Table Of ContentA Molecular Implementation of the Least Mean Squares Estimator
Christoph Zechner∗1 and Mustafa Khammash†2
1Department of Biosystems Science and Engineering, ETH Zurich, Basel, Switzerland
Abstract – In order to function reliably, synthetic molec- Inthepresentwork,weconsideranotherpowerfulclass
ular circuits require mechanisms that allow them to adapt of estimation schemes that are frequently used in adap-
to environmental disturbances. Least mean squares (LMS) tive signal processing and control theory. These schemes
schemes,suchascommonlyencounteredinsignalprocessing –termedleastmeanssquares(LMS)estimators[13,14]–
and control, provide a powerful means to accomplish that iterativelycomputethesolutionofageneralleastsquares
7
goal. In this paper we show how the traditional LMS algo- problemthroughagradient-basedparametersearch. This
1
rithm can be implemented at the molecular level using only iterative structure allows a circuit to estimate unknown
0
2 a few elementary biomolecular reactions. We demonstrate quantities in an adaptive fashion by processing measure-
our approach using several simulation studies and discuss its ments in realtime. In this paper we demonstrate how
n
a relevance to synthetic biology. LMS-type estimators can be realized using elementary
J biomolecularreactions. Ourworkisrelatedto[10],where
3 theauthorshaveproposedaDNA-basedgradient-descent
1 Introduction scheme, which is able to learn static linear functions. In
] this work, however, we focus on dynamical and possibly
N
Engineered circuits in living cells often exhibit poor ro- stochasticbiochemicalsystemsthatshallbe identifiedby
M
bustness and substantial variations from one cell to the a molecular LMS estimator.
. next [1, 2]. In extreme cases, they are found functional
o
in only a small fraction of cells in an isogenic popula- The remainder of the paper is structured as follows.
i
b tion, while others act unpredictably. A major cause for InSection2weintroduce the mathematicalnotationand
q- such behavior is that the biochemical components that models required to describe molecular circuits. In Sec-
[ constitute a circuit depend on factors in their molecular tion 3 we introduce the concept of LMS estimation and
environment(or context)of the cell[3]. For instance, the present possible molecular realizations. In Section 4 we
1
rate at which a protein is expressed depends on the gene test the performance of the proposed circuits using sev-
v
2 dosage or the number of available ribosomes and so on. eralsimulationstudiesanddiscusshowtheymaybeused
0 Recently, progress has been made in taking into ac- in practical applications.
6
count such environmental factors into the modeling and
0
design of molecular circuits [4, 5, 6, 7]. This can tremen- 2 Biochemical Reaction Networks
0
. dously improve the faithfulness of computational models
1
and in turn the predictability of rationally designed cir- Weconsiderwell-mixedmolecularreactionnetworkscom-
0
7 cuits. However, in practical scenarios, the origins and prising K molecular species Z = (Z1,...,ZK)T that in-
1 propertiesofpotentialdisturbancesarebarelyknownand teractwitheachotherthroughLreactionchannelsofthe
: hardto anticipate duringdesigntime. Fromthis pointof form
v
i view, it seems barely realistic to tune a circuit in silico Z−h−i−(Z−(−t)⇀) Z+νi, (1)
X suchthatitactsrobustlyunderallpossibleperturbations
with i as the reaction index, Z(t) as the abundance of Z
r that it may encounter in the real environment of a cell.
a at time t, hi as a rate function determined by the law of
A viable alternative is to employ adaptive design prin-
mass-action and ν as the stoichiometric change associ-
i
ciples in which a circuit continuously senses and adjusts
ated with reaction i. Throughout this paper, we follow
itselftochangingenvironmentalconditions. Thisrequires
the convention to denote molecular species as boldface
molecular circuits that learn and make inference about
symbols. Note that we use the same symbol also to refer
their surroundings. A few attempts have been made re-
to the circuit that those species constitute.
cently to devise such circuits in the form of chemical re- We describe the time-evolution of Z as a continuous-
action networks, for example to perform neural network
time Markov chain (CTMC) that can take into account
computations [8], to realize message passing inference [9]
the inherent randomnessof biochemical reactions [15]. It
or supervised learning [10]. Along these lines, we have
canbeshown[16]thatthemolecularabundanceZ(t)sat-
recentlyproposedamolecularimplementationofanopti-
isfies a stochastic integral equation of the form
mal filter that allowsone to estimate dynamically chang-
ing noise signals [11]. This estimator was derived under L t
Z(t)=Z + P h (Z(s))ds ν , (2)
a Bayesianoptimality criterion by employing a Kushner- 0 i(cid:18)Z i (cid:19) i
Xi=1 0
Stratonovich differential equation [12].
where P is an independent unit Poisson processes de-
i
∗[email protected] scribing the firings of reaction i. Eq. (2) is commonly
†[email protected] known as the random time change model.
1
Assuming the chemical species to be highly abundant, withα(n)asatuneablestep-size. Thechoiceofthelatter
molecular fluctuations become negligible and eq. (2) can usuallyinvolvesatradeoffbetweentherateofconvergence
be approximated by a deterministic rate equation of the andthesteadysateerrorofthescheme(i.e.,excesserror).
form Since we consider continuous-time dynamical systems,
L
d we seekfor aninfinitesimal variantofthe LMSalgorithm
Z˜(t)= h (Z˜(t))ν (3)
dt i i [17], i.e.,
Xi=1
with Z˜(t)≈Z(t). We will make use of equations (2) and dθˆ(t)=−α(t) ∂ J(θˆ) , (7)
(3) at a later in this manuscript to model stochastic and dt ∂θˆ (cid:12)(cid:12)
(cid:12)θˆ(t)
deterministic reaction networks, respectively. (cid:12)
(cid:12)
in which case α(t) can be understood as a rate at which
the scheme adapts.
3 A Continuous-Time LMS Algorithm
Suppose a circuit requires knowledge about certain envi-
Target system
ronmentalfactorsθ. Forexample, θ couldbe the number
of phosphotases available to the circuit. However, these
factors are typically not accessible directly by the circuit
butonlyindirectlythroughavailableintermediatesY. In
the example above, Y couldbe a proteinthat is targeted Y(t)
e(t)
+
by that phosphatase, for instance.
-
The idea is now to use a second molecular circuit X X(t)
Adaptive system
thatisable toidentify the dynamicsofY throughasuit-
able adaptation scheme. We assume here that X and Y
areequivalentintheir structurebuthavedistinctparam-
eters θˆ and θ, respectively. The goal of the adaptation
scheme is to adjust the parametersθˆsuchas to minimize
the discrepancy between the measured output Y(t) and
the output of X (termed X(t)). The resulting optimal LMS scheme
parameters θˆ∗ then represent an estimate of θ.
Figure 1: Schematic illustration of an adaptive molecular cir-
A suitable and analytically convenient metric to as-
cuit. The goal is to construct a biomolecular circuit X that ad-
sess the discrepancy between Y(t) and X(t) is the mean
justsitsparameterstomimickthoseofatargetsystemY. While
squared error
those parameters are inaccessible by the circuit, it can measure
the output Y(t) of the target system. This output is compared
J(θˆ)=E (X(t)−Y(t))2 . (4)
to the output of X to construct an error signal e(t), which is in
(cid:2) (cid:3) turn used tofind theoptimal parametersof X.
According to this measure, we seek for the set of param-
eters that minimizes J(θˆ), i.e.,
θˆ∗ =argmin J(θˆ), (5) For the sake of simplicity, we make a few simplifica-
θˆ tions related to X and Y before deriving the scheme. A
moregeneralscenario,however,willbesubjectofafuture
in which case θˆ∗ is referred to as the least mean squared manuscript. First, we assume that θ (and correspond-
(LMS) estimator. The closed-form solution of this opti- inglyθˆ)containsonlyasingleparameterthatneedstobe
mizationproblemcanbe foundin certainspecific scenar- identified from Y(t). Furthermore, we allow only Y(t) to
ios,forinstanceinthecaseoflinearsystemdynamics[14]. becorruptedbymolecularnoise,andreactionsassociated
Inmostscenarios,however,(5) isanalyticallyintractable with X(t) are assumed to evolve deterministically (e.g.,
and one has to minimize J(θˆ∗) numerically. Note that through appropriate rescaling of the associated reaction
iterative schemes may be beneficial even when (5) is an- rates).
alytically tractable, because it gives X the flexibility to Under these assumptions, the gradientof J(θˆ) is given
readapt to changes in θ as will be shown later in this by
manuscript.
A common strategy to minimize J(θˆ) is to employ a ∂ ∂
J(θˆ)= E (X(t)−Y(t))2
gradient-basedmethod that, at eachiteration,moves the ∂θˆ ∂θˆ
parameters θˆalong the direction of the steepest descent, (cid:2) (cid:3)
∂
giving rise to the well-known LMS algorithm. This algo- =E (X(t)−Y(t))2
(cid:20)∂θˆ (cid:21)
rithm is usually used in a discrete-time scenario, for in- (8)
∂
stance when operated on a digital signal processing unit. =E 2(X(t)−Y(t)) X(t)
Insuchcase,ateachtimeiterationn,thealgorithmwould (cid:20) ∂θˆ (cid:21)
update the parameters using the relation ∂
=2E (X(t)−Y(t)) X(t) .
(cid:20) ∂θˆ (cid:21)
∂
θˆn+1 =θˆn−α(n) J(θˆ) (6)
∂θˆ (cid:12)(cid:12) Since we assume that X(t) is deterministic, (8) further
(cid:12)θˆn
(cid:12)
(cid:12)
2
simplifies to can be represented by
∂ ∂ f
J(θˆ)=2E (X(t)−Y(t)) X(t) A+B−↽⇀−O
∂θˆ (cid:20) ∂θˆ (cid:21) b
=2X(t) ∂ X(t)−2E[Y(t)] ∂ X(t) (9) O+C−λ−b−/⇀f D,
∂θˆ ∂θˆ
=2X(t)S(t)−2E[Y(t)]S(t), assuming b,f >>λ.
Specific implementations of the derived LMS adapta-
with S(t) := ∂ X(t) as the sensitivity of X(t) with re- tion scheme will be given in the subsequent section.
∂θˆ
spect to θˆ. Note that the particular formof this sensitiv-
itydependsonXandhowθˆentersitsdynamics. Specific 4 Case Studies
examples will be given later in Section 4.
Inthissectionweprovideseveralnumericalandanalytical
Now, plugging eq. (9) into (7) yields a dynamic equa-
examples to demonstrate our molecular LMS estimation
tion for the LMS estimator θˆ(t), i.e.,
framework. In all of the examples, we will consider a
target process
d
θˆ(t)=−2α(t)X(t)S(t)+2α(t)E[Y(t)]S(t) ρ φ
dt (10) ∅−⇀Y −⇀∅, (13)
=−α˜(t)X(t)S(t)+α˜(t)E[Y(t)]S(t),
with ρ and φ as the process parameters that we aim to
identify using an LMS scheme. As indicated earlier, we
with α˜(t)=2α(t).
restrict ourselves to the case of a single unknown param-
eter, meaning that we either have θ ={ρ} or θ ={φ}.
3.1.OnlineAdaptation. Eq. (10)providesthedesired
continuous-time solution of the LMS problem. However, 4.1.Self-adjusting birth-rate. We first consider the
in its current form it is not adaptive, meaning that it as- case where the birth-rate ρ is unknown to the circuit.
sumes known (and fixed) statistics of Y (i.e., the output The goal is to construct a corresponding adaptive circuit
mean E[Y(t)]). In practice, however, such statistics are X
oftenunknownandtheymightalsovaryovertime. Using ∅−⇀θˆ X−⇀φ ∅, (14)
LMS estimation, this problem can be bypassed by esti-
matingE[Y(t)]online fromavailablemeasurementsY(t). whose birth-rate θˆ adapts to that of Y. To accomplish
This way, the required statistics are extracted directly this, we require a molecular implementation of relation
from data, which in turn allows the scheme to readapt (11). The firststepis to derivethe particularformofthe
when θ changes. A common and simple approach is to sensitivity function S(t) from the dynamics of X. The
approximate E[Y(t)] by the current value of Y(t) such latter is given by the rate equation (3) which in this case
that reads
d
X(t)=θˆ−φX(t). (15)
dθˆ(t)=−α˜(t)X(t)S(t)+α˜(t)Y(t)S(t) (11) dt
dt
Differentiating both sides of eq. (15) with respect to θˆ
and we adopt this strategy also in the present work. A yields
graphicaldepiction of the online LMS scheme is depicted d ∂ d
X(t)= S(t)=1−φS(t). (16)
in Fig. 1. dt∂θˆ dt
Fortunately, this equation is already in the form of a
3.2.Molecular Implementations. The goalisnowto valid rate equation. In particular, it describes the time-
synthesize eq. (11) using biochemical reactions. How- evolution of a birth-death process
ever, in its present form eq. (11) is incompatible with
mass-actionratelawsbecause it containsa negative(i.e., ∅−⇀1 S−⇀φ ∅. (17)
degradation) flux that does not depend on the current
value of θˆ(t). To account for this, we choose the adap- Inconjunctionwith(11),the overalladaptivesystemcan
tation rate to be proportional to θˆ(t), i.e., α˜(t) := λθˆ(t) be implemented through reactions
and thus,
∅−⇀θ Y
dθˆ(t)=−λθˆ(t)X(t)S(t)+λθˆ(t)Y(t)S(t). (12) Y −⇀φ ∅
dt
θˆ −⇀1 θˆ+X
While (12) is now in principle compatible with mass-
φ
actionkinetics,itinvolvestrimolecularreactions,thatare X−⇀∅
(18)
hardormaybeimpossibletorealizeinpractice. However, ∅−⇀1 S
the trimolecular reaction can be composed from two bi-
φ
molecular reactions with appropriately chosen rate con- S−⇀∅
stants. For example, the reaction
S+θˆ+X−⇀λ S+X
A+B+C−⇀λ D S+θˆ+Y −⇀λ S+2θˆ+Y.
3
We first studied the adaptation performance of (18) 0.3
as a function of the tuning parameter λ in an idealized
noise-free scenario. In this case, the network from (18) is 0.2
described by the rate equations
0.1
d
Y(t)=θ−φY(t) (19)
dt
0
d X(t)=θˆ(t)−φX(t) (20) 0 20 40 60 80 100
dt Time
d θˆ(t)=−λθˆ(t)X(t)S(t)+λθˆ(t)Y(t)S(t) (21) true value
dt
d Figure 2: Convergence ofestimated birth-rateasafunction of
S(t)=1−φS(t). (22) λ. Theadaptivecircuitfrom(18)wassimulatedusingparameters
dt
θ =0.2 and φ= 0.01. Small λ lead to slow convergence, while
This equation was simulated for different values of λ as λ > φ3/(4θ) = 1.25e −6 cause overshooting and oscillatory
depicted in Fig. 2. The results indicate a tradeoff that is behavior.
associated with the choice of λ: too small λ lead to slow
convergence of the scheme, while too large λ cause the
40 0.4
adaptation scheme to “overshoot” the target value and
exhibit oscillations. 30 0.3
In order to analyze the convergence properties of (18), X(t)20 0.2
we performed a local stability analysis of (22) based 10 0.1
on linearization. Noting that Y(t) and S(t) evolve au-
0 0
tonomously,wecanreplacethosevariablesbytheirsteady 0 200 400 600 0 200 400 600
Time Time
state values Y∞ =θ/φ and S∞ =1/φ, respectively. This
True Estimated
yields the reduced system
Figure 3: Algorithm performance in the presence of molecular
d
X(t)=θˆ(t)−φX(t) (23) noise. The adaptive circuit from (18) was simulated using the
dt stochastic simulation algorithm to account for molecular noise.
dθˆ(t)=−λθˆ(t)X(t)+ λθθˆ(t) (24) We assume that the target value θ changes spontaneously at
dt φ φ2 certain time points tocheck thecircuit’s ability to readapt. The
parameters used for the simulations were θ =0.2, φ=0.01 and
which has equilibrium points X∞1 = θˆ∞1 = 0 and X∞2 = λ=1e−6.
θ/φ and θˆ2 =θ. The respective Jaccobians are given by
∞
−φ 1 −φ 1
A1 =(cid:18) 0 λφθ2(cid:19) A2 =(cid:18)−θφλ 0(cid:19). (25) vioarluoefSθˆ(∞t),=m1e/aφn)indgoetshantotθcwhiallngreemthaienaassymthpetootnilcybsethaabvle-
The eigenvalue λθ/φ2 of A1 is always positive meaning equilibriumpoint. However,theadaptationlawsimplifies
that for initial conditions θˆ(0) > 0, the system will not to
converge to that equilibrium point. From A2 we find d λ λ
θˆ(t)=− θˆ(t)X(t)+ θˆ(t)Y(t)
eigenvalues
dt φ φ
(27)
λ
φ4−4θλφ+φ2 − φ4−4θλφ+φ2 = θˆ(t)(Y(t)−X(t)),
µ1 =−p 2φ µ2 =− p 2φ . φ
(26)
whichisstructurallyequivalenttoaspecificcontrolmotif
For any λ > 0, the real part of both eigenvalues is
that has been studied previously [19, 20]. In particular,
negative, meaning that θ is the only stable equilibrium
it was shown to act as an integral controlcircuit exhibit-
of the adaptation scheme. However, we find that for
ing robust perfect adaptation [21]. This points out the
λ > φ3/(4θ), the system will have complex eigenvalues,
potential use of our adaptive estimation framework for
indicatingoscillatorybehavior. Thiscanalsobeseenfrom
studying robustness in biological networks.
in Fig. 2, especially for the case λ=3e−6.
Wenextanalyzedthecircuit’sperformanceinthepres-
4.2.Self-adjusting death-rate. We further show how
ence of molecular fluctuations and spontaneous changes
the LMS adaption can be used to identify the target cir-
inθ. We usedthe modelfromeq. (2)andstochasticsim-
cuit’s death rate θ = {φ}. The corresponding sensitivity
ulations [18] to simulate the reaction network from (18).
S(t) can be shown to satisfy
The results from Fig. 3 show that the birth-rate θ and in
turn the system output Y(t) is accurately trackedby the
d
molecularLMSscheme. Adetailedandquantitativeerror S(t)=−X(t)−θˆS(t). (28)
dt
analysis in the presence of molecular fluctuations will be
subject of future work. There are two issues associated with the above equation.
Remark: We want to point out another interesting First, it depends on the tuning parameter θˆ, which will
propertyofthis particularLMSestimator. Replacingthe change over time due to the adaptation scheme. Corre-
sensitivityS(t)byapositiveconstant(e.g.,itsstationary spondingly, the value of S(t) will be different from the
4
actual sensitivity ∂ X(t). While this will have an im- found that for any λ > 0, only the non-zero equilibrium
∂θˆ
pact on the convergence rate of the LMS scheme, it does point is stable. For any λ > θ4/(4ρ2), the adaptation
not affect its steady state behavior (see analyticalresults exhibit oscillatory behavior, which should be taken into
below). The second problem is that S(t) is incompati- consideration when designing this circuit. These results
ble with mass-action rate laws due to the negative de- are confirmed in Fig. 4, for which we simulated (35) for
pendency on X(t). In order to address this problem, we three different values of λ.
−
consider the equation for S (t) = −S(t), which is given
by 0.02
d
S−(t)=X(t)−θˆS−(t) (29)
dt
and correspondingly use the LMS update rule
0.01
d
θˆ(t)=−λθˆ(t)X(t)S(t)+λθˆ(t)Y(t)S(t)
dt (30)
=λθˆ(t)X(t)S−(t)−λθˆ(t)Y(t)S−(t). 0
0 50 100
Time
Overall, the adaptive circuit is given by the reactions
true value
ρ
∅−⇀Y
Figure 4: Convergenceofestimateddeath-rateasafunctionof
Y−⇀θ ∅ λ. Theadaptivecircuit from (31)wassimulated forθ=0.1and
φ=0.01anddifferentvaluesofλ. Forλ>θ4/(4ρ2)=2.5e−7,
ρ
∅−⇀X theadaptation scheme exhibits oscillatory behavior.
θˆ+X−⇀1 θˆ
(31)
X−⇀1 X+S
θˆ+S−⇀1 ∅
5 CONCLUSIONS
S+θˆ+Y−⇀λ S+Y
S+θˆ+X−⇀λ S+2θˆ+X. LMSschemesprovideapowerfulandversatileframework
for adaptive estimation. In this work we have shown
Similar to the previous section, we performed simula- how simple LMS-type estimators that usually run on a
tions to check the adaptation performance of the circuit computer can be implemented biochemically for the pur-
as a function of λ under idealized noise-free conditions. pose of synthetic biology. Such algorithms would allow
This allowsus to describe the adaptivecircuitby the dif- a circuit to make inference about its environment and
ferential equations facilitate adaptive behavior. We have shown by simula-
tion that the LMS circuit is able to accurately estimate
d
Y(t)=ρ−θY(t) (32) and track unknown parameters of a birth-death process.
dt We arecurrently extending the proposedschemeto more
d
X(t)=ρ−θˆ(t)X(t) (33) general scenarios when multiple parameters are to be es-
dt timated simultaneously and when both X and Y are ar-
d
θˆ(t)=−λθˆ(t)X(t)S(t)+λθˆ(t)Y(t)S(t) (34) bitrary, possibly multivariate stochastic circuits.
dt
We anticipate several important potential applications
d
S(t)=−X(t)−θˆ(t)S(t). (35) of the presented framework. In [11], optimal filters (such
dt as the Kalman filter [22]) were employed to design noise-
cancelling synthetic circuits. To this end, the LMS ap-
We again performed a local stability analysis of the
proachprovidesanattractivealternativetooptimalfilter-
differentialequationstoinvestigatetheconvergenceofthe
ingdue toits genericandsimple structure. Inthefuture,
circuit. Inthiscase,thesensitivityS(t)iscoupledtoX(t)
and θˆ(t) and thus, has to be included in the dynamic wewillextendtheapproachtononlinearandmultivariate
modelsandprovideanin-depthanalysisofitsproperties.
analysis. Aftereliminatingtheequationcorrespondingto
Y(t), we obtain a three-dimensional system
d ACKNOWLEDGMENTS. Thisprojectwasfinanced
X(t)=ρ−θˆ(t)X(t) (36)
dt with a grantfrom the Swiss SystemsX.ch initiative, eval-
d λρ uated by the Swiss National Science Foundation.
θˆ(t)=−λθˆ(t)X(t)S(t)+ θˆ(t)S(t) (37)
dt θ
d
S(t)=−X(t)−θˆ(t)S(t), (38) COPYRIGHT INFORMATION. This article was
dt
presented and published at the 2016 IEEE 55th Confer-
whichhas equilibria at the originandat the point X∞3 = ence on Decision and Control (CDC) in Las Vegas. In
ρ/θ,θˆ3 =θ and S3 =−1/θ2. For compactness, we skip the original version, Figure 1 was unreferenced and θ in
∞ ∞
explicit expressions of the respective Jaccobian matrices eq. (28) shouldhave been θˆ. Inthe presentversion,both
andeigenvalues. However,asinthepreviousexample,we issues have been corrected.
5
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