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OPEN Adaptation with transcriptional regulation

received: 18 October 2016 accepted: 10 January 2017 Published: 24 February 2017

Wenjia Shi1, Wenzhe Ma2, Liyang Xiong1,3, Mingyue Zhang1,3 & Chao Tang1,3,4
Biochemical adaptation is one of the basic functions that are widely implemented in biological systems for a variety of purposes such as signal sensing, stress response and homeostasis. The adaptation time scales span from milliseconds to days, involving different regulatory machineries in different processes. The adaptive networks with enzymatic regulation (ERNs) have been investigated in detail. But it remains unclear if and how other forms of regulation will impact the network topology and other features of the function. Here, we systematically studied three-node transcriptional regulatory networks (TRNs), with three different types of gene regulation logics. We found that the topologies of adaptive gene regulatory networks can still be grouped into two general classes: negative feedback loop (NFBL) and incoherent feed-forward loop (IFFL), but with some distinct topological features comparing to the enzymatic networks. Specifically, an auto-activation loop on the buffer node is necessary for the NFBL class. For IFFL class, the control node can be either a proportional node or an inversely-proportional node. Furthermore, the tunability of adaptive behavior differs between TRNs and ERNs. Our findings highlight the role of regulation forms in network topology, implementation and dynamics.

Current biology has moved into a quantitative era. Mathematical models are increasingly used in researches to help elucidating underlying mechanisms of biological processes. Among different models, the biological network is a natural way to represent complicated biological regulations, and straightforward to translate into a mathematical model. One interesting feature of network model is that the network topology and network function are related. For example, bistable1,2 and excitable systems3 often have positive feedback loops whereas oscillating systems often come with negative feedback loops4,5. The relationship has significant meaning to biology research because it shines light on understanding the complex regulation diagrams and predicts new regulations. Our previous study has shown that there is a relationship between biological function and network topology but it is not one-to-one mapping, instead, a small set of different network topologies can lead to the same function. For example, the adaptation networks has either negative feedback loops or incoherent feedforward loops6. Other groups have also shown similar properties of function-topology relationship in other biological systems7–13. One remaining question of this kind of study is whether specific regulation forms or rules could change the function-topology relationship. Different reactions such as phosphorylation, degradation and gene regulation have different time scales and regulation characteristics. Mathematically, different regulations are represented with different function forms. For example, enzymatic reactions follow Michaelis-Menten forms while gene regulations follow Hill functions.
Adaptation exists in a broad range of biological systems. Typical examples of adaptation include the adaptation of signal transduction pathway14,15, adaptation of neuron activity16,17, stress response18–20, bacteria chemotaxis21–23 and homeostasis24. In this work, we still use adaptation as the model system to study function-topology relationship of biological networks. A typical adaptation process contains two parts, a pulse phase indicating the sensing of stimulus, where we define a quantity- response to represent it, and a recovery phase indicating adaptation to the environment, where adaptation error is defined (Fig. 1A). Perfect adaptation is achieved if the output of the system recovers to the exact original value before stimulation. In our previous work, by investigating the whole three-node ERNs, we found two classes of network topologies that are capable of perfect adaptation: a negative feedback loop with a buffer node (NFBLB) and an incoherent feedforward loop with a proportioner node (IFFLP)6. Both of these two classes have an intermediate node which serves as a controller to determine the mechanism of adaptation. In the NFBLB class, the intermediate node is an integral feedback controller which

1Center for Quantitative Biology, Peking University, Beijing 100871, China. 2Department of Systems Biology, Harvard Medical School, Boston, Massachusetts, United States of America. 3School of Physics, Peking University, Beijing 100871, China. 4Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China. Correspondence and requests for materials should be addressed to C.T. (email: tangc@pku.edu.cn)

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Figure 1. (A) Functional characterization of adaptation used in this study. (B) Models of transcriptional regulation with different logics. (C) Two families of perfectly adaptive networks among ERNs. The node A (white circle) and C (black circle) act as the input and output node, respectively. The node B (grey circle) is a control node which plays the role of an integrator in the NFBLB family and a proportioner in the IFFLP family. Right side is the transcriptional regulation model of IFFLP with AND logic.

buffers the change of output by integrating the error of the output node. In the IFFLP class, the intermediate node is a proportioner which balances the influence of the input on the output by responding to input signal proportionally (Fig. 1C).
Gene transcriptional expression changes in cellular adaptation to short- or long-term environmental changes, with extensive regulation occurring at the transcriptional level25,26. So here, we used the framework of enzymatic adaptation6 to study the general design principles of gene regulatory networks for adaptation and explored

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the differences affected by regulatory rules. We studied three different types of transcriptional regulation logics. For all the rules, we obtained the same two general classes of adaptive core topologies: negative feedback loops (NFBLs) and incoherent feedforward loops (IFFLs) as in the ERNs. However, there are important differences between the adaptive TRNs and ERNs, in terms of how the control node gets involved and tuning of the response.

Methods Model Construction.  Gene expressions are regulated by other genes’ products that work as transcription
factors (TFs). Commonly more than one TF binds to the respective gene regulatory sequences27, with the involvement of RNA polymerase complex, to determine the transcriptional activity. Simple logic model can be used to model multiple TF regulations28–31. In the following text, in characterizing a transcriptional network topology, every node stands for the transcriptional gene product and each link represents a transcriptional regulation by the gene product (as a TF) from which the link originates. The transcription activity is a function of the concentration roareecfpgtTiruvFelassats.tiioIoonrnnyeitqsteeurrramemtgisousonlgan,(te1gwd)e,intwfhxehoxtirh.levddextxaHidlsliettlnhlhoefeutmegnsecanttxheiieompnrraaogltedpAuori ocf=tdcsouAuncinntcAi+edoinnKentrnrrgaaaottnieoudonngfrcegIhjgea=nunelgaxIetnje.o+KdτfnxKdtinhesegdtrhgeaenednohaetatipilnofr-gnoli.dtfheGueocxtfii.tstWhhaiaesfcuagtnseisvncuatemitopioenrnotohdafaunatcdlgtl.eTjnthFe

fx

=

dx dt

=

vxGx(g A1,

g A2,

...,

g I1,

g

I

...)
2

−

x τx

(1)

Here we considered three logics for multiple TFs (Fig. 1B): (1) AND logic, where gene expressions are only turned on when all the activators are at high concentrations and the repressors are at low concentrations; (2) AND&OR logic, in which turning on any of the activators is enough to turn on the downstream genes, provided that the repressors are low; (3) Competitive Inhibition logic, in which the relative weight of repressor and activator together determines the downstream gene activation. Here, activators and repressors compete for the same binding sites, and the repressors decrease the effect of activators instead of blocking gene expression. The IFFLP modeled with AND logic is shown in Fig. 1C as an example.

Results and Discussion Computationally Searching for Circuits Capable of Adaptation with Transcriptional Regulation. 
Searching for adaptive topologies can be achieved by two complementary approaches. One approach is computational enumeration, which is feasible for relatively small-size networks. The other one is theoretical analysis around the system’s steady state, which can give rigorous conditions for perfect adaptation.
We firstly computationally searched all possible networks with three nodes: the input (A), the output (C), and the control (B) nodes. Each node can regulate three nodes (two other nodes and self), so that each network contains up to nine links (positive, negative or none regulation). We have 16038 networks in total (there are 19683 topologies in the whole three-node network space, but the topologies that have no direct or indirect links from the input to the output are excluded). Each node has a maximal production rate v and a decay rate τ as parameters, and each regulatory link has a Hill coefficient n and an activation/repression threshold K as parameters. For each network, 10,000 sets of parameters and three transcriptional regulatory logics are used in the ordinary differential equations (ODEs) simulation. During our simulations, an architecture is referred as a functional solution of perfect adaptation when it has: adaptation error <0.005, response >0.2 (Input changes from 0.06 to 0.6). The main results do not change with different criteria for function (Fig. S1).

Adaptive TRNs Have Different Features and Parameter Constraints from ERNs.  Our first question is whether all the functional solutions also converge on the NFBLB and IFFLP families. We separated the functional solutions of all three logics into two families: the NFBL family, which includes all solutions that contain a NFBL but no IFFL; the remaining family, the rest topologies in functional solutions except for the NFBL family. Interestingly, the topologies in the remaining family all contain the skeleton of an IFFL. Thus the core families to achieve adaptation both in TRNs and ERNs are all NFBLs and IFFLs. However, when we looked into more topological details, we found there are also some differences.
We analyzed the simulation results under AND logic as an example. 425 out of 16,038 topologies are functional, including 206 (48.47%) topologies belonging to the NFBL family and 219 (51.53%) the IFFL family (Fig. 2A and B). We separately clustered the networks from these two families using Hamming distance between architectures (Fig. 2C). Each column represents one specific regulation, and each row represents one architecture. The motifs extracted from each sub-cluster (listed on the right of each panel) indicate that: IFFLs work as a core structure and are very tolerable on additional regulations, and all the topologies in the NFBL family contain an auto-activation loop on regulatory node B (Fig. 2C).
To clearly figure out the differences between adaptive networks among TRNs and ERNs respectively, we then compared our clustering results with the simulation of ERNs (data from Ma et al., Fig. 2C), we found:

(1)	 The minimal solutions contain three links within three nodes. That is to say, neither one-node nor two-node network is capable of performing adaptation in both these two regulatory conditions.
(2)	 NFBLs and IFFLs are two families that can achieve adaptation in both these two regulatory conditions. (3)	 An auto-activation on the buffer node in the NFBL family is necessary for TRNs but optional for ERNs. This
auto-activation results in a special kind of adaptive network: negative feedback loop with an exponential buffer node (NFBLEB) which helps NFBLs buffer the adaptation error in a logarithmic way6 (detailed example can be seen in next section). Meanwhile, all the negative feedback loops in both these two regulation conditions go through the buffer node rather than feedback from the output node to the input node directly.

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(A)
NFBLEB NFBLB NFBL family

Type1 IFFL Type2 IFFL Type3 IFFL Type4 IFFL

IFFL family

(C)
NFBL+ other motifs ( 168 )

ERNs

NFBLEB

(B)

100%

26.94%

Type2 22.83%
7.76% Type3

42.47% Type4

Type1

NFBLEB IFFL 48.47% 51.53%

IFFL+ other motifs ( 224 )
IFFL + NFBL Type3 IFFL

A=>A A=>B A=>C B=>A B=>B B=>C C=>A C=>B C=>C

NFBLB

A=>A A=>B A=>C B=>A B=>B B=>C C=>A C=>B C=>C

Type1 IFFL

NFBL+ other motifs ( 206 )

TRNs

IFFL + other motifs ( 219 )
IFFL + NFBL
Type2 IFFL

NFBLEB

Type4 IFFL

Type3 IFFL

A=>A A=>B A=>C B=>A B=>B B=>C C=>A C=>B C=>C

A=>A A=>B A=>C B=>A B=>B B=>C C=>A C=>B C=>C

Type1 IFFL

Activation Activation

Inhibition Inhibition

No regulation Activation or Inhibition

Figure 2.  Comparison of adaptive networks with gene regulation and enzymatic regulation. (A) Categories of adaptive motifs. (B) Motifs compositions of adaptive solutions in TRNs. Bars in green and pink backgrounds are for NFBL and IFFL families, respectively. (C) Clustering results of adaptive TRNs (AND logic) and ERNs. The network motifs associated with each of the sub-clusters are shown on the right.

(4)	 All four types of IFFLs are adaptive in TRNs (Fig. 2C), while in ERNs there are only two (type 1 and 3) that are adaptive. The simulation results of the other two transcriptional logics agree with the first 3 conclusions above, but differ
in the 4th one: AND&OR logic has 3 types of adaptive IFFLs and Competitive Inhibition logic has only two types of adaptive IFFLs as in the enzymatic regulation (Fig. S2). Mechanisms for Transcriptional Adaptation.  Distinct topological features between TRNs and ERNs lead us to investigate the origin of these differences analytically. We addressed this question by performing a line(Bpa,rrCoav,niIad)l,eyddsBitsh/odaftt t=thh eefBts(ryAasn,tesBmc,rCihp)at,sioaannsadtledsayCdsy/tdesmtt a=tse. f)TC.(hTAeho,eBrde,teCivc)iaalctlyaio,nnthbΔeeeAlqi,nuΔeaatBrioiaznnesddfoΔarrCoaunfnryodtmhtrhteheeie-rnssottedeaaeddnyyesttswatatoet,rekwAdh*Ae, nB/d*tth, =aeni fndAp(CAu*t, changes from I to I + ΔI, satisfy the following linearized equation:

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

d∆A dt
d∆B dt
d∆C dt



=



∂f A ∂A ∂f B ∂A ∂f C ∂A

∂f A ∂B ∂f B ∂B ∂f C ∂B

∂f A ∂C ∂f B ∂C ∂f C ∂C



⋅

 ∆∆∆ABC 

+



∂f A ∂I 0 0



⋅

∆I

(2)

The requirement for perfect adaptation is ΔC*  =  0, which means that after the input change the output value

exactly returns to its original state. All the topologies satisfied the requirement fall into two classes that both have

htinharvFeeiegnn. o3oBrdeeNsstFrwiBcittLhiofinanms(Foilinyg).t ;h3a)en:s(dp1(e)2cN)ifFIicFBFfLoLsrsmwwitoihtfht∂h∂feBfB/s∂/y∂BsBt =e <m  0 ’0s.

At least one NFBL is required in this family (colored loop (colored loop in Fig. 3B IFFL family)6. These conclusions function except that they should have stable steady states

(see Supplementary materials for one and two- node system’s derivations). Thus the B node equation (equa-

tion (3)), is the key

which point.

we denote as B-equation, plays a very important role for achieving The mathematical form of B-equation determines how the system

paecrhfieecvteasdtahpetaretiqouniraesm∂efBn/t∂oBf

perfect adaptation both in topological design and parameter constraints.

fB

=

dB dt

=

vB

G (A,

B,

C)

−

B τB

(3)

Exponential Buffer Node in the Negative Feedback Loop.  In B-equation (equation (3)), there is a linear decay

term. For NFBLs, the condition for production term of B-equation also

pcoernfteacitnasdthapetvaatiroianbrleeqBusiroetsh∂aftBi/t∂cBa n=b 0e,

which can be factored out:

satisfied

robustly

if

the

fB

=

B

⋅

k

⋅

G (A,

C)

−

1 τB



(4)

The way to achieve this is to have node B positively regulating NFBL (Fig. 4A) with AND logic, the ODEs of the system are:

itself

with

nBB = 1

and

B ≪ KBB.

As

an

example

of

dA dt

=

v

A

I

n

IA

I nIA + KInAIA

−

A τA

dB dt

=

vB



B

BnBB nBB + KBnBBB

C

C nCB nCB + KCnBCB



−

B τB

dC dt

=

vC



AnAC AnAC + KAnCAC

KBnCBC BnBC + KBnCBC



−

C τC

(5)

If the buffer equation for

node B can

BbewaoprpkrsoxwimithatBed ≪b yK:BB

and

in

a

non-cooperating

form

(the

Hill

coefficient

being

1,

the

rate

dB dt

=

vB



B K BB

C

C nCB nCB + KCnBCB



−

B τB

=B

⋅

(G (C)

− 1)

(6)

where G(C) is a function of only C. So in steady state G(C*) = 1 and C* = constant, independent of the input. The

node B integrates the relative difference between the output activity C and its input-independent steady-state

( )value in a
role of an

eloxgpaornitehnmtiafloirnmte:gBra(tto) r=ofBth⁎ee∫aτ1dBapCCt(⁎ta) tniCoBn−1erdrto(ra.sAsullmthinegadKaCpBt ≫ive CNfoFrBsLismsphlairceittyh)i.sHcehraer,ancotedreisBticp.lays

the

Inversely Proportional Node in the Incoherent Feedforward Loop.  For transcriptional IFFLs, adaptation is achieved by a balance between the transcriptional production rate change caused by two signal-transmitting pathways acting on node C and the linear decay of C. Thus at steady state the production rate should maintain constant that is independent of the input to balance the unchanged decay term, which means the co-regulators of output, node A and B must establish certain relationship to satisfy the above requirement. In the three-node network, the B-equation undertakes the task to establish this relationship (Fig. 3C). In the case of A activating B, a robust proportional relationship can be established with the regulatory TF working with A ≪ KAB:

fB

=

dB dt

=

vB An

An + KAnB

−

B τB

≈

vB KAnB

An

−

B τB

(7a)

At steady state, k′A*n = B*, where k′ = ship can be established with A ≫ KAB:

.vBτB

K

n AB

In the case that A inhibits B, a

robust inversely

proportional

relation-

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Figure 3.  Theoretical analysis of TRNs. (A) The requirement of zero adaptation error around a stable steady

state results in the equations that are shown for one-, two- and three-node networks. One- and two- node

systems cannot satisfy the requirement, so there are no topologies to achieve adaptation in these two systems.

Two kinds of three-node topologies can satisfy the requirement: NFBL and IFFL. (B) The NFBL family achieves

fpcioegnrufdreeict.tiSoaondaa∂tpfltBea/at∂isoBtn o=nw 0ei,tNhthF∂eBftBLe/r∂imsBrs =eiqn 0utiahrneeddditenhteetrhJmaisciofnabaminaitnly|Jd.|TectwoerrormeNsinFpBaonnLtds|wJt|o o<du i0lfdf(ersreteasnbutilltfietiynedrabemqacuokirrleeomnoepegnsatat)si.vWceo|ilJto|h,rewtdhheiinchthcean

lead to can be

a smaller adaptation error. satisfied in the TRN model

No feedforward loop with the buffer node

can be present in B auto-activating

itthsieslffawmitihlyH. Tilhlecocoefnfidciiteinont 1∂. fB/∂B = 0

(C) For

The this

fIaFmFLilyfa, mthielylianckhsiceovelosrpeedrifnectthaedfaigputarteioanrewnietche∂ssfaBr/∂yBto≠be0

and also |J| < 0, which present and constitute

aimn IpFliFeLs .∂TfBw/o∂Bo p<p 0o.sing

regulations on C need to be cancelled out which requires certain input-independent relationship between

A and B at their steady state. The proportionality relationship with A activating B or inverse proportionality

relationship with A inhibiting B can be established by the equation of node B.

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Figure 4.  The amplitude of response changes with parameters’ perturbations in two adaptation motifs.

Each dot shows the result with a perturbation of the corresponding numbered parameter (circle and triangle

for increase and decrease respectively). The performance of the system without perturbations is marked

with the red star. Parameters are grouped into colored groups. Purple, blue and yellow for Hill coefficients,

transcriptional perturbed with

thresholds and 10% and other

basic dynamic parameters, respectively. (A) parameters with 50% increase and decrease.

The (B)

TNhFeBILFEFBLIsPysstyesmte.mn.BBAilsl

the

parameters are perturbed with 50% increase and decrease.

fB

=

dB dt

=

vB

An

KAnB + KAnB

−

B τB

≈

vBKAnB An

−

B τB

(7b)

pArtospteoardtiyosntaalten,oAd*enB(*I F=F kL′P, w) ohreraenki′n=vervsBeKlyAnpBτroBp. Torhteisoenraellnatoiodnes(hIFipFsLlIePa)d.

to incoherent feed-forward loops with a Here, the node B can be a proportioner

or an inverse proportioner, whereas in enzymatic regulation, B can only be a proportioner in IFFLP6.

Let us analyze one specific IFFLIP in detail (Fig. 4B). Assuming the AND rule, the ODEs of the system are:

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fA

=

dA dt

=

v

A

I

n

IA

I nIA + KInAIA

−

A τA

fB

=

dB dt

=

vB

An

KAnBAB AB + KAnBAB

−

B τB

fC

=

dC dt

=

vC

AnAC AnAC + KAnCAC

BnBC BnBC + KBnCBC

−

C τC

with A ≫ KAB, the equation of node B becomes:

fB

=

vB

KAnBAB AnAB

−

B τB

and thus at steady state we have:

(8) (9)

vBτBKAnBAB = A⁎nAB B⁎ When the TFs A and B satisfy A ≪ KAC and B ≪ KBC, we have:

(10)

fC

=

vC

AnAC KAnCAC

BnBC KBnCBC

−

C τC

(11)

which, combining with equation (10), leads to a steady state solution ofC⁎ if

=

A⁎ nAC

vBAτ⁎BnKAnABABB nBC

K

nAC AC

KBnBCC

v

C

τ

C

.

C∗

is

a

constant,

nABnBC = nAC

(12)

Note that the Hill coefficients do not have to be 1, as long as they satisfy equation (12), the system will adapt. This more relaxed condition on Hill coefficient differs from that of the adaptive IFFLPs with enzymatic regulation where the Hill coefficients have to be 16.

Tuning Response of Adaptive TRNs.  For an adaptive system, the transient response to an input change transduces

information to the downstream molecules or pathways. Thus, the ability of tuning the response of an adaptive

network is important. In the analysis above, we noticed that the adaptive NFBL and IFFL families have different

parameter constraints both within TRNs and compared with ERNs. For the NFBLEB system, only the parameters

itnhethIeFFauLtIoP-asyctsitveamti,ocnotnesrtmraiinntBs -oefqmuaotrieonp,aKraBmB aentedrns BaBrearreeqcruiitriceadl

for ensuring small adaptation errors. to achieve adaptation. To investigate

While for the tuna-

bility of the response for these two families, we performed a single parameter perturbation analysis. Specifically,

we increased or decreased only one parameter each time by 20% or 50%, and monitored the adaptation behavior

(Fig. 4). We (arbitrarily) grouped all the parameters into three groups: Hill coefficients n (purple), transcriptional

thresholds K (blue), and the basic dynamic parameters (yellow), which include half-life τ and maximal produc-

tion rate v (Fig. 4).

As can be seen from Fig. 4A, for the NFBLEB system, a small adaptation error can be maintained with respect

to changes of many parameters, with the exception of small adaptation errors (grey region in Fig. 4A, which

nisBsBh. Wowenzoonomtheedriinghtht oonseapdairfafemreentetrsccahlaen).gTehs ethmatorsetseuflfticeidenint

iptaorraBm.eTthereswfoeratkuenrininghtihbeitrieosnpofrnosme anreodτAe

aBnadftveAr.

The the

idnepcuret acsheaonfgτeAleanavdevsAmdoecreretaismeeCf,obruntoadlseo

decrease the inhibC to increase tran-

sdbieeegcnritnelyan,sitenhigun.sTBch,owrnohturiiglbehuattnehsientdocreaecarlasereagsoeefroKrfeABsCp,soltohnwessese.dTpohewertnsuetrchboeantsdiiogbnnesaslltetgraarvoneusmmp oiissrseτioBt,inmvfBreaofnmodrnKthoAdeCe.oDAuteptcoureCtatsaoinntdrgadτneBscioerrenavtslByelseinaBdcarstettaohsaee

and thus contribute to a larger response. The Hill coefficients can also alter the response by changing the reaction

dynamics. For example, response. However, the

the increase of Hill coefficient

ngAroCusppesehdoswuspwtheaekterarntusineanbtiilnitcyrethasaenotfhCe ,otthhuesr

contributes to an increase two groups. In summary,

of in

this specific system, the concentration of the inhibitor B is sensitive to tune for the response. We rank the efficiency

mofoturenuinpgsttrheeamresinpothnesesiagmnaolninggpfaereadmbaectekrlogroopu, pths:eKmAoC r>e  eKffCiBc >ien KtBtCo;

tτuAn >e  tτhBe >re τsCp(ovnasne d(Inτ

have similar effects). The the loop A ->C->B->C,

A is the first to receive the signal, B is the second one, C is the last one that needs to integrate A and B).

In Fig. 4B, for the IFFLIP system, a small adaptation error cannot be maintained with respect to changes of

many parameters. We also zoomed in those parameter changes with relative smaller adaptation error for the

IFFLIP system (grey region in Fig. 4B). The Hill coefficient group performs the worst because they should satisfy

certain relationship mentioned in the previous section. Not all the transcriptional thresholds tolerate poorly

iaancdtjtiuhvseattepiroentrhtouanrtbcCaatnaionmndsat,ihanulttsahiponruoagmhsmotthaeelClya’asdrtaerpaatnlalstriieeonqntueiinrrercdorertaoasnewd(oatrnukdnienvitccheeervtreaerisnspar)oe.ngMsieoe.naTns.hwAehmdileeoc,nroegwatsihneegomtfo,KKtBhBCeCsiipns ehaeipdboisttiueopnntBiba’ysl

tnehfofeidcteioeAnlet,rfaBonracltewuonafyinspgamrthaameinreteatseipnrospnaestretl.ouLwrabrcagoteinorcnτeBnfootrrraKvtBiBoCinn. ,cTrsheoeaisbteaisssBiec,atdshyyunfsoaBrmBwicatoiptaswrfoaomrrkleowtneigrtseh,rBetsi ≪mpee KctiBaoClbl,yweτihnBi,hcvhibB,citτoeCndtabrniybdAuvtCteislaltrioet

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Negative Feedback Loops

A B
C

A B
C

A B
C

A B
C

A B
C

A B
C

A
B C

A B
C

AA

A

A

B C

B C

B C

B C

Incoherent Feedforward Loops

A
B C

A
B C

A
B C

A
B C

A
B C

A
B C

Logic

AND

AND&OR

Competitive Inhibition

Parameter constraint

Unconstrained

X<<K

X>>K

At least one of the two purple lines should satisfy X>>K X is the concentration of regulating TF

Figure 5.  Minimal topologies of perfect adaptation with three regulatory logics. Yellow, pink and green tags represent AND, AND & OR and Competitive Inhibition logics, respectively. Colored lines represent constraints on the parameters as indicated in the legend.

reaches a low concentration and cannot activate node C. It leaves more time for C to transiently increase and leads tfAeownaeeleradrpgsaetrroarmseasetpitsoefrnyssbeino. tvthAhAaen I≫dFF τKLAAIbBPe(hasysasvatenepminoshopirbleyirtfwoorirtomhf pendeordwtueerBlbl)aaattinotdunnAsian ≪tgm tKhaAienCrt(eaasisnpaionnngasaectstimhvaaatnollrtahodefanNpotFadtBeioLCnE)Be. rIsrnyossru,tebmmemc.aTaurhsyee, Hill coefficient group is the worst option to tune. Tuning the binding affinity of TF B of gene C, or changing the half-lives or maximal transcriptional rate of protein B and C can improve the IFFLIP system’s response. The Roles of Transcription Logic.  When a node is transcriptionally regulated by more than one link, different transcription logics have different mathematical forms, which may have different consequences on the topological requirement for adaptation. Following the theoretical analysis, we derived the minimal design table with three transcriptional logics (Fig. 5). Every minimal topology is labeled with yellow, pink and green

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tags representing its feasibility in AND, AND&OR, and Competitive Inhibition logics, respectively. There are 12

NFBLs and 4 IFFLs in total, but not all of them are feasible for all logics.

For the NFBL family, there is one common feature that all NFBLs have an auto-activation of the buffer node B

wFoorrkthinegAwNitDh&BO ≪R KaBnBdaCndomnBpBe =tit 1iv.1e2InNhFiBbiLtsioanreloaglliccsa,pnaoblaecotifvaactihoinevoinngnpoedrefeBctoatdhaepr ttahtaionnthweitahuttho-eaActNivDatiloongicis.

allowed. This is because with these logics (Fig. 1B), it is hard to factor out variable B with two or more activation

terms in the B-equation. For the Competitive Inhibition logic, the requirement that each node should have at least

one activator (Fig. 1B) further reduces the number of feasible topologies.

For the IFFL family, there are total of 4 topologies (Fig. 5). All 4 IFFLs are adaptable for the AND logic, two of

which with proportional mechanism and the other two with inversely-proportional mechanism. For the AND&OR

logic, node C can have at most one activating regulation, otherwise the summation of regulations from nodes A and

B are hard to cancel out. This leaves 3 IFFLs for this logic. For the Competitive Inhibition logic, the simplest way to

achieve a constant output is to establish a linear relationship between An and Bn from B-equation (the Hill coeffi-

cient n can be 1), and then to have the two nodes A and B regulating C oppositely (one activating and one inhibit-

isntega)d. Iynsttahtee.reTghiiosnscAe ≫na rKioACoannlyd/woorrBk ≫s w KitBhC,tthheeprreogpuolarttiioonnsalfrmomechnaondiessmA,

and B cancel out, making C a so only two IFFLs are feasible

constant at for perfect

adaptation with the Competitive Inhibition logic (Fig. 5 and see Supplementary materials for detailed derivations).

Discussion

Generating a comprehensive function-topology map can supply a complete design table as well as illustrate the

underlying mechanism to achieve the function32,33. Nature provides a versatile toolbox for biochemical reactions

and regulations. In this study, we focused on a well-studied function, perfect adaptation to investigate the conse-

quence of different regulation types and rules on topology, parameters constraints, and other functional features.

We found that similar to the enzymatic networks, the topologies of the transcriptional adaptive networks belong

to two general classes: negative feedback loop (NFBL) and incoherent feed-forward loop (IFFL). However, there

are several distinct features for the adaptive TRNs. First, an auto-activation loop of the buffering control node

with Hill coefficient 1 is necessary for the NFBL class. This is more restrictive compared with the adaptive ERNs of

the NFBL class in which this loop is optional. The reason behind the auto-activation loop is that in TRNs there is

always a (linear) decay term in each rate equation, including the one for the control node B. In order for the NFBL

motif linear

to satisfy factor in

the adaptation B. Whereas, in

condition ERNs the

s∂wfBi/t∂chBf =ro 0m, tahcetiavcattievdattioonintaercmtiviantetdhestraatteeceaqnubateiodnonsheobuyldotahlseor

contain a enzymes,

so it is not necessary to have an auto-activation loop on B. On the other hand, there are more distinct topologies

available for the adaptive TRNs of the IFFL class, in comparison to that of ERNs. For TRNs, the control node can

be either a proportional node or an inversely-proportional node, although the tunability of the response in the

IFFLIP class is quite limited. Adaptive TRNs also have fewer restrictions on Hill coefficients.

Biological systems often respond to signal changes in multiple time scales, e.g. with fast reactions at the begin-

ning and changes in gene expression at later stages34. In the budding yeast Saccharomyces cerevisiae, when facing

with osmotic stress, cells adapt through the accumulation of glycerol35. They first close glycerol channels and

rearrange metabolic activities in cytosol within minutes to promote glycerol accumulation, and then express

more than 300 genes, including the cytosolic glycerol synthesis genes, GPD1 and GPD2, at a time scale about

30 minutes and longer36. This adaptation process involves both enzymatic and transcriptional regulations34,37,38.

Previous studies focused on the enzymatic regulations39,40, while it remains unclear what the role of transcription

in this adaptive system is. Our study may provide some clues for transcriptionally involved adaptation systems.

From a synthetic biology point of view, type1 IFFL circuits with a proportional control node have been con-

structed and shown to perform adaptation41,42. It would be interesting to see if IFFLIP circuits can also be con-

structed and achieve the desired function.

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Acknowledgements
We thank Yuansheng Cao, Yao Zhao, Shanshan Qin, Weiqian Zeng, Yuhai Tu, Donggen Luo, Ping Wei, Xiaojing Yang, Chunbo Lou and Kim Sneppen for helpful discussions, Iain Bruce for detailed proofreading of the manuscript. The work was supported by Chinese Ministry of Science and Technology (2015CB910300) and National Natural Science Foundation of China (91430217).
Author Contributions
Conceived and designed the projects: C.T., W.M., W.S. Performed analytic analysis: W.S., W.M., L.X., M.Z. Developed the computational models: W.S., W.M. Conducted the simulation: W.S. Wrote the manuscript: W.S., W.M., L.X., C.T.
Additional Information
Supplementary information accompanies this paper at http://www.nature.com/srep Competing financial interests: The authors declare no competing financial interests. How to cite this article: Shi, W. et al. Adaptation with transcriptional regulation. Sci. Rep. 7, 42648; doi: 10.1038/srep42648 (2017). Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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