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A Computational Study of Liposome LogicTowards Cellular Computing From the Bottom Up

James Smaldon · Francisco J. Romero-Campero · Marian

Gheorghe · Cameron Alexander · Natalio Krasnogor

the date of receipt and acceptance should be inserted later

Abstract Modeling and simulations feature promi-nently in “top-down” synthetic biology, particu-larly in the specification, design and implemen-tation of logic circuits through bacterial genomereengineering. In this paper we present a set oftools for the specification, modelling and analysisof “bottom-up” liposome logic, also called vesiclecomputing.

Liposome logic makes use of supra-molecularchemistry constructs, e.g. protocells, chells, etc., toencapsulate logical functionality. In particular weanalyse the scalability of the techniques presentedwhen the liposome logic complexity increases fromrelatively simple NOT gates and NAND gates toSR-Latches, D Flip-Flops all the way to 3 bit rip-ple counters. The approach we propose consists ofspecifying, by means of P systems, gene regula-tory network-like systems operating inside proto-membranes. This P systems specification can beautomatically translated and executed through amultiscaled pipeline composed of Dissipative Par-ticle Dynamics (DPD) simulator and Gillespie’sStochastic Simulation Algorithm (SSA). Finally,

J. Smaldon and F.J. Romero CamperoSchool of Computer Science, University of Nottingham, Ju-bilee Campus, Wollaton Road, NG8 1BB

M. GheorgheDepartment of Computer Science, University of She!eld

C. AlexanderSchool of Pharmacy, University of Nottingham

N. KrasnogorSchool of Computer Science, University of Nottingham, Ju-bilee Campus, Wollaton Road, NG8 1BBE-mail: Natalio.Krasnogor@Nottingham.ac.uk

model selection and analysis can be performed througha model checking phase.

This is the first paper we are aware of thatbrings to bear formal specifications, DPD, SSAand model checking into the problem of modelingtarget functionality in chells. Potential chemicalroutes for the laboratory implementation of thesesimulations are also discussed.

1 Introduction

Just as an electrical engineer can construct circuitsfrom modules with common inputs and outputswithout consideration of internal module construc-tion, the standardisation of biological componentsproposed by T.F. Knight, D. Endy, R. Weiss andothers (Endy 2005; Knight 2003; Heinemann andPanke 2006; Serrano 2007), and exemplified in theMIT biobricks project (Shetty et al 2008), may al-low a bioarchitect to construct biological systemswith pre-specified phenotypes in a more scalableway. One important application within the fieldof Synthetic Biology is Cellular Computing. Cel-lular Computing (Amos 2004) seeks the construc-tion of genes, signals and metabolic regulation net-works within organisms1 which, by implementingboolean logic gate circuits (Weiss et al 1999), couldaccomplish specific computational tasks (Tan et al2007).

In this computing paradigm, individual cellsperform a small part of a computation in a highlyasynchronous fashion with communication takingplace only between cells which are within a short

1 In what follows we will call these networks BiologicalRegulatory Networks (BRN).

distance from one another (so called amorphouscomputation (Abelson et al 2000)). In particular,cellular logic NOT and AND gates were first char-acterised in detail by Ron Weiss et al. (Weiss andBasu 2002) in-vitro and with “bioSPICE”, an ODEbased modelling technique. Since then, several smallscale systems have been constructed, either in thelab or in simulation (Amos 2004). Other examplesof in-vitro implementations of cellular computingsystems include band detectors, coupled oscilla-tions (Basu et al 2005, 2004) and, more recently,a solution to the three vertices Hamiltonian pathproblem (Baumgardner et al 2009).

By creating modular logic gates that behavein well characterised ways, it might be possible toabstract away some of the biological detail whendesigning more complex synthetic biology systems(Andrianantoandro et al 2006). In doing so, the be-haviour of a composite system becomes more pre-dictable and designs can be constructed and pro-totyped in-silico before attempting to implementthem in the lab.

Thus far cellular computing has only been in-vestigated from the “top down” perspective, thatis, by modifying existing organisms through the in-corporation of synthetic biological regulatory ne-toworks (BRNs). Little, if any, attention has beenpaid to how such distributed computation mightbe implemented from the “bottom up” perspec-tive, that is, by using protocells (Rasmussen et al2008) or “chells” (artificial chemical cells (Croninet al 2006)) rather than fully fledged biologicalcells. Although highly innovative, trying to usebacteria to perform amorphous computation is liketrying to build a glider by knocking out parts outof a jumbo jet. The problem is, both technicallyand philosophically, one of managing complexity:bacterial cells have evolved for millions of yearsand they carry too much evolutionary baggage. Be-fore any useful and general computation –ratherthan specific as exemplified by the above men-tioned references– can be achieved, this unwantedcomplexity must be tamed.

In contrast, the approach we suggest in this pa-per retains all the advantages of amorphous com-puting at the nanoscale (e.g., redundancy, mas-sive parallelism, asynchronous local processes, self-organisation, etc.) but by starting from the bottom-up with engineered components of pre-specified andlimited complexity, one avoids unnecessary biolog-ical nuisances from the start. Moreover, by turningto chemical cellular-like constructs, compartmen-talisation and orthogonality are maximised while

crosstalk minimised, thus the approach we proposemight provide a route to, e.g., more reliable pro-grammable drug delivery systems (Pasparakis andAlexander 2008; Gardner et al 2009).

In this paper we present a set of tools for thespecification, modelling and analysis of “bottom-up” liposome logic, also called vesicle computing.As explained above liposome logic makes use ofsupramolecular chemistry constructs, e.g. proto-cells, chells, etc., to encapsulate logical function-ality. The vesicle computing approach proposed isrelated to the e!ort to build a semi-synthetic pro-tocell (Luisi et al 2006), however, the goal in thiscase is to create a chemical automaton that is auseful platform for design and implementation ofcellular computing circuits, rather than attempt-ing to reproduce exactly the properties of livingsystems2. Liposome logic could then be used fordesigning the next generation of smart drug deliv-ery systems going beyond of the current state ofthe art (MacDiarmid et al 2009).

It is also proposed that vesicles could be usedfor encapsulation and implementation hiding, ashierarchical (i.e. nested) structures of membranescould be created with clearly defined inputs andoutputs, that create boundaries around function-ality just as organelles contain specific functionsin eukaryotic cells. This kind of compartmental-isation will enable interference between BRNs tobe minimized, and the need for multiple promotersequences/transcription factors to be reduced. Weanalyse the scalability of the techniques presentedwhen the liposome logic complexity increases fromrelatively simple NOT and NAND gates to SR-Latches, D Flip-Flops all the way to 3 bit ripplecounters.

The approach we propose consists of specifying,by means of P systems, BRN-like systems operat-ing inside proto-membranes. This P system spec-ification can be automatically translated and ex-ecuted through a multiscaled pipeline composedof Dissipative Particle Dynamics (DPD) simulatorand Gillespie’s Stochastic Simulation Algorithm(SSA). Finally, model selection and analysis can beperformed through a model checking phase. Thisis the first paper we are aware of that brings tobear formal specifications, DPD, SSA and modelchecking into the problem of modeling target func-tionality in chells. Potential chemical routes for thelaboratory implementation of these Liposome logicsimulations are also discussed.

2 Although there will, of course, be some synergy betweenthe two

2

2 Methods

In this section we describe the proposed modelingpipeline as depicted in Figure 1. Liposome logicmodeling starts with the specification of the logiccircuitry using P systems. The P system specifica-tion is then executed through DPD or an advancedGillespie’s Stochastic Algorithm implementation.The decision of whether to execute the model inDPD or directly through SSA depends on the timeand lengths scales of interest and also on whetherphysical volumes should be modelled explicitly, i.e.geometrically, or implicitly, i.e. topologically. If thelength/time scales are large and the volume is onlytopologically represented then SSA is used. A fur-ther analytical level is a!orded by the use of modelchecking techniques. In what follows we describethe P system specification formalism, DPD, SSAand model checking.

Fig. 1: Computational pipeline for Liposome logic. Comput-ing circuitry is specified through P systems that could thenbe interpreted and simulated either through a DPD simu-lator or, if the time and length scales are larger, througha stochastic simulation algorithm (e.g. Gillespie’s SSA). Afurther analysis can be performed using model checking.

2.1 P systems as a Specification Framework

P systems (Paun 2002) constitute a recently de-veloped specification framework bringing into sys-tems and synthetic biology methodologies from for-mal rewriting systems distributed over multicom-partmentalised regions. Our modelling approachbased on P systems falls within the classification ofcomputational, rule-based, modular and discrete-stochastic modelling frameworks. In this work, weuse a variant called stochastic P systems speciallysuitable for the scalable and parsimonious speci-fication of cellular systems exhibiting evident lev-els of stochasticity (Perez-Jimenez and Romero-Campero 2006).

The main components of a stochatic P sys-tems are objects, representing molecular species;compartments defined by membranes containing

multisets of objects and rewriting rules specificallyassociated with each compartment describing themolecular interactions taking place in and betweendi!erent compartments.

Formally, a stochastic P system is a construct

! = (O,L, µ,Ml1 ,Ml2 , . . . ,Mln , Rl1 , . . . , Rln)

where:

– O is a finite alphabet of objects specifying themolecular species in the system.

– L = {l1, . . . , ln} is a finite set of labels identi-fying compartment types.

– µ is a membrane structure containing n ! 1membranes defining compartments arranged ina hierarchical manner. Each membrane is iden-tified in a one to one manner with a label in Lwhich determines its type.

– Mli for each 1 " i " n, is the initial multiset ofobjects over O placed inside the compartmentdefined by the membrane with label li in theinitial state of the system.

– Rli = {rli1, . . . , rli

kli

}, for each 1 " i " n, is

a finite set of rewriting rules associated withthe compartment with label li # L and of thefollowing general form:

o1[o2]lic$% o!1[o

!

2]li (1)

with o1, o2, o!1, o!

2 multisets (potentially empty)of objects over O representing the molecularspecies and the stochiometries involved in themolecular interaction represented in the rule.The label li # L identifies the compartmentwhere the interaction takes place. These multi-set rewriting rules can potentially change boththe inside and outside of (proto)membranes.An application of a rule of this form replacessimultaneously the multisets o1 outside mem-brane li and o2 inside membrane li by the mul-tisets o!1 and o!2, respectively. A stochastic con-stant c is associated specifically with each rulein order to compute how often the rules are ap-plied and the time elapsed between rule appli-cations according to Gillespie’s theory of stochas-tic kinetics (Gillespie 2007). More specifically,rewriting rules are selected according to an ex-tension of Gillespie’s well known Stochastic Sim-

ulation Algorithm (SSA) (Gillespie 2007) to themulticompartmental structure of P system mod-els (Romero-Campero et al 2009; Perez-Jimenezand Romero-Campero 2006).

3

Cellular phenotypes arise from the orchestra-tion of the interactions between di!erent molecu-lar modules acting as discrete entities whose func-tionalities are up to certain point separable fromone another (Hartwell et al 1999). The interactionmodality in cellular systems is an intensed researchfield in systems and synthetic biology which is un-raveling specific modular patterns in BRNs (Alon2007). Biological modularity is thus one of the cor-nerstones of synthetic biology (Andrianantoandroet al 2006) and its relevance for systems and syn-thetic biology has been recently emphasized byMallavarapu et al. (Mallavarapu et al 2009). Inthis work we follow a modular modelling approachwhereby models are incrementally, parsimoniouslyand hierarchically built by combining virtual partsthat are available from a library. This library com-prises a set of elementary modules that specify bio-logical regulatory-like networks as well as modulesdescribing the regulation of specific gene promot-ers widely used in synthetic biology.

A P system module is defined as a set of rewrit-ing rules, each of the form in (1), for which someof the objects, stochastic constants or the labelsof the compartments involved might be variables.This facilitates reusability and parsimony in thedevelopment of models. Large models can be spec-ified by integrating commonly found modules thatare then further instantiated with specific valuesobtained experimentally. Formally, a P system mod-ule M is represented as M(V,C, L) where V speci-fies object variables, which can be instantiated us-ing specific names of molecular species like genesand proteins, C are variables for the stochasticconstants associated to the rewriting rules, whichcan be instantiated using specific a"nities betweengenes and proteins, half lifes for degradation pro-cesses, etc. and finally L are variables for the la-bels of the compartments involved in the rules thatmight represent di!erent cell compartments, e.g.,cytoplasm, lysosome, cellular membrane, etc., ordi!erent (proto-)cells altogether.

In the next section the DPD and SSA simula-tion techniques are described in detail.

2.2 Dissipative Particle Dynamics

Simulations at small length and timescales wereperformed using a self-developed mesoscopic mod-elling framework based on the Dissipative ParticleDynamics (DPD) technique. Our framework en-ables easy specification of large scale models inDPD, and vesicles and other bilayer structures that

form over the course of the simulation can be ex-tracted and stored for later recombination into newinitial states for further simulations. This featureallows for the combinatorial bootstrap of compu-tationally expensive simulations. For example, itis possible to (1) simulate the formation of vesi-cles and (2) save these emergent structures as tothen (3) create a new initial state for a simulationcontaining those vesicles with the internal volume,perhaps modified to contain particles representinggenes, proteins etc.

DPD is a coarse grained particle simulationtechnique, in which each particle represents sev-eral molecules of a given molecular species, ratherthan a single atom. By dispensing with the detailsof individual atoms, the short length and timescaleprocesses can be averaged out, allowing simula-tion for much larger length and time-scales thanis possible with other particle dynamics methods.Simulations are formed by filling a volume withparticles and integrating the equations of motionto calculate the particle positions and velocities ateach time step. Three forces act between particlesin a symmetric pairwise fashion, the Dissipativeand Random forces act as the thermostat in DPD,with the dissipative force removing energy fromthe system (whilst conserving momentum) and therandom force introducing energy in the system byproducing a brownian style motion between par-ticles, equation 2 shows the forces acting betweentwo particles i and j. The conservation of momen-tum in the system means that the hydrodynamicsare represented correctly.

Fij = FCij + FD

ij + FRij (2)

The conservative force FCij simply introduces a pa-

rameterisable repulsion between particles types whichdecreases linearly with distance reaching zero atthe cut-o! distance rc.

FCij = "(1 $

|rij |rc

) (3)

The " parameter sets the maximum repulsion fora given pair of types so for example the " param-eter will be set to a high value for the interactionbetween an oil and water particle, as these wouldbe immiscible, but to a smaller value for two waterparticles.

The Dissipative force acts as a drag force, slow-ing down particles that are approaching one an-other:

FDij = $#wD(rij)(rij · vij)rij (4)

4

Where # is the dissipative force parameter whichcontrols the magnitude of the force, rij is the dis-tance between particle i and particle j, rij is theunit vector point from particle j to particle i andvij is the relative velocity between particle i and jand wD is a weighting function described below.

The Random force introduces a randomised forcebetween each particle pair

FRij =

$wR(rij)%ij

&3rij&

dt(5)

Where $ is the random force parameter control-ling the magnitude of the force, %ij is a uniformlydistributed random number with unit variance andwR is the random weighting function described be-low. Polymers can also be represented in DPD withharmonic bonding and angle potentials.

FSij = k(rij $ r0) (6)

where k is the bond strength parameter, and r0

is the preferred bond length. preferred angles be-tween two bonds can be included with a harmonic3-body potential (Kranenburg et al 2003)

U! =1

2k!(% $ %0)

2 (7)

Where k! is the angle force strength parameter, %is the angle between the two bonds and %0 is thepreferred angle.

Espanol and Warren (Espanol and Warren 1995)investigated the statistical mechanics of DPD andfound that in order to maintain a correct and sta-ble temperature, the $ and # force parametersshould be set according to the following relation:

$2 =!

2#KbT (8)

where kBT is the required particle kinetic energy.The dissipative and random forces are coupled

with weighting functions wD(rij) and wR(rij). Oneof these functions may be chosen arbitrarily, andwe use the weighting proposed by Groot and War-ren (Groot and Warren 1997) for the random force:

WR(r) =

"

(1 $ r) when(r < 1)0 otherwise

(9)

The dissipative weight function is then derived fromthe following relation:

WD(r) = [WR(r)]2 (10)

Our implementation of DPD contains a colli-sion based artificial chemistry supporting first andsecond order reactions. A collision is considered to

have occurred if two particles come within a pa-rameterisable collision radius of one another. Inthe simulations presented here the collision radiusis set to the force interaction radius rc. Each re-action is assigned a rate cdpd, which is the rate atwhich colliding particles will react as a result ofthat collision per DPD time unit (the per collisionrate is therefore cdpd 'dt. If the types of the collid-ing particles match the reactant types for a reac-tion, then a pseudo random number is generated,and if this number of less than the rate then thereaction occurs and the types of the particles arechanged to represent the products of the reaction.In the case of first order reactions, for each particlethe reaction is attempted once per timestep, witha rate cdpd ' dt.

Groot and Warren gave a thorough explana-tion of the correct setting of the DPD parame-ters (Groot and Warren 1997) and described amethod for parameterising the conservative forcebased on the immiscibility of fluids, as well as anew integration method more suitable to the largertimesteps taken in DPD. The work by Groot andRabone (Groot and Rabone 2001) showed simula-tions of poration in a phospholipid membrane anddescribed the coarse graining procedure. These pa-pers define the de facto standard implementationof DPD, and our implementation of the algorithmis based on these works.

Despite the increased simulated length and timescales that DPD method permits, calculation of ex-plicit particle forces and velocities is computation-ally very expensive, and so simulations are typi-cally limited to milliseconds of simulated time andvolumes in the order of 0.1 cubic micrometres. Inorder to simulate the formation of DMPC vesi-cles, an implementation of DPD was created usingthe general purpose GPU CUDA programming en-vironment from Nvidia, producing a 50x speedupover the single processor implementation using anNvidia Tesla C1060 card.

Vesicles can be formed via a variety of di!erentmethods, including microfluidics (Tan et al 2006),centrifugation (Noireaux and Libchaber 2004), son-ication and spontaneous formation. Regardless ofthe route to formation, all vesicles are composedof amphiphiles which have a hydrophobic sectionwhich does not dissolve in water and a hydrophilicsection which is polar. In the presence of a polarsolvent such as water, the hydrophobic sections ofthe molecules move together such that the disrup-tion to the structure of the solvent is minimized.

5

This hydrophobic e!ect is the cause of spontaneousformation of micelles, vesicles and bilayers.

Clearly there is a large di!erence in the timeand length scales in which the BRN are normallysimulated and the timescales which can be cap-tured using the DPD method. However the use ofthe DPD method has some clear advantages overother less detailed techniques. Firstly, the vesiclecontainer and the emergent dynamics of the sys-tem are a result of the application of simple rules,e.g. the vesicle does not form due to any prespec-ified design, but as a result of minimization ofconfigurational energy of the lipids, just as realvesicles do. If the reaction rates of BRN modelscan be scaled so that the processes occur withintimescales that can be simulated in DPD, thenthis allows an exploration, at least in a qualita-tive sense, of systems where the BRN may pro-duce proteins which a!ect the membrane, eitherby production of proteins that have hydrophobicmoeties that could embed within the membrane(such as "-hemolysin) or by producing enzymesthat catalyse the formation of other lipids (whichmay form domains in the vesicle membrane, even-tually leading to fission). Also, as every particle inthe system has an explicit position, and the systemis not assumed to be mixing, unlike what occurswith the stochastic simulation algorithm (see nextsubsection), concentration gradients can arise andare captured within the model.

Moreover, as suggested in (Cronin et al 2006),both the P systems specification of a model andits execution through DPD adhere to the abstrac-tion that programmable living matter can be en-gineered through clearly identifying the compart-ment (C) that delimits the self from non-self, infor-mation (I) storage and processing that helps guidethe manufacturing of the compartment’s buildingblocks and the orchestration of metabolism (M)processes as the arbiters of energy and waste man-agement. That is, neither the P system nor theDPD simulations require that C, I,M be imple-mented in the way biology does but can indeed, fol-low a more chemical (rather than biological) routefor liposome logic (Pasparakisa et al 2009).

Figure 2 shows the formation of a vesicle frommodel DMPC amphiphiles in DPD. The dynam-ics of formation are as follows: the amphiphilesinitially aggregate into small micelles, which thenaggregate into larger micelles. Once the micellesreach a critical size, they become oblate (flattened)patches of bilayer membrane. If the bilayer mem-brane is large enough, then the membrane will be-

Fig. 2: The figure shows the process of vesicle formationfrom the initial state of the system were amphiphiles are dis-tributed randomly in solvent (top left, solvent not shown).The amphiphiles are pushed together into micelles (topright) which in turn join together to form large planar bi-layers (bottom left), these bilayers then begin to curl at theedges and fold over into a spherical vesicle (bottom right).

gin to fold inwards into a bowl shape, which con-tinues to curve until fusing at the top to form thespherical vesicle.

2.3 Stochastic Simulation Algorithm

Simulations of cellular logic systems for longer time-scales were performed with the MCSS toolkit (Romero-Campero et al 2008, 2009), a high performancemulticompartment stochastic simulator, support-ing simulation of stochastic P system models spec-ified in an XML format. This toolkit has at itscore an optimised implementation of the GillespieSSA. Stochastic discrete simulation techniques forbiological systems have a number of advantagesover ODEs at the cellular scale. Firstly, as ODEsare continuous and the concentrations of chemi-cal species within cell volumes can be very small,integration of the ODEs could result in concentra-tions which represent fractions of a molecule. Sec-ondly, as ODEs represent the dynamics of concen-trations of molecules rather than individual mol-ecules themselves, they do not capture the stochas-tic nature of the cellular volume (Gillespie 1977)and thirdly ODE models are typically more di"-cult to create and understand in comparison withexecutable biology (Fisher and Henzinger 2007),also called algorithmic systems biology (Priami 2009),methodologies. Moreover, stochastic models spec-

6

ified in, e.g., P systems are more amenable to for-mal computational analysis such as model check-ing.

2.4 Model Checking

Model checking is a well-established formal methodfor analysing the behaviour of various systems. Itnormally requires a computational model of thesystem, provided as a high-level formalism (such asa Petri net, process algebra or P system), and a setof properties of the same system, expressed usu-ally in temporal logic (LTL or CTL) (Kwiatkowskaet al 2009). A computational model associated toa system may consist of distinct parts, modules inthe case of the P system formalism (see Section 3),each one with a complex behaviour and generat-ing many states. The model allows to test and ver-ify certain hypotheses by executing the model andcomparing the outcome with experimental data(Fisher and Henzinger 2007). Knowing that somesystems are non-deterministic or probabilistic, theconclusions obtained are just limited to the num-ber of executions performed. In order to ascer-tain more general properties, model checking tech-niques are employed. These properties can be vali-dated for the entire system or for some componentsof it.

In probabilistic model checking, which will beused in this paper, the models are extended withquantitative information regarding the likelihoodthat some events will occur and the time they doso (Kwiatkowska et al 2009). The models referredto in this paper are continuous-time Markov chains

(CTMCs), where rates of negative exponential dis-tributions are assigned. The properties are still ex-pressed in temporal logic, but they show now somequantitative aspects. So, rather than verifying thatfor the NOT gate (see Section 4.1.2) “the proteinoutput always eventually reaches a certain level”we may check “what is the probability that theprotein output eventually reaches a certain level”.More than this, using rewards we can ask questionslike “what is the maximum protein output of theNOT gate”. Such questions will be formulated in aspecific temporal logic called continuous stochasticlogic (CSL) (Kwiatkowska et al 2009).

Model checking is very e!ective in verifying cer-tain hypotheses regarding the system when morethan one execution is possible and when only in-complete data is available and through the newcharacteristics revealed, the model checking ap-

proach may suggest new experiment to confirm orreject hypotheses (Fisher and Henzinger 2007).

3 P System Specification of Liposome

Logic Models

In this section we describe the P sytems specifi-cations for liposome logic circuits. These specifi-cations are the first step in the proposed method-ological pipeline shown in Figure 1.

3.1 A P System Specification for the Repressilator

Logic gates in cellular computing are constructedfrom networks of gene regulation in prokaryoticgenomes. In prokaryotes, genes are sometimes ar-ranged into operons, sequences of DNA contain-ing a promoter region which is recognised by RNApolymerase enzymes, an operator region which isrecognised by gene transcription factors, and oneor more gene sequences (see Figure 3).

Fig. 3: The operon in prokaryote genomes, the promoter re-gion is recognised by RNA polymerase, which binds to thepromoter to initial transcription. The operator is recognisedby transcription factor proteins which alter the rate of geneexpression, the operator may control the expression of mul-tiple genes.

The liposome logic simulations presented in thispaper are based on the repressilator reported byElowitz and Leibler (Elowitz and Leibler 2000).The repressilator is a ring oscillator built fromthree genes. Figure 4 shows a schematic diagramof the repressilator network. The system includesthree di!erent genes, LacI, &cI and TetR, with theprotein expressed from each gene acting as a re-pressor which binds to the promoter of the nextgene, and reduces the rate of transcription.

The authors present a stochastic model of therepressilator, in which all three genes and promot-ers have identical properties in terms of the ratesof binding etc. The repression of the gene is rep-resented by a cooperative binding of the repressorprotein to the gene promoter, (reactions 11 and12):

G + R1nm!1s!1

$$$$$$$% GR (11)

GR + R1nm!1s!1

$$$$$$$% GRR (12)

7

Fig. 4: The Repressilator, the system is composed of threedi"erent genes, LacI, !cI and TetR. The protein expressedfrom each gene inhibits the next, so for example the LacIproteins inhibit TetR expression, and TetR proteins inhibit!cI expression.

Where G is the NOT gate promoter and gene, Ris the repressor protein which binds to the geneoperator and represses transcription of the gene,M is transcribed mRNA from the gene G, and Ois the expressed protein from G, translated fromM.

The repressor proteins also decomplex from thegene promoter, and these are modelled with reac-tions 13 and 14. It should be noted that the rate ofdecomplexation when both repressor proteins arebound to the sequence is greatly decreased whencompared with the rate when only a single proteinis bound.

GR224s!1

$$$$% G + R (13)

GRR9s!1

$$$% GR + R (14)

Reactions 15 to 18 represent transcription and trans-lation. Transcription when the gene promoter isunrepressed occurs 1000 times more frequently thanwhen the gene is repressed.

G0.5s!1

$$$$% G + M (15)

M0.167s!1

$$$$$$% M + O (16)

GR5"10

!4s!1

$$$$$$$% GR + M (17)

GRR5"10

!4s!1

$$$$$$$% GRR + M (18)

mRNA and protein degradation occurs with a half-life of 120 seconds and 600 seconds respectively(reactions 19 and 20).

O0.0012s!1

$$$$$$% (19)

M0.0058s!1

$$$$$$% (20)

These reactions specify a stochastic model ofthe behaviour of one gene in the repressilator model.If the repressing protein is considered as the inputto the system, and the expressed protein the out-put, then the behaviour of the model mimics that

of a NOT logic gate, which outputs a high sig-nal when the input is low, and a low signal whenthe input is high. The gene operon has two oper-ator regions. A repressor protein can then bind tothese regions and repress the gene. When only oneoperator is occupied by a repressor protein, the re-pressor is more likely to decomplex from the genethan when both promoters are occupied, and it isthis cooperative binding which causes a “switch-like” transition between the high and low outputstates of the logic gate (Amos 2004), as a cer-tain threshold of repressor concentration must bereached within the cell volume before both opera-tors become occupied. The e!ect can be magnifiedby increasing the number of operators which coop-eratively bind repressors, or by using oligomer pro-teins which must bind together before being ableto bind to the gene. As we are interested in themodular assembly of variable depth logic circuits,we define a P system module (see formal defini-tion in section 2.1) that, by using the repressilatorcircuitry, encodes a NOT gate module:

NOTGate({R,G,M,O},{c1, c2, c3, c4, c5, c6, c7, c8, c9, c10}) =#

$

$

$

$

$

$

$

$

$

%

$

$

$

$

$

$

$

$

$

&

[G + R]c1$% [GR], [GR + R]

c2$% [GRR],

[GR]c3$% [G + R], [GRR]

c4$% [GR + R],

[GR]c5$% [GR + M ], [G]

c6$% [G + M ],

[GRR]c7$% [GRR + M ], [M ]

c8$% [M + O],

[M ]c9$% [], [O]

c10$$% []

'

$

$

$

$

$

$

$

$

$

(

$

$

$

$

$

$

$

$

$

)

The module’s variables {R,G,M,O}, includingthe continuous ones {c1, . . . , c10}, can be instanti-ated with di!erent promoters, genes, proteins andkinetic constants as to represent specific systems.Also note that the square brackets indicate thatthe reactions take place inside a specific (proto)membrane or compartment. Indeed to simplify thenotation we have taken out the compartments’ namevariables as we use only one compartment in thisstudy.

To construct the P system module represent-ing the repressilator, we start by deriving from theNOTGate module a specific instantiation namedNG, as to avoid specifying the stochastic rate con-stants each time (which are the same unless oth-erwise specified):

8

NG({R,G,M,O}) =#

$

%

$

&

NOTGate({R,G,M,O},{1, 1, 224, 9, 5 ' 10#4, 0.5, 5 ' 10#4, 0.167,

0.0058, 0.0012})

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Three or more NG modules can be connectedtogether in sequence, with the output of the lastconnected as the input of the first gate, to pro-duce a ring oscillator. Ring oscillators made from

Fig. 5: Ring oscillator built from three not gates.

N gates can be constructed as follows, for any oddinteger N greater than or equal to three:

RON({G1, · · · , GN ,M1, · · · ,MN ,

O1, · · · , ON}) =#

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NG(ON , G1,M1, O1),

NG(O1, G2,M2, O2),

· · ·NG(ON#1, GN ,MN , ON )

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Therefore when N = 3 the original Repressilatormodel, named RO3, is reproduced.

3.2 A NAND Gate

By creating two copies of the same gene, with dif-ferent promoter regions, a NAND gate can be cre-ated (Figure 6).

Fig. 6: A Nand Gate built from two NOT gates. The inputsto the gate are two repressor proteins labelled X and Y, andthe output protein is labelled Z.

The NAND gate is defined by the followingmodule, note that the gene, mRNA and outputprotein are the same for both NG modules, but

the input repressor is di!erent (R1 for one geneand R2 for the other).

NAND({R1, R2, G1,M1, O1} =*

NG({R1, G1,M1, O1})NG({R2, G1,M1, O1})

+

It should be noted that constructing a NANDgate in this way produces two distinct output levelswhen the gate output is high, in the first case whenneither input to the gate is present, both genes aretranscribed. However when the gate is presentedwith a single input one gene is repressed and thegate output, whilst still representing a logic valueof True or high, produces roughly half the amountof protein than it does when no input is present.

3.3 A Set-Reset Latch

Fig. 7: Set Reset Latch constructed from two NAND gates.

Two NAND gates can then be connected tocreate a Set-Reset Latch (Figure 7), the output ofeach gate is connected to the input of the other,and the state of the latch can be switched by hold-ing the remaining set or reset inputs high for ashort period. The Latch acts as a simple one bitmemory which can be set or reset by expressing theappropriate protein that represses the gene of therelevant NAND gate. The Latch module is builtfrom two NAND gates

SR $ Latch({R1, R2, G1, G2, O1, O2}) =*

NAND({R1, O2, G1,M1, O1})NAND({R2, O1, G2,M2, O2})

+

3.4 A D type Flip-Flop

Latches can then be connected to NAND and NOTgates to construct a D type flip-flop, as shown inFigure 8.

A D Flip-Flop takes a data input, indicatingwhether the flip-flop should be set or reset, anda clock input. The output of the flip-flop will be

9

Fig. 8: The D Flip-Flop built from two latches, four NANDgates and a NOT gate.

Fig. 9: A 3 bit counter connected to a 5 gate ring oscillator.

the last active input when the clock was still high,and so the output is fixed when the clock inputgoes from high to low. Each flip-flop stores a singlebit, and can be coupled in sequence to make largermemories. The D-Flip Flop can also be convertedto a toggle flip-flop by connecting the Q outputto the D input. Therefore each time a clock pulseoccurs, the gate will toggle between the Set andReset states. The module configuration for the Dflip flop is shown below:

DFlipF lop({G1, · · · , G9,

DInput, ClockInput,M1, · · · ,M8, O1, O2}) =#

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NG({ClockInput,G9,M9, O9})NAND({Dinput, ClockInput,

G5,M5, O5})NAND({O5, ClockInput,

G6,M6, O6})SR $ Latch({O5, O6, G1, G2,

O1, O2})NAND({O1, O9, G7,M7, O7})NAND({O9, O2, G8,M8, O8})SR $ Latch({O7, O8, G3, G4,

O3, O4})

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3.5 A 3 bit Ripple Counter

A toggle flip-flop can in turn be used to build ripplecounters, simple counters in which the the Q out-put of one flip flop is connected to the clock inputof the next flip flop. Figure 9 shows a diagram of

three flip flops connected together to form a 3 bitripple counter, with a 5 NOT gate ring oscillatoracting as the system clock, when the clock is high,the state of the first flip-flop is toggled to producea high output, connected to the clock input of thenext flip-flop, which is then also toggled. The firstflip-flop remains in the logic high state until thenext clock pulse, upon which it toggles to the lowstate, causing the state of the second flip-flop to befixed high. As the output of each flip flop togglesat half the rate of it’s clock input, the output ofthe first flip flop is high for one clock cycle, andthen low for one clock cycle. the second bit highfor two cycles and low for two cycles, and the thirdbit high for four cycles and low for four cycles.

The counter module is constructed from thefollowing modules:

Counter $ 3bit({G1, · · · , G28,

M1, · · · ,M35, O1, · · · , O8}) =#

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DFlipF lop({G1, · · · , G9,

ClockInput,O2,M1, · · · ,M9, O1, O2})DFlipF lop({G10, · · · , G18,

O1, O4,M10, · · · ,M18, O3, O4})DFlipF lop({G19, · · · , G27,

O3, O6,M19, · · · ,M27, O5, O6})5GateClock({G28, · · · , G35,

O5, O8,M28, · · · ,M35, O7, O8})

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4 Experimental Results

The liposome logic circuits specified in the pre-vious section were simulated under various di!er-ent conditions using DPD, SSA or both (see Fig.10) which shows a diagram of the relationship be-tween the di!erent modules and indicates whichwere simulated with DPD and which were simu-lated using SSA. The results of these simulationsare presented below.

4.1 Dissipative Particles Dynamic Results

4.1.1 Forming the Vesicles

Our model amphiphiles in DPD are based on pa-rameters from Kranenberg et al (Kranenburg et al2004) in which the authors investigate a numberof di!erent coarse grainings of DMPC like am-phiphiles. Figure 11 shows the structure of themodel amphiphiles. Each amphiphile has two hy-drophobic tails and a hydrophilic headgroup of

10

Fig. 10: The diagram indicates which modules were simu-lated using which simulation technique (e.g. DPD or SSA),and the relationship between the di"erent models simulatedusing the pipeline, an arrow pointing from one module toanother indicates that the module at the arrow’s source isused by the other module.

three particles. Angle forces maintain a rigidity inthe tail chains with a preferred angle of 180 de-grees and a force strength of 6KbT , and hold thetail particles apart with a preferred angle of 90degrees and force strength of 3KbT .

Fig. 11: Schematic diagram of the DMPC amphiphile. Thegreen particles make up the hydrophobic tail chains of theamphiphile, and are held together by Hookean spring forceswith a preferred distance of 0.7 units, and a bond angle forcemaintaining the angle between bonds are 180 deg . The twotail chains are held together by a bond angle force with apreferred angle of 90 deg

The alpha parameter matrix for the hydropho-bic, hydrophilic and water particles is shown inTable 1.

Water Head Tail

Water 78 75.8 110Head 75.8 78 110Tail 110 110 78

Table 1: The alpha parameters type matrix for the DMPCpolymer. A value of 78 produces the correct compressibil-ity for water at room temperature, larger values indicate arepulsion between particle types.

The physical length and timescales in the DPDsimulation can be ascertained by performing themapping described by Groot and Rabone. The unitlength in the simulation is set to the force interac-tion radius, and all beads have the same mass in

the simulation and occupy the same volume, equiv-alent to three water molecules (( 90A3). Since theunit cube density parameter ' is set to 3, mean-ing on average there will be three particles in eachunit cube, the side length of each cube, and there-fore the physical interpretation of the unit lengthis

3&

270A3 = 6.4633A. The physical interpreta-tion of the time step is based on calculation of theself-di!usion constant of the simulated DPD fluid,which is then mapped to the same value for wa-ter at room temperature resulting in a time unitlength ( of ( 88ps. Typical simulated times are2500( (220ns) to 100000( (8.8µs) in volumes of( 34nm3.

All DPD simulations in this work were per-formed with $ = 3,# = 4.5 and ' = 3 and theGroot Warren intergrator was used with & = 0.65and the timestep length dt = 0.05

For the vesicle computation simulations in thiswork a vesicle was formed which was composedof 5825 DMPC molecules, encapsulating a coreof 58550 solvent particles. The vesicle was thenplaced within a simulation space of 50r3

c , and thevolume which was external to the vesicle mem-brane was filled with solvent particles such that thecorrect density (3 particles per r3

c ) was achieved.For each NOT gate in the module a solvent par-ticle within the vesicle core was chosen at randomand replaced with a particle representing the gene.

4.1.2 Liposome logic in DPD

The NOT gate model can be implemented in DPDas a set of first and second order reactions. How-ever, the rates in the original model are specifiedover timescales of the order of seconds, with thedynamics of the system only observable over min-utes/hours. In order to observe the model dynam-ics within DPD timescales, the reaction rates arerescaled to occur within the DPD timescale. Forthe first order reactions this is a straightforwardprocess, as long as all the first order reaction ratesare scaled equivalently. However, for the secondorder reactions, the situation is more complex, asthe stochastic rate constant is the rate at whicha reactant pair will collide and react. This rateis determined by the physical properties of thesystem (e.g. temperature, reactant mass etc.) andthe probability that the colliding particles will bein the correct orientation for the reaction to oc-cur. Therefore to scale the second order reactionscorrectly, it is necessary to determine the rate atwhich particle pairs collide within the vesicle. The

11

collision rate was determined directly from simula-tion (data not shown) and was found to be 0.0002collisions per DPD time unit. Thus the NOT gatemodel is instantiated as

NOTGate({P,R,G,M,O},{1(#1, 1(#1, 1(#1, 0.0402(#1, 5 ' 10#4(#1,

0.5(#1, 5 ' 10#4(#1, 0.167(#1, 0.0012(#1,

0.0058(#1})

For the other reactions, the rate constants wererescaled by changing the unit of time from secondsto DPD time units ((). The consequence of this forthe second order reactions representing binding ofrepressor to gene, is that the rate of collisions isreduced as the number of collisions per time unitis much smaller in DPD. Therefore the actual re-action rate in DPD is shown below.

[R][G] ' 0.0002 (21)

Where [R] and [G] indicates the number of repres-sors and genes in the simulation respectively. Theconsequence of this is that the rate at which repres-sors bind to the gene is reduced by a factor of 5000in comparison with the other scaled rates. Notethat the rates of the decomplexation rules whichrepresent the repressor protein unbinding from thegene have been rescaled to 1(#1 and 0.0402(#1

(the rates in the original model were 224s#1 and9s#1 respectively), as the original rates were toofast to be represented in the rescaling, and so werereduced by a factor of 224.

The e!ect of this alteration somewhat miti-gates the reduction of the complexation rate, andthe reduction of the binding rate relative to theadjusted decomplexation rate is reduced by a fac-tor of 22.3. The e!ect of these changes will be thatthe repression of the gene occurs more slowly, andboth the decomplexation and complexation reac-tions occur more slowly in comparison to the firstorder reactions, but the qualitative structure of themodel should be maintained.

4.1.3 The e!ect of Encapsulation on logic gate

dynamics

The first experiment involved placing the NOTgate inside the vesicle membrane, and comparingthe results of simulation in which the NOT gatewas not encapsulated within the membrane (e.g.allowed to di!use freely within the full 50rc vol-ume.) to show the e!ect that the encapsulationhas on the second order reaction rates.

The result of simulating the NOT gate within avesicle with no input signal connected to the gate,for 5000( is shown in Figure 12. The NOT gategene is expressed when the gate has no input andthe amount of protein rises until an equilibriumbetween expression and degradation is reached, atan output level of ( 10000 proteins.

Fig. 12: Time series of the protein output from a gene repre-senting a NOT gate, encapsulated within a vesicle showingthe mean number of proteins present in the volume overthe course of the simulation, with the error bars showingthe estimated standard error.

Figure 13 shows the results of simulating theNOT gate with a high input signal (i.e. a gene pro-ducing the NOT gate input repressor protein wasadded to the system) the gate and input gene par-ticles were encapsulated within a vesicle and thenumber of expressed proteins recorded each timeunit to produce a time series (continuous blackline). Simulations were also performed in whichthe NOT gate and input gene particles were notplaced within the vesicle, but instead were able todi!use freely within the entire simulated volume(dotted black line).

At the start of the simulation the NOT gategene is initially expressed, until the amount of re-pressor (which is being concurrently expressed fromthe input gene) reaches the threshold required tofully repress the NOT gate gene, causing the amountof output protein to drop as the protein and mRNAdegrade and are not replenished, so that typicallyless than 50 proteins remained by the end of thesimulation.

The dotted line in Figure 13 shows the resultof simulating the high input model for the casewere the input and NOT gate genes were not en-capsulated within the vesicle. The mean number

12

Fig. 13: Output protein levels for simulation of NOT gateplaced within a vesicle and di"using freely in the simu-lated volume, averaged over 10 runs (error bars indicateestimated standard error). The continuous grey and blacklines show the time series for the input and output pro-teins for the NOT gate placed within a vesicle. The dottedgrey and black lines show the input and output proteins ofthe NOT gate with an input present when system is notencapsulated within a vesicle.

of proteins expressed at the peak of expressionwas greater by ( 1000 particles when compared tothe output time series for the encapsulated gate,and the peak was reached later in the simulation,indicating the transition of the NOT gate (mod-ule NG) from the high to low output state oc-curred more slowly. The mean time series for theinput repressor protein in the encapsulated andnon-encapsulated NOT gate simulations are shownby the grey continuous and grey dotted lines re-spectively. Once the repressor protein levels havereached an equilibrium, there is a di!erence of over( 1000 proteins between the encapsulated and non-encapsulated equilibrium value. Correspondinglythere is a di!erence between the levels of outputtedmRNA when the NOT gate was and was not en-capsulated, Figure 14 shows that the mean mRNAoutput when the gate was not encapsulated peakedat slightly less than 60 molecules, whereas the mRNAoutput for the encapsulated gate peaked at around43 molecules. As the collisions occur more frequentlyin the vesicle volume, the gene becomes fully re-pressed more quickly, and so the peak level of mRNAoutput is reduced.

Figure 15 shows the output protein levels fromthe NAND gate model built from two NOT gates(model NAND in section 3.2). Four time series areshown, one for each possible combination of sig-nal inputs to the gate. The continuous line showsthe output for the gate when both of the genes for

Fig. 14: The time series for the transcribed mRNA fromthe NOT gate gene, averaged over 10 runs. The continu-ous black line shows the output level of transcribed mRNAwhen the NOT gate was encapsulated within the vesicle,whereas the dashed line shows the mRNA time series whenthe NOT gate was di"using freely throughout the entirevolume.

the input signals (labelled X and Y in the figure)were present, the output level rises initially un-til enough of the input proteins is present to fullyrepress both genes in the NAND gate, at whichpoint the output signal drops to zero. The dottedline shows the case where there was no input sig-nal to the NAND gate, the level of output proteinreaches an equilibrium value of around 17500 pro-teins. The dashed and dot-dashed lines show thecase were one of the input signal genes was present,in both of these cases, one of two NOT gates whichmake up the NAND is repressed, and so the out-put protein levels reaches an equilibrium value ofaround 8000 proteins, which is roughly half theoutput level when neither input was present.

4.1.4 The encapsulated repressilator

The second vesicle computing experiment involvedthe encapsulation of the repressilator within thecore of a self assembled vesicle (module RO3). Timeseries from simulations of the increased decomplex-ation rate repressilator model, encapsulated withina vesicle are shown in Figure 16, the expressed pro-tein levels for each of the three NOT gates in therepressilator (shown for for three runs of the simu-lation) can be seen to oscillate. The increased de-complexation rate of the repressors from the genewhen compared to the original repressilator modelmeans that the period of oscillation is not quitelong enough to allow all of the transcription factorto degrade, and so the amount of each transcrip-

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Fig. 19: Snapshots from a simulation of the repressilator within a vesicle were taken every 2500" , the vesicle membraneis composed of hydrophobic tail chains (green) and hydrophilic head groups (red), a small micelle was trapped within thevesicle when it formed and is visible in each image. The vesicle was sliced so that the inner volume is visible (note thatsolvent particles are not shown). The images show (from left to right) the initial vesicle condition, high concentrations of theoutput protein expressed from the first NOT gate, the second NOT gate, and the third NOT gate (note the concentrationgradient visible in the last image).

Fig. 15: The time series of protein output levels from thesimulation of the NAND gate, the output of the gate isshown in response to 4 di"erent combinations of inputs la-belled X and Y

tion factor drops to around 1000 proteins. Figure17 shows the results of simulating the model wherethe decomplexation rates were only scaled, and notincreased. The decomplexation of repressor fromgene occurs less frequently in this model and sothe oscillations have a longer period, allowing thetranscription factors to degrade completely beforethe next cycle of the oscillation and the period ofthe oscillation is increased. Figure 19 shows theimages from the inner volume of vesicle, with theparticles representing the di!erent output proteinsfrom each of the three NOT gates given di!erentcolours, each of the images are captured at thepoint in the simulation were the respective proteinis being expressed.

Fig. 16: Simulations of the repressilator model with in-creased rate constants for the decomplexation of repressorsfrom the promoter.

4.1.5 Immiscible Repressors

The third vesicle computing experiment involvedthe same initial configuration as the previous re-pressilator experiment (module RO3), but the al-pha parameters for the proteins were modified slightlyto examine the case where the output protein isslightly hydrophobic, and also less miscible withother proteins. The e!ect of this should be to cre-ate three distinct protein phases, which may meanthe dynamics of the repressilator will be altereddue to the non-uniform concentrations of repres-sors. Table 2 shows the alpha parameter vector foreach repressor protein in the system.

The results from simulations of the repressila-tor with increased " parameters between the re-pressor proteins expressed from each NOT gategene are shown in Figure 18. Because the tran-scription factors are now hydrophobic and do not

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Fig. 17: Simulations of the repressilator model within a vesi-cle in DPD, the parameters are rescaled versions of thosefrom the elowitz model such that the dynamics can be ex-amined within DPD timescales.

Fig. 18: Simulations of the repressilator model with hy-drophobic repressor proteins.

Solvent Gene R1 R2 R3

Solvent 78 78 85 85 85Gene 78 78 78 78 78

mRNA 78 78 78 78 78R1 85 78 78 85 85R2 85 78 85 78 85R3 85 78 85 85 78

Table 2: # parameters for immiscible repressors.

mix with the solvent, the volume is no longer ho-mogeneous, causing the dynamics of the repressi-lator to be altered. The period of the oscillationsis no longer steady as the gene might not di!useinto an area that contains a high concentrationof proteins that repress it. The repressor proteinsalso form distinct phases which tend to move to-wards the boundary between the vesicle membraneand the solvent, so that contact between hydropho-

bic repressor and solvent is minimised. The resultof this movement was a bulging deformation ofthe normally spherical vesicle shape, this e!ect isshown in Figure 20. Deformation of the membranemay be interesting to those working on the prob-lem of causing vesicle fusion, as the deformation ofthe membrane will create areas of increased ten-sion due to the elasticity of the membrane, whichmay increase the likelihood of fusion if two suchvesicles were to come into close contact (Shillcockand Lipowsky 2005). This result also illustrates thesort of system dynamics that can be observed inDPD rather than in other less detailed simulationtechniques.

Fig. 20: Hydrophobic repressor domains form within thevesicle, and deform the membrane: The image on the leftshows the surface of a vesicle which has been deformed bythe formation of phases within it. The image on the rightshows a slice through the same vesicle, the output proteins(coloured orange, blue and purple) have formed phases inthe vesicle core and are pressing against the membrane.

4.2 Stochastic Simulation Algorithm Results

If more complex logical circuits need to be simu-lated, or simulations for long length/timescales arerequired, we can abstract away the molecular andthree dimensional detail of DPD and use instead astochastic simulation algorithm to simulate deeperlogic circuits that capture compartments’ topolo-gies but ignores their detailed geometries. The re-sults of the SSA experiments are now described.

4.2.1 Oscillator Frequency

We extended the Elowitz models with increasingnumbers of NOT gates, to investigate whether in-creasing clock periods would match the theoreticalestimates for silicon gates, and if there are limitsto the number of gates which can be connected to-gether in this way. The oscillator models were con-

15

structed from 5,7,9,11,21,31,41 and 51 gates mod-ules (RO3,RO5,etc.) and simulation of each oscil-lator was performed for 2 days of simulated time.

The formula for calculating the frequency of aelectronic ring oscillators built from any odd num-ber of NOT gates is shown in Eq. 22:

1

2nTp(22)

Where n is the number of logic gates, and Tp isthe propagation delay of each gate. We determineif this formula accurately calculates the frequencyof the oscillators built from logic gates by calculat-ing the propagation delay for the gates, and cal-culating the oscillator frequency from the outputdata, and then compare with the value from theformula.

The propogation delay of the NOT gate was de-termined to be 766.46+-1.95 seconds, by simulat-ing an NOT gate with the initial number of inputrepressor proteins set to the mean equilibrium out-put for the gate (11983+-47.29), with a constantinput of repressor protein also present. The pro-pogation delay was determined as the mean num-ber of seconds for the NOT gate output to fall tohalf of it’s original level. The results from simu-lation of oscillators with 1,3,5,7,9,11,21,31,41 and51 NOT gates are shown in Figure 21, the figureshows that the relationship between the numberof NOT gates and oscillator frequency is similar toequation 22 until the number of NOT gates is 11although the frequency is reduced by between 0.3and 1 microhertz. For oscillators with more than 11NOT gates the standard deviation of the frequencyis increased, and the shape of the curve no longerfollows the predictions from equation 22. Lookingat the data for each individual run showed thatfor 21 NOT gates and above, the oscillator wasdecreasingly likely to settle into a stable oscilla-tion. Table 3 shows the number of oscillators inthe 10 runs which were unstable for the di!erentnumbers of NOT gates.

4.2.2 The e!ect of RNAP and Ribosomes

The behaviour of the RO51 oscillator was also ex-amined in a more detailed model where the tran-scription and translation explicitly included poly-merase and ribosomes, The number of polymer-somes and ribosomes were at realistic levels for abacterial or large vesicle volume.

When RNAP and ribosome interactions are in-cluded explicitly in the model, the e!ect is that

Fig. 21: The figure shows the frequency of oscillation inµhz for oscillators constructed from 1,3,5,7,9,11,21,31,41and 51 NOT gates. The blue line shows the the oscilla-tor frequencies observed in simulation, each point is themean frequency of 10 simulations of the oscillator, the er-ror bars show the standard deviation. The red line showsthe frequency calculated from equation 22 for the di"erentnumbers of gates.

Number of NOT Gates Unstable Count

3 05 07 09 011 021 131 541 751 9

Table 3: The number of unstable oscillations observed dur-ing 10 runs of oscillators composed of di"erent numbers ofNOT gates.

there is a global constraint on the rate of transcrip-tion and translation. Figure 22 shows the levels offree RNAP and ribosomes for a simulation of the51 gate oscillator model modified to include RNAPand ribosome interactions explicitly, the model wasinitialised with 35 RNAP and 350 ribosomes. Theresult shows that when the oscillator is functioningthe average number of RNAP in use is slightly lessthan one, and the average number of ribosomesin use is around 25. However the inclusion of theRNAP and Ribosomes did not alter the transcrip-tion rate significantly.

4.2.3 The 3-bit Ripple Counter

The counter models were simulated in MCSS forsimulated time periods of either 2 or 3 days, withthe number of molecules of each chemical species

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Fig. 22: Time series for free RNA polymerase (RNAP) andRibosome (Rib) proteins in a simulation of the 51 gate os-cillator model.

recorded at every 3 minutes of simulated time toproduce a time series for each chemical species inthe simulation.

Figure 23 shows the time series for the simula-tion of the 3-bit counter with a 3 gate ring oscil-lator as the clock, and Figure 24 shows the resultsof simulating the same counter with a 5 gate ringoscillator.

Fig. 23: Time series for 3-bit counter model with 3-gateclock as input, proteinout2 is the clock signal, proteinG8

is the output of the first bit of the counter, proteinG18 theoutput of the second bit of the counter and proteinG26 isthe output of the third bit.

The time series show the output protein levelsfor each bit of the 3-bit counter. In the case of thecounter connected to a 3-gate clock, it is likely thatthe propagation delay of the flip flops is greaterthan the time between clock pulses, and so theoutput of the first counter bit (proteinG18) does

Fig. 24: Time series for 3-bit counter model with 5-gateclock as input, proteinout4 is the clock signal, proteinG8

is the output of the first counter bit, proteinG18 the out-put of the second bit of the counter and proteinG26 theoutput of the third bit.

not always indicate that the flip flop was correctlytoggled by the clock input. When the counter isconnected to a lower frequency clock (constructedfrom 5 NOT gates), the dynamics of the outputof the first counter bit have a much more consis-tent period and number of period of high outputis roughly 1/2 the number of input clocks as ex-pected. Figure 25 shows the clock input and firstbit output overlayed for the 5 gate clock model.The figures shows that there is a clear correspon-dence in each case between the high level of eachbit and the triggering of the output of the next bit,the counter is therefore functioning as intended.Note that when the counter reaches its limit (7in this case) it simply overflows and the counterstarts from zero again.

5 Model Checking

We focus our analysis on two of the simplest partsin our study, namely the NOT gate and the NANDgate, that are subsequently used to construct therest of the models. In order to asses their perfo-mances we applied formal analysis on their dy-namics using simulative probabilistic model check-

ing. More specifically, the behaviour of our P sys-tem models were translated into CTMCs and thenanalysed using the probabilistic model checker PRISM(Kwiatkowska et al 2002). Due to the complex-ity of the models under study the complete statespace was not constructed, but, instead, ensem-bles of multiple simulations or trajectories in thestate space were generated and the corresponding

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Fig. 25: Overlayed time series for protein output levels, thetop figure shows the clock input level overlayed with thebit-0 output for the counter, the middle figure shows thebit-0 output overlayed with the bit-1 output, and the bot-tom figure shows the bit-1 output overlayed with the bit-2output.

properties, expressed in the temporal logic CSL(Kwiatkowska et al 2002), were checked againstthem.

In the analysis that follows 1000 simulationswere used to produce an estimate p of the answerp to a query. This resulted in a precision of 0.1 with

a confidence of 0.01 which determines the accuracyof the estimate according to the following formula.

P [ |p $ p| > precision ] < confidence

5.1 NOT Gate

In the case of our molecular NOT gate we studiedthe accuracy of its behaviour with respect to thegeneral specification of a NOT gate and the speedof its reponse when provided with some input mol-ecules.

5.1.1 Expected number of output proteins in the

long run for di!erent values of input proteins.

We examine whether or not this basic buildingblock behaves as expected. That is, in the pres-ence of low values of input proteins, high levels ofoutput proteins should be produced and viceversa,when high amounts of input proteins are provided,no output protein should be synthesized.

In order to investigate this, the following in-

stantaneous reward formula was formulated anda reward corresponding to the number of outputproteins was associated to each state in the corre-sponding continuous time Markov chain.

R = ? [ I = 6000 ]

The property was analysed at the time instantI = 6000 seconds. Figures 26 and 27 show usinglinear and logarithmic scale, respectively, how forlow number of CI proteins the number of outputproteins in the long run is high. Whereas an in-crease in the number of input proteins producesa sharp decrease to zero in the number of outputproteins. The transition from high to low outputoccurs at around 150 input proteins. These resultsare in agreement with the general specification ofa NOT gate.

]

Fig. 26: Expected number of output proteins for di"erentnumber of initial input proteins (linear scale)

18

Fig. 27: Expected number of output proteins for di"erentnumber of initial input proteins (logarithmic scale)

5.1.2 Expected propagation time or response time.

We analyse how fast our molecular device respondsto its input by determining the time expected toreach half way between the initial and the finalstate once input proteins are introduced in the sys-tem. This property is normally termed propagation

time or response time.

The following reachability reward formula wasconsidered in order to investigate the propagationtime of the NOT gate.

R = ? [ F proteinOut < 5000 ]

This type of query accumulates, over a trajec-tory, the rewards associated with each state timesthe time spent in that state until a state fulfill-ing the corresponding formula is reached. Since wewant to accumulate the time spent in each stateover a given trajectory a reward equal to one is as-sociated to each state in the corresponding CTMC.

The property whose reachability needs to beanalysed is the output protein descending belowthe threshold of 5000 molecules, which is half ofthe the initial number of output proteins, whichwas of the order of 104. In Figure 28 we can observethat a low number of input proteins leads to a veryslow response, whereas an increase in the numberof input molecules produces a fast decay in thepropagation time. Interestingly, our study showsthe existence of a threshold for the input proteinsaround 150 for which any further increase does notproduce an acceleration in the response.

From these two properties we can conclude thatfor our NOT gate there exists a threshold of around150 input proteins. Below this number our molec-ular device produces a high number of output pro-teins. By contrast, if a number of input proteinsabove this threshold is provided to the system,then no output proteins are synthesised. Moreover,this threshold of 150 proteins provides the optimalinput value with respect to the propagation time,

Fig. 28: Expected propagation time for di"erent number ofinitial input proteins

as an increase in the input beyond this level doesnot produce a faster response.

5.2 NAND gate

Similar to the previous case for the NAND gate westudy properties that determine the accuracy ofthe behaviour of our genetic design when compareto the general specification of a NAND gate.

5.2.1 Expected behaviour of the NAND gate.

In the presence of both inputs our molecular deviceshould synthesise no output proteins whereas inany other case, that is, presence of only one inputor absence of both inputs, output proteins shouldbe detectable.

The following instantaneous reward property isused to determine the number of output proteinsin the long run, time instant I = 6000, for di!erentvalues of the two input proteins.

R = ? [ I = 6000 ]

0100

2000 3000 50 100 150 200 400250 INPUT1300 350 400 500450 500

INPUT2

1000

2000

3000

4000

5000

6000

7000

8000

9000

Fig. 29: Expected number of output proteins in the longrun for di"erent number of input proteins

Note that since the NAND gate is a compo-sition of two identical NOT gates with the same

19

parameters as the one analysed above the thresh-old of 150 input molecules is also evident in thebehaviour of this gate, Figure 29. This determinesfour di!erent regimes in the behaviour of the gate.When INPUT1 < 150 and INPUT2 < 150 theoutput is maximal. For INPUT1 < 150 and INPUT2 >150 (similarly for INPUT2 < 150 and INPUT1 >150) the output is produced at a half maximallevel. Finally, no output proteins are sinthesisedwhen INPUT1 > 150 and INPUT2 > 150.

5.2.2 Probability of the absence of a detectable

level of output proteins.

In order to get a more detailed intuition of the be-haviour of the NAND gate we estimated the prob-ability of a non-detectable level of output proteinsin the long run for di!erent values of the two inputproteins. The detectable level was fixed to 500 out-put proteins. For this we used the following tran-

sient probability formula.

P = ? [ true U[6000,6000] proteinOut < 500 ]

050

PROBABILITY

0.0

0.1

0.3

0.2

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10050

150100200150

250200250 300

INPUT1300 350350

400INPUT2 400

450450500500

Fig. 30: Probability of the absence of a detectable level ofoutput proteins for di"erent levels of both inputs

Figure 30 shows the sharp transition aroundthe threshold of 150 input proteins from a de-tectable level of output proteins to an undetectableone.

6 Potential Routes to a Chemical

Implementation

The potential power of the vesicle computationmethod and the use of compartmentalisation inthe DPD simulations o!er intriguing possibilitieswithin a chemical context. The “bottom-up” ap-proach allows for many further molecular systemsto be invoked than those currently used in biol-ogy, sophisticated though these already are. For

example, logic gates have been constructed from avariety of non-biological systems and have used in-puts/processes ranging from photoelectron trans-fer and fluorescence through to gel swelling andelectrical signals (Asoh and Akashi 2009; de Silvaand Uchiyama 2007; Gunnlaugsson et al 2000; Jamesand Shinkai 2002; Yoshida and Yokobayashi 2007;Magri 2009; Pischel 2007). Abiotic small moleculesystems generally rely for their logic processingon binding events such as host-guest interactions,which lead to a perturbation in the electronic orconformational state of the molecule, which in turnare converted to signals. Combinations of di!erentinputs (e.g. pH, ion binding) on to molecules withmore than one potential host-guest interaction orconformational change lead to multiple logic oper-ations and functions such as Adders and Subtrac-tors built from AND, XOR, INH and OR gates.Small molecule logic systems of this type can alsobe coupled to non-chemical inputs, such as light,enabling their use in energy interconversion andsignal transduction. In this way, a number of pro-cessor elements in the size range of a few nm havebeen developed, with obvious advantages in minia-turisation compared to top-down machining or litho-graphic fabrication methods.

However, potentially much more powerful op-erations are possible when multicomponent coop-erative or interfering interactions are used. Intro-duction of multiple binding or reporter elementsonto polymer chains enables a further level of so-phistication in processing information. This is be-cause each interaction, for example at a receptorsite, on a polymer chain is inherently coupled to itsnearest neighbour on the chain. This can be pos-itive or negative in terms of the next interaction,and thus enhancement or thresholding e!ects canbecome apparent. Natural logic systems such asDNA, RNA already exploit these e!ects in bind-ing or repression of binding as described above,but recent studies have also shown simple logic cir-cuits can be derived from host-guest interactions insynthetic polymers (Pasparakis et al 2009). Con-formational changes in these polymers resultingfrom temperature-driven phase transitions causechanges in functional group accessibility which re-sult in “switching” of signalling. The system canbe reset with pH or temperature, leading to ANDand INH functions. The cooperativity of hydrogen-bonding solvent interactions drives the phase tran-sition, and this is a property related to the balanceof entropic and enthalpic factors governing poly-mer solubility and is fundamentally “polymeric” in

20

origin. These factors combine to produce the over-all e!ect, i.e. switching of binding “on” or “o!”,but because the phase transition is tuneable throughthe choice of chemistries in the polymer, other“states” of switching are possible. For example,by connecting polymer chains together in such away that one component undergoes a phase tran-sition while another does not, a simple “on-o!”solubility change can become a unimer-to micelleor unimer-to vesicle switch (Sundararaman et al2008). This can be considered as an alteration inthe symmetry of the system, as chemical speciesable to interact with the unimers in isotropic so-lution become distinct from each other dependenton whether they are inside or outside the micellaror vesicular compartments which form during thepolymer phase transition. Functionality that be-fore the transition was identical becomes stronglydirectional on the inner and outer surfaces of thevesicle, while concentration and di!usion gradientsare generated.

Ultimately then, it is possible to envisage so-phisticated information processing circuits that couldbe formed using synthetic polymer vesicles in ex-actly analogous ways to the repressilators describedin Section 4.1.4 for proteins and gene circuits insideliposomes. Interactions of a synthetic polymer witha ligand (a “repressor”) in the aqueous interior ofa vesicle could lead to a change in solubility of thepolymer which drives it towards the hydrophobicinterior of the vesicular membrane. Incorporationin the membrane of a reagent capable of, for ex-ample, reacting with or sequestering the ligand,will return the polymer back into the vesicle inte-rior, but only in the case where the ligand remainsaccessible. This can be a function of the degreeof binding as the interaction of multiple weaklyhydrophobic ligands will, eventually, lead to anoverall change in solubility of the polymer-ligandcomplexes. If the synthetic polymer solubility istuned such that at a certain binding threshold itthen becomes membrane-inserting or membrane-traversing, a “flip-flop” operation becomes possi-ble, dependent on starting concentrations. Thesein turn will be set by the conversion of unimers tovesicles, generating multiple feedback loops. Thuseven for quite simple synthetic polymer constructs,it is theoretically feasible, if not yet fully experi-mentally tractable, to put in place chemical imple-mentations of the computational simulations andmolecular logic operations described earlier.

In the above discussions, we have focussed onwater as the solvent in which the chemistries take

place. It is also possible to consider other solventsin which micelle, vesicles and other containers form,thus the metabolism and information processingcould be far removed from existing biological en-tities. If the rules of macromolecular phase tran-sitions, vesicle formation and molecular associa-tion/dissociation can be derived for other solventsystems, there is no reason why sophisticated log-ics and synthetic biologies should not emerge innon-aqueous environments.

7 Conclusions

In this paper we have presented our investigationsof vesicle and cellular computing. These simula-tions were performed through a novel use of theDissipative Particle dynamics technique, and throughthe use of Gillespie’s SSA to perform large scalesimulations of the the digital logic gate abstrac-tion for GRN design. In the first case, vesicles wereself-assembled in DPD and used as containers forthe repressilator. We showed the possibility of us-ing vesicles to encapsulate functionality by plac-ing one vesicle containing a GRN within another,with the same/genes promoters used in each vesi-cle. Some important issues were not dealt with inthis work however. One of which is how might sucha system be formed in vitro? Although multilam-melar vesicles form in the lab for certain vesicleformation techniques, forming unilammelar vesi-cles which contain GRNs is currently an area ofactive research in the field of protocells, and it isnot clear how GRNs could be reliably encapsulatedin more complex structures.

8 Acknowledgements

Natalio Krasnogor and Cameron Alexander wouldlike to acknowledge the EPSRC for funding projectsEP/E017215/1, EP/G042462/1 and the BBSRCfor their support for the “SynBioNT: A SyntheticBiology Network for Modelling and ProgrammingCell-Chell Interactions” (BB/F01855X/1) .

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