introduced to the  system, we add the following equation to the model (2): d[ES2] / d t  =  k ES2+ [E][S2] − k ES2 − [ES2]  ,  (4) and modify Eqs. (2e) and (3a) to be d[S2] / d t  =  k s 2 ,TL [ s 2 m :R] − k ES2+ [E][S2]+  k ES2 − [ES2]  (5) and [E] = [E] tot − [ s 2:ES1] f  ( s 2) − [ y :ES1] f  ( y ) − [ES1] − [ES2]  ,  (6) respectively. (Notationally, references to ‘ x ’ and ‘ X ’ havebeen replaced with ‘ s 2 ’ and ‘ S2 ’.) Simulated and experi-mental results are shown in Fig. 9. We note that (1) coreRNAP sequestration by a secondary sigma factor has a morepronounced effect on the circuit output than does the ribo-some loading, and particularly at higher gene concentrations,and (2) this sequestration, unlike the untranslated RNA andnoninteracting protein cases, results in fluorescent proteinproduction rates that are sublinear for the 2.5 hours of theexperiment. Time (mins) 206010014001020    [   Y   ]   (      μ    M   ) Secondary sigma factor 0.1 nM1 nM Fig. 9. Simulation and experimental results for implementation of thesecondary sigma factor circuit schematized in Fig. 8. Solid lines aresimulation results, and shaded areas indicate the standard deviation (n=2)of reporter concentration as determined by fluorescence for  [ s 2] tot  = 0 . 1 nM and  [ s 2] tot  = 1  nM. VI. DISCUSSIONWe have presented a detailed model for a simple two-generegulatory biocircuit operating  in vitro  that makes explicit theimportant functional roles played by RNA polymerase, sigmafactors, and ribosomes and that provides insight into howthese resources are shared between components. This model,with support from our cell-free experiments, demonstratesthat even a single noninteracting protein-coding gene addedat a low concentration can introduce significant crosstalk through ribosomal loading. Additional simulations suggestthat the performance of the circuit is insensitive to changesin RNAP concentration but highly sensitive to ribosomeconcentrationat physiologically-relevantlevels of componentDNA. We also show that ribosome utilization effects may bedifficult to avoid in any natural circuit of even minimal com-plexity; an elimination of these effects would require eitheran exceedingly low level of circuit DNA or a substantialmodification of the ribosome binding affinities. Lastly, weshow through a simple extension of the model and supportingexperiments that the presence of a constitutively-expressingalternative sigma factor gene decreases circuit output viasequestration of the core RNAP by the sigma factor.The model proposed here is a foundational one thatmay be easily expanded to include any number of genes.Of course, any extension of the model would lead to in-creases in the dimensionality of the state and parameterspaces and bring it further into the regime of the well-known ‘parameter problem’ [23]. However, the robustnessof biological systems would lead us to suspect that anyrealistic biological model would not be particularly sensitiveto specific values of parameters such as rate constants—indeed, order-of-magnitude approximations are often suffi-cient to explain and predict system behavior. Should betterparameter estimates be required, there exists a large andgrowing number of computational tools specifically designedfor this purpose [24]. In addition, technological advances inmicrofluidics and experimental platform miniturization aremaking high-throughput and quantitative measurements in-creasingly feasible; for example, the parallel characterizationof a large number of independent biomolecular associationand dissociation rates has recently been demonstrated [25].Such computational and experimental methods are compat-ible with our modeling framework and TX-TL breadboardsystem.APPENDIX  A. Methods Preparation of the TX-TL extract was as described pre-viously [15], [16]. The deGFP reporter construct was alsodescribed in that work. The noninteracting protein (TetR)was expressed from a  P LlacO − 1  regulatory part composedof a promoter specific to  σ 70 flanked by two  lac  operators.The secondary sigma factor  σ 28 was expressed from an OR2-OR1-Pr regulatory part composed of a  σ 70 -specific promoterflanked by two lambda Cl operators. The untranslated RNAgene was expressed off plasmid pAPA1256 from [26].Data were collected over two separate experimental runsusing a Victor X3 plate reader set at 29 ◦ C. Measuredfluorescence values were converted to concentrations usinga predetermined calibration curve and plotted with an 8minute offset to account for the time between the mixingof breadboard components and the start of data collection.Simulations were done using Mathematica.  B. Model parameters The model parameter values used are listed in Table I.Values were taken from [16], [27], and references therein,with the following exceptions and notes: •  Transcription rates  k i,TX  were assumed to be equal to(the previously-measured)  k y,TX . •  Translation rates  k i,TL  were assumed to be equal to (thepreviously-measured)  k y,TL . •  RNA degradation rates  k im,deg  were assumed to beequal to (the previously-measured)  k ym,deg . •  When only a dissociation constant  K  d  ( =  k − /k + ) couldbe found, the on-rate ( k + ) was taken to be  3  ×  10 7

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Feb 5, 2019
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