Prediction of cellular burden with host-circuit models
Evangelos-Marios Nikolados, Andrea Y. Weiße, Diego A. Oyarzún
PPrediction of cellular burden with host-circuit models
Evangelos-Marios Nikolados , Andrea Y. Weiße , Diego A. Oyarz ´un , School of Biological Sciences, University of Edinburgh, UK; [email protected] Department of Infectious Diseases, Imperial College London, UK; [email protected] School of Informatics, University of Edinburgh, UK; [email protected]
Summary
Heterologous gene expression draws resources from host cells. These resources includevital components to sustain growth and replication, and the resulting cellular burden is a widely recog-nised bottleneck in the design of robust circuits. In this tutorial we discuss the use of computationalmodels that integrate gene circuits and the physiology of host cells. Through various use cases, weillustrate the power of host-circuit models to predict the impact of design parameters on both burdenand circuit functionality. Our approach relies on a new generation of computational models for mi-crobial growth that can flexibly accommodate resource bottlenecks encountered in gene circuit design.Adoption of this modelling paradigm can facilitate fast and robust design cycles in synthetic biology.
Keywords
Cellular burden; growth models; whole-cell modelling; gene circuit design; syntheticbiology; resource allocation
The grand goal of Synthetic Biology is to engineer living systems with novel functions. The approachrelies on the combination of biological knowledge with design strategies from engineering sciences[1, 2, 3, 4]. Engineering principles, such as modularity and standardisation, have led to gene circuitswith a wide range of functions such as cellular oscillators [5, 6], memory devices [7] and biosensors[8, 9]. As synthetic biology matures into an engineering discipline of its own, mathematical modellingis playing an increasingly important role in the design of biological circuitry [10]. Moreover, model-based design offers opportunities for other fields such as computer-aided design [11], control theory[12] and machine learning [13] to contribute with new methods and protocols for gene circuit design.The success of the celebrated “design-build-test-learn” cycle [14] relies on the availability of goodquality models for circuit function. A major drawback of current modelling frameworks, however, is a r X i v : . [ q - b i o . M N ] A p r ost-circuit modellingthe implicit assumption that biological circuits function in isolation from their host. This simplificationlimits the predictive power of circuit models and slows down the iterations between system design,testing and characterisation. In reality, gene circuits interact with their host in many ways, includingthe consumption of molecular resources such as amino acids, nucleotides or energy, as well as usingmajor components of the genetic machinery such as polymerases and ribosomes.Competition for a limited pool of host resources produces a two-way interplay between syntheticcircuits and the native physiology of the host [15]. This interplay is commonly known as burden andperturbs the homeostatic balance of the host, resulting in slowed growth, reduced biosynthesis andthe induction of stress responses [16]. Since such effects can impact circuit behaviour, they createfeedback effects that can potentially break down circuit function [17, 18, 19]. As a result, individualmodelling of circuit parts and their connectivity is not sufficient to predict circuit function accurately.In a seminal study on host-circuit interactions, Tan and colleagues [20] studied a simple circuitconsisting of T7 RNA polymerase that activates its own expression in Escherichia coli . Contrary towhat standard mathematical models would predict, the circuit displayed bistable dynamics. The au-thors show that synthesis of the polymerase produced an indirect, growth-mediated, positive feedbackloop, which when included in their model was able to reproduce the observed bistability. This studywas the first empirical demonstration that growth defects can drasticlly change circuit function. Anumber of subsequent works have focused on the sources and impact of burden on gene circuits. Forexample, Ceroni et al showed that genes with weaker ribosomal binding strength are less taxing on thehost resources [21]. Other works have focused on strategies to mitigate burden. An and Chin built agene expression system that combines orthogonal transcription by T7 RNA polymerase and translationby orthogonal ribosomes [22]. The system reported in [23] allows to allocate resources among com-peting genes, while [24] built libraries of promoters that tune expression of burdensome proteins anddecrease cellular stress. The work by Shopera et al showed that negative feedback control can reducethe cross-talk between gene circuits [25]. Another strategy for reducing burden was proposed in [26]using an orthogonal ribosome for translation of heterologous genes. A particularly attractive strategyis to exploit burden to improve functionality. For example, Rugbjerg and colleagues increased metabo-lite production by coupling pathway expression to that of essential endogenous genes [27], while [28]employed stress-response promoters to build a feedback system with increased protein yield.2ost-circuit modellingAs a result of the increasing interest in cellular burden and host-circuit interactions, the modellingcommunity has devoted substantial attention to improving models for gene circuits and their interactionwith a host. A key challenge is to find a suitable level of model complexity with enough detail todescribe tunable circuit parts but without excessive granularity that makes models impractical. At oneend of the complexity spectrum, a number of works have proposed simple resource allocation modelsfor the interplay between circuit and host genes [29, 30, 31]. Using different modelling approachesand assumptions, these models generally predict a linear relation between expression of native andheterologous genes. Increases in the expression of one gene causes a linear drop in the expressionof another gene, as a result of a limited abundance of ribosomes for translation. At the other endof the spectrum, the whole-cell model of
Mycoplasma genitalium [32] was an ambitious attempt todescribe all layers of cellular organization under a single computational model. A subsequent workdemonstrated the use of the whole-cell model in conjunction with gene circuits [33]. Yet to date suchwhole-cell models have not been built for bacterial hosts commonly employed in synthetic biology,and their high complexity prevents their systematic use in circuit design and optimization.A number of approaches have sought to find a middle ground between model complexity andtractability. Inspired by the widely established “bacterial growth laws” [34, 19], Weiße and colleaguesbuilt a mechanistic growth model for
Escherichia coli [35]. The model uses a coarse-grained partitionof the proteome to describe how cells allocate their resources across various gene expression tasks.It accurately predicts growth rate from the interplay between metabolism and gene expression, andcan be extended with a wide range of genetic circuits. Applications of the Weiße model include thedesign of orthogonal ribosomes [26], the addition of extra layers of regulation [36] and its extensionto single-cell growth dynamics [37]. Most recently, Nikolados et al employed the model to study theimpact of growth defects in various exemplar circuits [38].In this tutorial we describe how mechanistic growth models can be employed to simulate genecircuits together with the host physiology (Fig. 1). In Section 2 we first revisit the bacterial growthlaws and explain the core principles of the mechanistic growth model. In Section 3 we present howto extend the growth model with heterologous genes. We illustrate the methodology with a number oftranscriptional logic gates in Section 4. We conclude the chapter with a perspective for future researchin the field. 3ost-circuit modelling cellular hostcircuit parts
RBSgenespromotersexpression translation resourceusagecircuitfunctiongrowthdefects designspace host-circuit model parameter 1 p a r a m e t e r time p r o t e i n ribosomes db l t i m e Figure 1:
Host-circuit modelling.
Integrated host-circuit models provide a quantitative basis to studythe impact of design parameters on circuit function and genetic burden on their host.
We begin by describing the bacterial growth laws that form the basis for most current models forgrowth. Our focus is on coarse-grained models that describe cell physiology using lumped variablesrepresenting aggregates of molecular species. We deliberately exclude whole-cell models [32] andgenome-scale models [39], both of which have been discussed extensively in the literature [40, 41, 42]and so far have found relatively limited applications in gene circuit design.
Bacterial growth has been an active topic of study for many decades. The celebrated work of Nobellaureate Jacques Monod provided a key quantitative description for growth [43], based on the obser-4ost-circuit modellingvation that bacteria in batch cultures exhibit several phases of growth:•
Lag phase: cells do not immediately start to grow after nutrient induction, as they first mustadapt to the new environment; RNA and proteins are produced as the cell prepares for division.•
Exponential phase: cells duplicate at a constant rate, so that their number grows exponentiallyas N ( t ) = N t/τ with τ being the average doubling time. Equivalently, the number of cellscan be expressed as N ( t ) = N e λt , where λ = log 2 /τ is the growth rate.• Stationary phase: cell replication stops because an essential nutrient has been depleted fromthe batch. The number of cells remains constant during this phase.•
Death phase: cells begin to die, resulting in a decreasing cell population.The vast majority of studies on bacterial growth focus on the exponential phase, and to date thisremains the best characterised growth phase. A widely empirical model for exponential growth isgiven by Monod’s law, which relates the instantaneous growth rate and the substrate concentration: λ = λ max ss + K s , (1)where s is the growth substrate, λ max is the maximum growth rate possible in the substrate and K s isthe substrate concentration for which growth rate is half maximal. The relationship in Eq. (1) is knownas Monod’s law and describes the hyperbolic dependence of the growth rate λ on the concentration ofa growth-limiting nutrient s in the medium.Measurements of bacterial cells growing at different rates [44, 45] have revealed a central roleof ribosome synthesis in maintaining exponential growth [46, 47]. In particular, the ribosomal massfraction, φ R , has been shown to increase linearly with growth rate [48, 44]. This is the second growthlaw, described mathematically as: φ R = φ min R + λκ t , (2)where φ minR is an offset term and κ t is a phenomenological parameter related to protein synthesis.The third growth law relates to growth inhibition. It has been shown that sublethal antibiotic dosestargeting ribosomal activity produce a negative linear relation between growth rate and the ribosomalmass fraction [19]. Mathematically, this growth law can be described by: φ R = φ maxR − λκ n , (3)5ost-circuit modellingwhere the parameter κ n describes the nutrient capacity of the growth medium and φ maxR is the maximumallocation to ribosomal synthesis in the limit of complete translational inhibition.Taken together, Equations (1)–(3) provide a remarkably simple description of exponential growth.Yet a common caveat of such descriptions is their lack of explicit links between phenomenologicalparameters and the molecular processes that drive growth. Some works have indeed found quantitativedescriptions of model parameters in terms of intracellular properties [34, 19]. However, another strandof research has moved away from phenomenological models toward mechanistic descriptions of cellphysiology [49, 50]. Notably, earlier work by Molenaar and colleagues [51] proposed a model thatintegrates metabolism and protein biosynthesis into a resource allocation model. Key assumption inthat approach is that microbes adjust their proteome composition to maximize growth. This leads togrowth predictions that rely on an optimality principle, without the need of a mechanistic descriptionof how cellular constituents contribute to growth and replication. The mechanistic model in [35] describes bacterial growth based on first principles. The model re-produces the bacterial growth laws and, at the same time, contains detailed mechanisms for nutrientmetabolism, transcription and translation. It employs a partition of the proteome similar to an earlierwork [51], but it does not require the assumption of growth maximization. The model is versatileand can predict how cells reallocate their proteome composition under various types of perturbations,including nutrient shifts, genetic modifications and antibiotic treatments.The model combines nutrient import and its conversion to cellular energy with the biosyntheticprocesses of transcription and translation. In its basic form, the model includes 14 intracellular vari-ables: an internalised nutrient s i ; a generic form of energy, denoted a , that models the total pool ofintracellular molecules required to fuel biosynthesis, such as ATP and aminoacids; and four types ofproteins: ribosomes p r , transporter enzymes p t , metabolic enzymes p m and house-keeping proteins p q .The model also contains the corresponding free and ribosome-bound mRNAs for each protein type,denoted by m x and c x respectively, with x ∈ { r, t, m, q } . The model can be described by the chemicalreactions listed in Table 1. From these reactions we model the cell as a system of ordinary differentialequations, describing the rate of change of the numbers of molecules per cell of a particular species.6ost-circuit modelling transcription translation nutrientsenergyribosomes proteome enzymes metabolism Figure 2:
Mechanistic model for bacterial growth.
The model predicts growth rate from the allo-cation of two cellular resources (energy and ribosomes) among the various processes that fuel growthand replication [35].Next we explain in detail how the model equations are built.Table 1: Chemical reactions in the mechanistic growth model [35]. transcription dilution/degradation ribosome binding dilution translation dilutionribosomes φ w r −→ m r m r λ + d m −−−→ φ p r + m r k b − (cid:42)(cid:41) − k u c r c r λ −→ φ n r a + c r v r −→ p r + m r + p r p r λ −→ φ transporter enzyme φ w t −→ m t m t λ + d m −−−→ φ p r + m t k b − (cid:42)(cid:41) − k u c t c t λ −→ φ n t a + c t v t −→ p r + m t + p t p t λ −→ φ metabolic enzyme φ w m −→ m m m m λ + d m −−−→ φ p r + m m k b − (cid:42)(cid:41) − k u c m c m λ −→ φ n m a + c m v m −→ p r + m m + p m p m λ −→ φ house-keeping proteins φ w q −→ m q m q λ + d m −−−→ φ p r + m q k b − (cid:42)(cid:41) − k u c q c q λ −→ φ n q a + c q v q −→ p r + m q + p q p q λ −→ φ nutrient import s v imp −−→ s i internal nutrient s i λ −→ φ metabolism s i v cat −−→ n s a energy molecules a λ −→ φ The environment, or growth medium, of the cell contains a single nutrient described by the constantparameter s . A transport protein p t is responsible for the uptake of the external nutrient at a fixedconcentration, which once internalised, s i , is catabolised by a metabolic enzyme p m . The dynamics ofthe internalised nutrient obey: ˙ s i = v imp − v cat − λs i . (4)7ost-circuit modellingSimilarly to the bacterial growth laws described in Section 2.1, the growth rate is denoted by λ . Allintracellular species are assumed to be diluted at a rate λ because of partitioning cellular contentbetween daughter cells at division. Nutrient import ( v imp ) and catabolism ( v cat ) are assumed to followMichaelis-Menten kinetics: v imp = p t v t sK t + s , v cat = p m v m s i K m + s i , (5)where v t and v m are maximal rates, while K t and K m are Michaelis-Menten constants. Since trans-lation is known to dominate energy consumption [48], the model neglects other energy-consumingprocesses. Using c x to denote the complex between a ribosome and the mRNA for a protein p x , thetranslation rate for every protein obeys v x = c x γ ( a ) n x . (6)The parameter n x in Eq. (6) is the length of the protein p x in terms of amino acids, and the term γ ( a ) represents the net rate of translational elongation. Assuming that each elongation step consumes afixed amount of energy [35], the net elongation rate depends on the energy resource by: γ ( a ) = γ max aK γ + a , (7)where γ max is the maximal elongation rate and K γ is the energy required for a half-maximal rate.From Eq. (6) we can compute the total energy consumption by translation of all proteins and get adifferential equation for the net turnover of energy: ˙ a = n s v cat − (cid:88) x(cid:15) { r,t,m,q } n x v x − λa, (8)where the sum over x is over all types of protein in the cell. Overall, energy is created by metabolizing s i and lost through translation and dilution by growth. The positive term in Eq. (8), determines energyyield per molecule of internalized nutrient from Eq. (4). The parameter n s describes the nutrientefficiency of the growth medium.In rapidly growing E. coli , it is known that transcription has a minor role in energy consumption[52]. We therefore model transcription as an energy-dependent process, but with a negligible impactin the overall energy pool. If w x,max denotes the maximal transcription rate, the effective transcriptionrate has the form w x = w x,max aθ x + a , (9)8ost-circuit modellingfor all proteins except housekeeping ones, i.e. x ∈ { r, t, m } . We assume that the transcription ofhousekeeping mRNAs is subject to negative autoregulation so as to keep constant expression levels invarious growth conditions: w q = w q,max aθ q + a (cid:124) (cid:123)(cid:122) (cid:125) energy-depedenttranslation ×
11 + ( p q /K q ) h q (cid:124) (cid:123)(cid:122) (cid:125) negativeautoregulation . (10)In Eqs. (9) and (10), the parameter θ x denotes a transcriptional threshold, while K q and h q are regula-tory parameters. The differential equations for the number of mRNAs ( m x ) are therefore: ˙ m x = w x − ( λ + d m ) m x + v x − k b p r m x + k u c x , (11)where x ∈ { r, t, m, q } . In Eq. (11), mRNAs are produced through transcription with rate w x , whilemRNAs are lost through dilution λ and degradation with rate d m . At the same time, mRNAs bind andunbind with ribosomes, so that the ribosome-mRNA complexes ( c x ) follow ˙ c x = − λc x − v x + k b p r m x − k u c x , (12)where k b and k u are the rate constants of binding and unbinding. Translation contributes with a positiveterm to Eq. (11) and a negative term to Eq. (12). The differential equations for protein abundance aretherefore: ˙ p x = v x − λp x , x ∈ { t, m, q } . (13)We note that Eq. (13) applies to all proteins except free ribosomes. The equation for free ribosomes p r includes an additional term: ˙ p r = v r − λp r + (cid:88) x ∈{ r,t,m,q } ( v x − k b p r m x + k u c x ) . (14)Through Eq. (14) the model accounts for competition among different mRNAs for free ribosomes, aswell as ribosomal autocatalysis. Ribosomal transcripts sequester free ribosomes for their own transla-tion, and the pool of free ribosomes can increase as a result of translation of new ribosomes and, at thesame time, the release of ribosomes engaged in translation of non-ribosomal mRNAs.9ost-circuit modellingFinally, it can be shown (details in [35]) that under the assumption of constant average mass, thespecific growth rate can be computed in terms of the total number of ribosomes engaged in translation: λ = γ ( a ) M × (cid:88) x ∈{ r,t,m,q } c x , (15)where M is the constant cell mass.Overall, Eqs. (4)–(15) constitute the core of the mechanistic growth model. Equations (8) and (14),in particular, model the availability of energy and ribosomes, both regarded as cellular resources sharedbetween metabolism and protein biosynthesis. The model contains 22 parameters. For E. coli , someparameter values were mined directly from the literature and others were estimated with Bayesianinference on published growth data [35, 19]. The parameter values are shown in Table 2. We note thatwe have assumed that all components of the proteome are not subject to active degradation. As weshall see in the next sections, the core model can be extended with gene circuits of varying complexity.Table 2: Model parameters for an
Escherichia coli host, taken from [35]. Units of aa correspond tonumber of amino acids per cell. parameter value parameter value s (molecules) M (aa) n r θ r
427 (molecules) γ max K γ v t
726 (min -1 ) K t v m -1 ) K m w r,max
930 (molecules / min) w m,max , w t,max w q,max
949 (molecules/min) d m -1 ) K q h q θ q , θ t , θ m n q , n t , n m
300 (aa/molecules) k b − molecules − ) k u -1 ) In this section we discuss how to extend the mechanistic growth model with heterologous circuitgenes. The extended model can be employed for predicting the impact of genetic parameters, such aspromoter strengths or gene length, on the growth rate of the host strain and the resulting heterologousexpression levels. We first describe the steps needed to extend the model, and then illustrate the ideaswith a simple model for an inducible gene. This is a simple example that contains all the elementsneeded by more complex circuits.
The extension of the model requires three steps:
Step 1: add new model species.
First, we include mass balance equations for the expression ofeach heterologous gene. This requires three additional species per gene: the transcript, the mRNA-ribosomal complex and the protein, all of which follow dynamics similar to Eqs. (11)–(13): ˙ p c i = v c i − ( λ + d p ) p c i , ˙ m c i = w c i − ( λ + d m ) m c i + v c i − k cb ,i p r m c i + k cu ,i c c i , ˙ c c i = − λc c i + k cb ,i p r m c i − k cu ,i c x − v c i , (16)where the superscript c denotes heterologous species and the subscript i denotes the i th heterologousgene. The ribosomal binding parameters k cb ,i and k cu ,i are specific to each gene and can be used, forexample, to model different ribosomal binding sequences. The translation rate v ci is modelled similarlyas that of native genes in Eq. (6): v ci = c c i n c i × γ max aa + K γ , (17)with n ci being the length of the i th circuit protein. Likewise, the transcription rate is similar to Eq. (9): w c i = w cmax ,i aθ c + a R i , (18)where w cmax ,i is the maximal transcription rate. Note that we have included an additional term R i to model regulatory interactions by other genes. Complex circuit connectivities can be modelled by11ost-circuit modellingsuitable choices of the function R i . Later in Section 4 we exemplify this with models for transcriptionallogic gates. Step 2: modify allocation of resources.
Second, we include the additional consumption of energyand ribosomes in the model. Starting from the resource equations in Eqs. (8) and (14), we write: ˙ a = n s v cat − (cid:88) x n x v x − (cid:88) i n c i v c i (cid:124) (cid:123)(cid:122) (cid:125) energy consumptionby foreign genes − λa, (19)(20) ˙ p r = v r − λp r + (cid:88) x ( v x − k b p r m x + k u c x ) + (cid:88) i ( v c i − k cb ,i p r m c i + k cu ,i c c i ) (cid:124) (cid:123)(cid:122) (cid:125) consumption of free ribosomesby foreign genes . Step 3: adjust growth rate prediction.
Third, we update the prediction of growth rate in Eq. (15)to include translation of heterologous genes: λ = γ ( a ) M (cid:18) (cid:88) x c x + (cid:88) i c c i (cid:124) (cid:123)(cid:122) (cid:125) ribosomalcomplexes (cid:19) . (21) Inducible expression systems are widely employed as building blocks of complex gene circuits. As anexample, we consider a reporter gene ( rep ) under the control of an inducible promoter, modelled bythe reactions in Table 3. Table 3: Reactions for an inducible reporter gene. transcription dilution/degradation ribosome binding dilution translation dilution/degradationREP φ w rep −−→ m rep m rep λ + d m,rep −−−−−→ φ p r + m rep k b − (cid:42)(cid:41) − k u c rep c rep λ −→ φ n r a + c rep v rep −−→ p r + m rep + p rep p rep λ + d p,rep −−−−→ φ The model contains mRNAs of the heterologous gene, which can reversibly bind to free ribosomesof the host, p r . Protein translation consumes energy ( a ) and, at the same time, proteins and other modelspecies are diluted by cell growth. In contrast to native proteins of the host, however, we assume that12ost-circuit modellingheterologous proteins are tagged for degradation by proteases, a strategy often employed to accelerateprotein turnover [53]. This active degradation is modelled by the parameter d p,rep in Table 3.We do not explicitly model the molecular mechanism for induction, as this will depend on theparticular implementation of choice. For example, in the tetR inducible system, the inducer anhy-drotetracycline (aTc) activates gene expression by reversible binding to the tetracyline repressor tetR ,whereas in the lac inducible system, the inducer Isopropyl- β -D-thiogalactoside (IPTG) binds to al-losteric sites of the lac repressor lacR . Instead, we lump the induction mechanism into an effectivetranscription rate, denoted as w rep in Table 3.Using the general circuit equations in (16)–(18) of Section 3.1, for the inducible gene Eq. (16)becomes: ˙ p rep = v rep − ( λ + d p,rep ) p rep , ˙ m rep = w rep − ( λ + d m,rep ) m rep + v rep − k b,rep p r m rep + k u,rep c rep , ˙ c rep = − λc rep + k b,rep p r m rep − k u,rep c rep − v rep . (22)The rate of reporter translation follows as in Eq. (23): v rep = c rep n rep × γ max aa + K γ , (23)where n rep is the length of the reporter in amino acids. Likewise, the transcription rate in Eq. (18)becomes: w rep = w max,rep × aθ c + a . (24)Note that in the transcription rate, the regulatory term is R i = 1 , because the inducible system doesnot contain any regulatory interactions.Before simulating the expression of the heterologous protein, we first need to obtain an estimatefor the proteome composition of the wild-type. This is required to initialize the host-circuit simulationswith a physiologically realistic cellular composition. To this end, we first simulate Eqs. (4)-(15) for the“wild-type model” until steady state. The results, summarized in Fig. 3A, show that host proteins aretranslated at different rates with most of the translating ribosomes bound to mRNAs of house-keepingproteins. However, a sizeable fraction is bound to ribosomal mRNA, highlighting how the growth13ost-circuit modellingmodel accounts for ribosomal autocatalysis. A closer look (Fig. 3A, bottom) reveals that translation-engaged ribosomes account approximately for two-thirds of the total ribosomal fraction in the form ofmRNA-ribosomal complexes, with one-third remaining free.Next, we simulate heterologous expression using the maximal transcription rate w max,rep in Eq. (24)to describe the effect of different gene induction strengths. As shown in the dose-response curve inFig. 3B, the model predicts that increased induction causes an increase in expression. We observe,however, that protein expression reaches a maximum at a critical induction strength and subsequentlydrops sharply for stronger induction. This reflects the limitations that resource competition imposeson the expression of a heterologous gene [38]. gene induction (mRNAs/min) e x p r e ss i o n ( o f m o l ec u l e s ) x10 translationrates house-keepingmetabolic ribosomal ribosomes free bound growth rate% of WT wild-type ribosomeshouse-keepinguptake enzymemetabolic enzymeheterologous protein A B
Figure 3:
Simulation of an inducible gene. (A)
Steady state translation rates and ribosomal abun-dance predicted for the wild-type
Escherichia coli model, parameterized as in Table 2. (B)
Predictedsteady state expression of a heterologous gene for increasing induction strength. The pie charts indi-cate translation rates and ribosomal abundance as in the left panel. The inset shows the predictedgrowth rate, relative to the wild-type. The induction strength was modelled with the parameter w max,rep in Eq. (24). The binding rate constant was set equal to the dissociation rate constant, sothat k b,rep = 1 × − min − molecules − , k u,rep = 1 × − min − . Transcript and protein half-liveswere set to two and four minutes, respectively [5], so that d m,rep = ln 2 / min − and d p,rep = ln 2 / min − .To understand the main source of the resource limitations, we use the model to explore the syn-14ost-circuit modellingthesis rates of the various components of the proteome. Because growth rate is linearly related to thetotal rate of translation (Eq. (21)), we can make direct conclusions for cellular growth as well. Asshown in Fig. 3B (inset), the model predicts a sigmoidal decrease in growth rate for stronger geneinduction. At low induction, expression of the foreign gene is mostly at the expense house-keepingproteins, while ribosomes, transporter and metabolic enzymes, show little decrease. This suggests thatthe host can compensate for this load through transcriptional regulation and repartitioning of the pro-teome (Fig. 3B). As the induction of the reporter gene increases, circuit mRNAs dominate the mRNApopulation, hence increasing the competition for free ribosomes. Finally, for sufficiently strong induc-tion, ribosomal scarcity leads to reduction of all proteins, which in turn leads to the drop in growthrate observed in Fig. 3B (inset). These results are in agreement with the widespread conception thatribosomal availability is a major control node for cellular physiology [19, 54, 55], with depletion offree ribosomes being the main source of burden for translation of circuit genes [21, 31]. There has been substantial interest in gene constructs that mimic digital electronic circuity [6, 56, 57].Cellular logic gates, in particular, have been used to produce desired behaviours in response to variousinputs such as temperature, pH and small molecules [58, 59, 60]. Multiple logic gates can be combinedto build larger information-processing circuits with advanced cellular functions [8].To illustrate our simulation strategy in more complex circuitry, here we build host-circuit modelsfor cellular logic gates based on transcriptional regulators [61]. We first build and simulate the modelsfor a NOT, AND and NAND gates shown in Fig. 4. To highlight the power of our approach for circuitdesign, we then use the host-circuit models to predict circuit function across the design space, usingcombinations of RBS strength and growth media. As discussed in Section 3.1, we model the circuitsby adding extra genes to the growth model and modifying the mass balance and growth rate equations.We model the circuit connectivity by choosing suitable regulatory terms R i in the transcription ratesin Eq. (18), and the gate inputs via the maximal transcription rate w cmax ,i .To compare our host-circuit simulations with those of traditional models, we built circuit-only15ost-circuit modellingmodels using mass balance equations for mRNAs and proteins: ˙ m c i = w c i R i − ( λ eff + d m ) m c i , ˙ p c i = k eff i m c i − ( λ eff + d p ) p c i , (25)where the subscript i denotes the i th circuit gene and we assume a constant dilution rate, λ eff = -1 , which is equal to the growth rate predicted by the model for the wild-type with a nutrientefficiency of n s = 0 . . The effective translation rates are fixed to k eff = k eff = 16 . min − and k eff = 0 . min − for the AND gate, and k eff = k eff = 13 . min − , k eff = 0 . min − , and k eff = 347 min − for the NAND gate. In all cases, we assume that mRNAs and proteins are activelydegraded with rate constants d m = ln 2 / min − and d p = ln 2 / min − . output gene 1 gene 2 input NOT A AND input 2input 1 output gene 1gene 2 gene 3 B AND gene 1gene 2 gene 3 gene 4
NOT NAND input 1input 2 output C Figure 4:
Logic gates based on transcriptional regulators. (A)
The NOT gate contains two genesconnected in cascade. Repression of gene 2 inverts the input signal. (B)
The AND gate contains threegenes, in which two transcriptional activators jointly trigger the expression of a third output gene. (C)
The NAND gate contains four genes and is the composition of an AND and a NOT gate. Circuitconnectivities are based on the implementation by Wang et al [61].
The NOT gate contains two genes in cascade, where gene 1 codes for a transcriptional repressor thatinhibits the expression of gene 2; the circuit diagram is shown in Fig. 4A. We first model the NOT gatein isolation using Eq. (25). We choose the regulatory functions R i as R = 1 , R = 11 + (cid:18) p c1 K c (cid:19) h . (26)16ost-circuit modellingThe choice of R models the inhibition of gene 2, and different inhibitory strengths and cooperativityeffects can be described by suitable choices of the threshold K c and Hill coefficient h . We fix K c =250 molecules and h = 2 .As shown in Fig. 5A, the isolated models correctly predicts the expected circuit function, withstronger induction of the input gene 1 gradually suppressing the expression of the output proteins ( p c ),with strong induction resulting in minimal output yield. In other words, the gate has high output onlywhen the input signal is low, in effect acting as an inverter of the input signal.To simulate the host-aware NOT gate, we follow the procedure outlined in Section 3.1. The host-aware simulations shown in Fig. 5B suggest that the function of the NOT gate remains largely unaf-fected by host-circuit interactions. For intermediate input levels, simulations predict an increase ingrowth rate of up to ∼
50% with respect to a basal case. Such apparent growth benefit is a consequenceof the circuit architecture (Fig. 4A): an increase in the input causes a stronger repression of gene 2 andthus relieves the burden on the host. But since the expression of the repressor coded by gene 1 alsoburdens the host, for high inputs the expression of gene 1 counteracts the growth advantages gained byrepression of gene 2, resulting in an overall drop in growth rate. o u t p u t ( m o l e cs . ) A B input (mRNAs/min) o u t p u t ( m o l e cs . ) input (mRNAs/min) g r o w t h r a t e ( % o f b a s a l ) isolated model input output01 10 NOT input (mRNAs/min) host-aware model basal Figure 5:
Host-aware simulation of a NOT gate. (A)
Gate output predicted by a model isolated fromthe cellular host. Inset shows the boolean truth table for the NOT gate. (B)
Output and growth ratepredictions from host-aware model of the NOT gate. Growth rate is normalized to a basal case.17ost-circuit modelling
The AND gate comprises two genes that co-activate a third output gene (Fig. 4B). As built in theoriginal implementation [61], the promoter for gene 3 is activated only when both the co-dependentenhancer-binding proteins, encoded by genes 1 and 2, are present in a heteromeric complex. Conse-quently, the regulatory functions for the AND gate are: R = 1 , R = 1 , R = (cid:18) p c1 K c (cid:19) h (cid:18) p c1 K c (cid:19) h × (cid:18) p c2 K c (cid:19) h (cid:18) p c2 K c (cid:19) h , (27)with K c =
200 molecules and h = K c = h = et al [61]. i n p u t ( m R N A s / m i n ) input 2 (mRNAs/min) input 2 (mRNAs/min) o u t p u t ( m o l e c u l e s x ) g r o w t h r a t e ( % o f b a s a l ) o u t p u t ( m o l e c u l e s x ) . output i n p u t ( m R N A s / m i n ) input 2 (mRNAs/min) i n p u t ( m R N A s / m i n ) AND output0011 0110 0010
A B isolated model
AND host-aware model
250 1000 basal input1 input2
Figure 6:
Host-aware simulation of an AND gate. (A)
Output predicted by a model isolated fromthe cellular host. Inset shows the boolean truth table for the AND gate. (B)
Output and growth ratepredictions from host-aware model of the AND gate across the input space. Growth rate is normalizedto the basal case in lower left corner of the heatmap.Simulations of the isolated model (Fig. 6A) show that, as expected, the gate has a high output onlywhen the input signals are high. This agrees with the expected truth table of the AND, shown in theinset of Fig. 6A. In contrast, simulations of the host-aware model, shown Fig. 6B, suggest a strongimpact of host-circuit interactions. The host-aware model predicts a bell-shaped response surface,where the output reaches a maximal value for an intermediate level of the inputs, beyond which the18ost-circuit modellingoutput drops monotonically. Such loss of function coincides with a drop in growth rate observed forincreased levels of either input, as seen in the right panel of Fig. 6B, and thus suggests a link betweengrowth defects and poor circuit function.
The NAND gate is the negation of an AND gate, and thus produces a low output only when bothinputs are high. As shown in Fig. 4C, the gate has four genes connected as the composition of an ANDand NOT gates. As with the previous two cases, we simulate the isolated model using Eq. (25). Theregulatory functions for the NAND gate are: R = 1 ,R = 1 ,R = (cid:18) p c1 K c (cid:19) h (cid:18) p c1 K c (cid:19) h × (cid:18) p c2 K c (cid:19) h (cid:18) p c2 K c (cid:19) h ,R = 11 + (cid:18) p c3 K c (cid:19) h , (28)with parameter values for R equal to those for R of the AND gate in Eq. (27), and parameter valuesfor R equal to those of R for the NOT gate in Eq. (26).As shown in Fig. 7, simulations reveal substantially different predictions between the isolated andhost-aware models of the NAND gate. The host-aware model predicts a complex relation betweeninputs and output that differs from the ideal response predicted by the isolated model. Host-awaresimulations produce the correct response across a range of the input space (Fig. 7B), but display sig-nificant distortions possibly caused by the loss-of-function of the AND component shown in Fig. 6B.The impact of host-circuit interactions can also be observed in the predicted growth rate, which sug-gests a growth advantage for intermediate levels of the inputs. This is a result of the architecture of theNOT gate, akin to what we observed in Fig. 5B. 19ost-circuit modelling . i n p u t ( m R N A s / m i n ) input 2 (mRNAs/min) o u t p u t ( m o l e c u l e s x ) g r o w t h r a t e ( % o f b a s a l ) o u t p u t ( m o l e c u l e s x ) output A B
NAND input 1 outputinput 2 isolated model host-aware model input 2 (mRNAs/min) input 2 (mRNAs/min) i n p u t ( m R N A s / m i n ) i n p u t ( m R N A s / m i n ) basal Figure 7:
Host-aware simulation of a NAND gate. (A)
Output predicted by a model isolated fromthe cellular host. Inset shows the Boolean truth table for the NAND gate. (B)
Output and growth ratepredictions from host-aware model of the AND gate across the input space. Growth rate is normalizedto the basal case in lower left corner of the heatmap.
In this final section, we conduct a series of simulations that mimic experiments commonly used incircuit design. These aim to explore the impact of design parameters and growth media on circuitfunction.
A number of studies have shown that RBS strength is a key modulator of cellular burden [21, 29, 31,30]. Here we examine the impact of RBS strengths on the AND and NAND gates from the previoussection. Using the notation in our model, see e.g. Eq. (16), we define the RBS strength as:RBS i = k cb,i k cu,i , (29)where k cb,i is the mRNA-ribosome binding rate constant (in units of min − molecules − ), and k cu,i istheir dissociation rate constant (in units of min − ).We simulated the AND and NAND gates with variable RBS strengths and gene induction strengths.As shown in Fig. 8A (left), the AND gate retains its function for increasing RBS strength. We observethat for the same induction, designs with stronger RBS lead to increased circuit yield. At the same20ost-circuit modellingtime, the simulations predict (Fig. 8A, left) a larger bell-shaped response surface, suggesting, that byincreasing RBS, we expect a slightly larger design space where the output can reach a larger maximalvalue for the same range of inputs. In all cases, however, after the output reaches a maximal value,we find a monotonic drop in circuit yield. The loss-of-function coincides with a drop in growth rateobserved in all designs (Fig. 8A, right), which becomes more pronounced with stronger RBS.As shown in Fig. 8B, the impact of RBS is more notable for the NAND gate. For designs withstronger RBS (insets Fig. 8B, left), but weak induction, the gate displays a behaviour akin to that of thebasal case. For intermediate induction, increasing RBS strength has more detrimental effects on thecircuit’s function. Specifically, the NOT component fails to fully repress the AND component, thusdistorting the region where the circuit is functional. However, further increase in RBS, greatly impairsthe system leading to near total loss-of-function across the entire response surface (insets Fig. 8B,left). Likewise, for stronger RBS and intermediate levels of the input, we observe loss of the growthadvantage gained by the NOT gate component (Fig. 8B, right). Bacterial growth is known to depend critically on the quality of the growth media. As a final illustrationof our approach, we used the host-aware models to explore the impact of media on the function of thetranscriptional logic gates. We model the quality of the media via the nutrient efficiency parameter n s in Eqs. (4) and (19), which determines the energy yield per molecule of internalized nutrient.Our simulations suggest that nutrient quality affects the quantity of output, but not the specificresponse of the AND gate (Fig. 9A). As the quality of the growth medium improves , the gene expres-sion capacity of the host increases and, as a result, we observe an increase operational range of thecircuit. However, this is not the case for the NAND gate, which displays a more complex behaviourfor low nutrient quality. As seen in Fig. 9B, richer media improve the function of the gate, comparedto the basal case (Fig. 7A). This is because an increase in nutrient quality improves the output of thegate’s AND component, which in turn leads to a stronger input for the NOT component, and hencestronger repression. On the contrary, poor nutrient quality leads to loss-of-function for the circuit. Asobserved in Fig. 9A, poorer media correspond to significantly decreased expression of the AND gate,which is also true for the AND component of the NAND gate. This translates to very weak input for21ost-circuit modelling i n p u t ( m R N A s / m i n ) input 2 (mRNAs/min) input 2 (mRNAs/min) A ou t pu t ( m o l e c u l e s x10 ) growth rate (% of basal) growth rate (% of basal) basal RBS X50 input 2 i n p u t input 2 i n p u t RBS X10 i n p u t ( m R N A s / m i n ) AND
AND R BS i n p u t ( m R N A s / m i n ) input 2 (mRNAs/min) B ou t pu t ( m o l e c u l e s x10 ) R BS NAND
RBS X10RBS X50 input 2 i n p u t input 2 i n p u t i n p u t ( m R N A s / m i n ) input 2 (mRNAs/min) basal RBS X50RBS X10 input 2 i n p u t input 2 i n p u t RBS X50RBS X10 input 2 i n p u t input 2 i n p u t Figure 8:
Impact of ribosomal binding site (RBS) strength. (A)
Output and growth rate predictionsfor the AND gate in Fig. 4B and three RBS strengths. (B)
Output and growth rate predictions for theNAND gate in Fig. 4C. RBS strengths were computed from Eq. (29) by simultaneously increasingthe binding rate constant k cb,i ∈ { − , − . , − . } and decreasing the dissociation rate constant k cu,i ∈ { − , − . , − . } in a pairwise manner for i = 3 (AND gate) and i = 4 (NAND gate).Gene induction strengths were varied in the range ≤ w c max,i ≤ mRNAs/min for i = 1 , inboth gates, and fixed w c max,3 = 375 mRNAs/min for the AND gate, and w c max,3 = 375 mRNAs/minand w c max,4 = 250 mRNAs/min for NAND gate.the NOT component, which in turn does not properly repress gene 4 (Fig. 4C), resulting in the loss ofgate functionality (Fig. 9B). 22ost-circuit modelling i n p u t ( m R N A s / m i n ) input 2 (mRNAs/min) A ou t pu t ( m o l ecu l es x10 ) input 2 i n p u t input 2 i n p u t i n p u t ( m R N A s / m i n ) input 2 (mRNAs/min) ou t pu t ( m o l ecu l es x10 ) input 2 i n p u t input 2 i n p u t B NAND
AND
AND n s = 0.6n s = 0.2n s = 1.0n s = 0.6n s = 0.2n s = 1.0 Figure 9:
Impact of growth media on circuit function. (A)
Simulations of the AND gate in Fig. 4Bin various growth media. (B)
Simulations of the NAND gate in Fig. 4C in various growth media. Inboth cases the nutrient quality parameter was set to n s ∈ { . , . , . } ; all other model parametersare identical to the simulations in Figures 6 and 7B. In this chapter we discussed host-aware modelling in Synthetic Biology. Starting from the three bacte-rial growth laws, we presented a deterministic model to simulate the single-cell dynamics of a bacterialhost [35]. We showed how to incorporate synthetic gene circuits into the host model, and used thismethodology to simulate host-aware versions of various gene circuits. Finally, we examined the im-pact of host-circuit interactions on the gates, for combinations of inputs, RBS strength, and growthmedia of different nutrient quality.While we focused on host-circuit competition for energy and free ribosomes, in practice genecircuits also consume other components that may become resource bottlenecks, such as RNA poly-merases and σ -factors for transcription, or amino acids and tRNAs for translation. Molecular speciesassociated with these processes can be readily incorporated into the growth model. For instance, in-stead of a single energy resource a , the catabolism of the internalised nutrient s i by the metabolicprotein p m , could also produce a pool of amino acids, which would then participate in the downstreamtranscription and translation processes. Explicit models of amino acid pools could be employed tostudy amino acid recycling after protein degradation, or global effects such as upregulation of tran-23ost-circuit modellingscription triggered by nutrient starvation [36, 62]. Such extensions, however, need to be dealt withcaution since they can increase model complexity, and ultimately obscure the relations between differ-ent sources of burden.A grand goal of Synthetic Biology is to produce target phenotypes through rational design of genecircuits. As with other engineering disciplines, predictive models are an essential step to acceleratethe design cycle, yet current models in synthetic biology are largely under-powered for this task. In-tegrated host-circuit models can effectively bridge this gap and offer a flexible framework to accountfor a wide range of resource bottlenecks. For example, recent data [63, 64] suggest highly nonlinearrelations between growth rate and heterologous expression and a sizeable burden caused by metabolicimbalances typically found in pathway engineering [65]. Such findings raise compelling prospects forthe integration of mechanistic cell models with large-scale characterization data, ultimately paving theway for more robust and predictable Synthetic Biology. References [1] E. Andrianantoandro, S. Basu, D. K. Karig, and R. Weiss, “Synthetic biology: new engineeringrules for an emerging discipline,”
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