Flexible controllability of interlocked feedback loops in biological system

In this paper, we study the behavior of interlocked feedback loops. We find that interlocked positive feedback loops which are classified into two classes based on their dynamic and biological functions could significantly enhance reliability and provide flexible controllability of toggle switches.

This paper is available online at http://stdb.hnue.edu.vn

FLEXIBLE CONTROLLABILITY OF INTERLOCKED FEEDBACK LOOPS IN BIOLOGICAL SYSTEM

Nguyen Cuong1∗, Hoang Do Thanh Tung1, Trinh Thi Xuan2

1Institute of Information Technology, VAST;2Faculty of Information Technology,

Hanoi Open University

∗Email: ncuong@ioit.ac.vn

Abstract The positive and negative feedback loops are ubiquitous basic elements

in regulatory biological complex networks Positive feedback loops are responsible

for bistability, creating discontinuous output response from continuous input and are regarded as reliable toggle switches for making all-or-none decisions Negative feedback loops are responsible for homeostasis and oscillations The feedback

loops hook up together and cooperate in biological complex networks rather than work alone In this paper, we study the behavior of interlocked feedback

loops We find that interlocked positive feedback loops which are classified into two classes based on their dynamic and biological functions could significantly

enhance reliability and provide flexible controllability of toggle switches The interlocking of a positive feedback loop with a negative feedback loop brings up relaxation oscillation with amplitude and frequency that could be controlled freely and independently Biological implications are also discussed

Keywords: Interlocked feedback loop, flexible controllability, tristability.

1 Introduction

The positive and negative feedback loops are ubiquitous basic elements in the regulatory biological complex networks such as transcriptional regulation networks [1], signaling transduction pathways [2], cell cycle regulatory networks [3-5] and circadian rhythm regulation networks [6]

The positive feedback loops are responsible for bistability This means that the positive feedback loop creates discontinuous output response from continuous input They are regarded as toggle switches for making all-or-none decisions [7] and self-perpetuating states [2] In certain cases, the positive feedback loop could make a one way switch or irreversible switch, and once the output turns on, it never turns off [7-9] However, the positive feedback loop alone does not guarantee bistability and high nonlinearity within positive feedback loops is required [9]

In negative feedback loops (NFL), output P somehow suppresses its own production through a feedback loop Typically, in negative feedback loops, output P activates its inhibitor X through a signal pathway including several components which introduce a time delay in a feedback loop The delayed negative feedback loop (dNFL) could be able to generate oscillation [7, 10] dNFLs are found in many biological systems such as signaling pathway [11], cell cycle regulation networks [3-5] and circadian clocks [6, 12]

However, in many biological systems, positive and negative feedbacks hook up and cooperate to work out the desired functions of the systems For instance, the Mitotic trigger and regulation of the cell cycle of various organisms is regulated by interlocked PFL and dNFL related to the Mitotic Promoting Factor (MPF) and its friends and enemies (APC) [3, 5, 13] The interlocked positive and delay negative feedback loops related to the CLOCK genes are also found in circadian regulation networks in a broad range of species [6]

Some studies have been made to address the role of interlocked feedback loops from perspectives such as robustness [14] [15] and noisy signal coping [16] However, the link between biological structure and biological function has not yet been discovered The question is, are there any new features of interlocked systems that did not exist as features

of single systems? And, what are the biological advantages and disadvantages of the new features, if any?

In this paper, we show that the interlocking of feedback loops makes the dynamic behaviors of single feedback loop more robust and brings new dynamic behaviors, such as tri-stability, controllability of amplitude and frequency of oscillation Those new features are utilized in real biological processes

2 Method

2.1 The wired diagrams

In a positive feedback loop (PFL), the output somehow promotes its own production There are two type of positive feedback loops, mutual activation (Figure 1A) and mutual inhibition (Figure 1B) In mutual activation, product P promotes its helper

TF which in turn promotes the production rate of output P , a so-called self-enhancing positive feedback loop (ePFL) In mutual inhibition, output P removes its inhibitors E to release itself from suppression, a so-called self-recovering positive feedback loop (rPFL)

In a negative feedback loop (NFL), the output somehow suppresses itself by activating an inhibitor X which in turn accelerates the degradation rate of output P (Figure 1C)

Figure 1 Wired diagrams of (A) a Self-enhancing positive feedback loop, (B) a Self-recovering positive feedback loop and (C) a negative feedback loop Solid and dashed arrows represent the

reactions and regulatory effect of components, respectively

2.2 The mathematical model

Output P is, in general, synthesized proportional to input S and the transcription factor TF and is degraded in proportion to inhibitor E and X and the output P itself The general ODE for P would be

dP

dt = ksS + keP − (kd+ krE + knX) P (2.1) Where ks and kdare synthesis and degradation rate constants The constants ke, kr

and kn represent the effect of TF, E, and X on output P and is referred to as positive and negative feedback strength By setting kr = kn = 0, kr = kn = 0, ke = kn = 0 or

ke = kr = 0, we get the equation for output P with ePFL, rPFL and NFL, respectively The activation and inhibition of helper TF and inhibitors E and X are governed

by Michaelis-Menten kinetics in order to utilize the nonlinearity[17] However, any other reaction types possessing high nonlinearity such as the hill-coefficient equation would do The ODE for transcription factor TF, enzyme E and inhibitor X would be

dT F

dt =

katfP (1 − T F )

Jatf + 1 − T F −

kitfT F

dE

dt =

kae(1 − E)

Jae+ 1 − E −

kieP · E

dX

dt =

 kaxP (1 − X)

Jax+ 1 − X −

kixX

Jix + X

 1

The constants of ka and ki are maximal activation and inhibition rates, and those

of Ja and Ji are Michaelis constants (relative to the total concentrations of TF) The transcription factor TF is more than half of maximum when P > θe = kitf/katf, and vice versa The ratio θe = kitf/katf is called the activation level of PFL The inhibitor

E is more than half of maximum when P > θr = kie/kae, and vice versa The ratio

θn= kie/kaeis called the activation level of PFL

The constant τ provides a time scale for the activation and inhibition of enzyme E The larger τis, the slower the activation and inhibition of enzyme E is

The parameter values are listed below: ks= 0.4, kp = 1, kd= 2, kn= 2, katf = 1,

kitf = 0.5, kae= 1, kie = 1, kax = 0.25, kix = 0.1, Jatf = Jitf = 0.01, Jae = Jie = 0.05,

Jax = Jix = 0.005, τ = 1 Changed parameters are specified in the text

3 Results

In positive feedback loops, output P somehow promotes its own production The output can elevate itself by promoting a helper which in turn elevates the output level or releases itself from suppression by suppressing its inhibitor

3.1 Self-enhancing positive feedback loop: signal amplifier

The typical example of a self-enhancing positive feedback loop is autocatalytic regulation in which output P promotes its own production by activating activator TF, Figure 2A

As shown in Figure 2B, the input-output response curves with different feedback strength are shown As long as input S increases, product P will increase monotonically When output P reaches the activation level of ePFL, P > re, it is able to turn on its helper, the TF However, if there is no feedback strength, ke = 0, there is no help from TF

to affect the rate of output P , thus PFL would continue to monotonically increase with input S, shown as a black, dashed-dotted line By increasing feedback strength ke, the transcription factor TF in turn enhances the synthesis rate of product P

If feedback strength is weak, the enhancement of synthesis rate is small and thus output P monotonically increases and is enhanced due to simple regulation (red line)

At stronger feedback strength, enhancement of the synthesis rate is larger and thus output P discontinuously increases, creating bistability or hysteresis (blue line) Bistability means that output P could be in an either low of high stable steady state (solid lines) at one input S Those stable steady states are separated by an unstable steady state, represented by a dashed line Output P abruptly changes from one stable state to other one

at two critical values, eSN1 and eSN2, called saddle node bifurcation The region bound

by SNs, eSN2 < S < eSN1 is called a bistable region

If feedback strength gets too strong, enhancement of the synthesis rate is strong thus making the bistability become irreversible bistability (magenta line) That means that once output P is enhanced, it never goes down even when input S is washed out, S = 0

The summarization is shown in Figure 2C, in which the phase diagram of input S and feedback strength ke is plotted The two saddle node bifurcations, eSN1 and eSN2, occur at smaller input S with stronger feedback strength When feedback strength is smaller than a critical value (cusp point), ke < kec, there will be no bistable region with any input S And the bistable region becomes irreversible when the feedback strength is larger than the critical value kei, ke > kei, when eSN2 goes to a negative part of input S This ePFL is referred to as an amplification module Input signal S is passed through and amplified The amount of amplification is proportional to the strength

of transcription factor TF, keTF The stronger the feedback strength is, the larger the amplification is

Figure 2 Self-enhancing and Self-recovering positive feedback loop (ePFL)

(A, C) Input-response curve of ePFL and rPFL with none (dashed-dotted), weak (red), strong (blue) and very strong (magenta) feedback strength, respectively: The solid and dashed black lines represent stable and unstable steady state of output P where eSN1 and eSN2 are saddle node bifurcation of ePFL (B, D) The Input-feedback strength phase diagrams show regions of monostability and bistability

3.2 Self-recovering Positive feedback loop: signal buffer

Typical examples of a self-recovering positive feedback loop are mutual inhibition and double negative feedback loops, in which output P removes its suppression by inhibiting suppressor enzyme E, Figure 2D

In this case, output P is synthesized proportional to input S The degradation rate is proportional to output P and the effect of suppressor E, (kd + krE)P , where

kd is degradation rate, kr represents the effect of enzyme E on the degradation rate, the so-called feedback strength Note that transcription factor E is suppressed when

P > rr = kar/kir, and vice versa See Figure 2E The ratio rr = kar/kir is called the deactivation level of rPFL

The input-output response curves with different feedback strength kr are shown

in Figure 2E As long as input S increases, product P is initially suppressed due to the suppression of enzyme E, which is proportional to feedback strength kr If there is no feedback strength, kr = 0, there is no suppression and thus output P monotonically increases as that of simple regulation, shown as a dashed-dotted line With the presence of feedback strength, the output increases but is suppressed When output is strong enough

to remove its suppression, P > rr, the enzyme is removed and the output is released If feedback strength is weak, the suppression on P is small and the recovery is continuous, shown as a red line Output P would be discontinuously and abruptly recurred (bistability) due to the simple regulation once the feedback is stronger, shown as a blue line

Obviously, feedback strength represents the suppression of P from its inhibitor The output is more suppressed with stronger feedback strength and therefore more input S is needed to produce output P The ratio rr = kar/kir determines when the inhibitor is removed

A summarization is shown in Figure 2F, in which the phase diagram of input S and feedback strength kris plotted As long as feedback strength increases, it is hard to remove the suppression on inhibiting enzyme E and thus more input S is needed to remove E The two saddle node bifurcations move to a larger input S Note that when feedback strength

is smaller than the critical value (cusp point), kr < krc, there will be no bistable region with any input S

This ePFL works as a buffer When the input signal increases, the output response is increased but buffered in an inactive state until it overrides its suppressor Thus, the output response is delayed from input S The delay strongly depends on suppressor capacity (feedback strength) kr The stronger the suppressor is, the more output is delayed by P

3.3 The interlocking of two positive feedback loops: monostability, bistability, and tristability

The interlocking of two positive feedback loops which show monostability and bistability would generate monostability, bistability and multistability

Obviously, each positive feedback loop has two important control parameters -feedback strength and the activation level Feedback strength determines how much the output is enhanced (kein ePFL) or suppressed (kr in rPFL) The activation levels, re and

rr, determine the working region of ePFL and rPFL with level of output P , respectively Therefore, there are two possibilities that the feedback loop in ePFL and rPFL will be activated far from each other, (i) re< rror (ii) re > rr, or (iii) the feedback loop in ePFL and rPFL can be cooperatively activated, re ≈ rr,

(i) First, consider the case that the ePFL activate at a smaller range of output P than that in rPFL, re < rr (See Figure 3A1-Figure 3A4) Output P in iPFL (shown in black)

is initially suppressed from simple regulation (shown as a dashed-dotted line) under the effect of inhibiting enzyme E, thus output P in iPFL initially follows that of rPFL (shown

in blue) When output P in iPFL reaches the activation level of ePFL, P > re, the TF turns

on to enhance output P The output P in iPFL continues to increase due to the activation level of rPFL, P > rr, at which time the inhibiting enzyme E is removed to release output P Therefore, there are two transitions of output P corresponding to the different working region of ePFL and rPFL The transition might be continuous (monostability)

or discontinuous (bistability) depending on the feedback strength Both transitions are continuous when feedback strengths ke and kr are weak (See Figure 3A1) One of them will be weak and the other will be strong and cause a continuous and a discontinuous transition (See Figure 3A2,3) If feedback strengths keand krare strong, there will be two discontinuous transitions (See Figure 3A4) The two discontinuous transitions might or might not overlap Once they overlap, mutltistability emerges Mutltistability implies that

at one input S there are more than two stable steady states separated by two unstable steady states Here, three stable steady states, a low, middle and high level of P , are separated by two unstable steady states

(ii) Consider the case that the ePFL activate at a larger range of output P than in rPFL, re > rm (See Figure 3C1-Figure 3C4) This means that as long as output P in iPFL (shown in black) increases and reaches an activation level of rPFL, P > rm, the inhibiting enzyme E is removed and output P reaccurs as simple regulation (shown as a black, dashed-dotted line) Output P in iPFL keep increasing until it reaches the activation level of ePFL, P > re„ and the TF turns on to enhance output P from simple regulation Therefore, there are two transitions of output P with corresponding to different working region of ePFL and rPFL Depending the feedback strengths ke and kr, the input-output response of P might have two continuous transitions (See Figure 3C1), a continuous and a discontinuous transitions (See Figure 3C2,3), or two discontinuous transitions (See Figure 3C4) Here we show a case where the two discontinuous transitions do not overlap and there is thus no mutltistability However, mutltistability could be obtained easily by tuning the activation level or feedback strength

(iii) When the ePFL and rPFL activate at the same range of output P , re ≈ rm, activation of ePFL causes activation of rPFL, and vice versa Thus they might cooperate with each other An interlocking of two PFLs with weak feedback strengths which show monostability could generate bistability (See Figure 3B1)

With an interlocking of two PFL with weak feedback strength and strong feedback strength, the bistable region of iPFL is moved and extended from that of a single PFL For instance, as seen in Figure 3B2, the bistable region of rPFL is moved and extended to the left under the effect of ePFL And the bistable region of ePFL is moved and extended to the right under the effect of rPFL (See Figure 3B3)

An interlocking of two PFL with strong feedback strengths in the bistable region

of iPFL is extended in both the left and right directions from that of a single PFL (See Figure 3C4)

Figure 3 (A1-A4, B1-B4, C1-C4) input-output response of output P at different activation levels of ePFL and rPFL with different feedback strengths as shown

Red, blue and black lines represent the input-output response of a single ePFL, rPFL and the interlocking of ePFL and rPFL (iPFL) The dashed-dotted line represents the input-output response curve of simple regulation For ePFL feedback strength, ke = 0.3 is weak and 1 is strong For rPFL feedback strength kr = 0.5 is weak while 1.5 is strong For the A’s: re = 0.5, rr = 0.8;

B′s: re = rr = 0.5; C′s: re= 1, rr = 0.5

3.4 The interlocking of a positive and a negative feedback loop: Relaxation oscillation with flexible controllability of amplitude and frequency

Oscillation comes from the cooperation of positive and negative feedback loops The negative feedback loop drives output P back and forth between low and high steady states which are from a positive feedback loop (See Figure 4A) Therefore, oscillation properties such as amplitude and frequency strongly depend on the feedback strengths of and the embedded time delay in feedback loops

Indeed, a too weak or too strong positive feedback strength pushes the system toward a stable fixed point In Figure 4C, D increases feedback strength kp and elevates the right branch of a reverted-N shape null-cline of output P (solid line) causing the oscillatory system to move toward the stable steady state If the positive feedback strength

is too low, the iPNFL turns out to be a single NFL and thus the iPNFL oscillation will

be shut down The bifurcation diagram in Figure 6A shows that with either a too weak (ke< ke1) or a too strong (ke2 < ke) positive feedback strength, the relaxation oscillation (shown as a blue line) is shutdown For (ke1 < ke < ke2), the iPNFL generates sustained oscillation with finite amplitude and frequency which are simultaneously varied as long

as positive feedback strength kechanges

Figure 4 (A, B) Cooperation of PFL and NFL generates relaxation oscillation (C, D) Dominant positive feedback strength leads to stable steady state (E, F) Dominant negative feedback strength leads to damped oscillation (G, H) Balanced positive and negative feedback

strengths maintain and enlarge amplitude of sustained oscillation.

A too weak or too strong negative feedback strength will also shut down iPNFL oscillation In Figure 4E-F, a negative feedback loop with overwhelming negative feedback strength kn strongly suppresses output P and the reversed N-shape is compressed, resulting in a damping of oscillation of output P (shown as a blue solid line)

If the negative feedback strength is too weak, the iPNFL turns out to be a single PFL with stable steady states rather than one which is oscillatory The bifurcation diagram in Figure

6B shows that with either a too weak (kn < kn1) or too strong (kn2 < kn) negative feedback strength, the iPNFL oscillation (blue line) is shutdown For (kn1 < kn< kn2), , the iPNFL generates sustained oscillation with finite amplitude and frequency which are simultaneously varied as long as negative feedback strength knchanges

It is shown that increasing positive feedback strength will increase the amplitude

of iPNFL oscillation but reduce the frequency (See Figure 6A) In contrast, increasing the negative feedback strength will decrease the amplitude of iPNFL oscillation but increase the frequency (Figure 6B) In both cases, the oscillation would vanish when positive or negative feedback strength is overwhelmed The contradiction of increment and decrement of amplitude and frequency with respect to single feedback strengths suggest that changing one feedback strength would unbalance the positive and negative feedback effects

To enlarge the amplitude, the positive and negative feedback strengths can be increased but they must be in agreement with each other in order to maintain the balance

of positive and negative feedback effect Indeed, by simultaneously increasing positive and negative feedback strength and keeping the agreement between them, the amplitude

of iPNFL oscillation (shown as a blue orbit) can be enlarged significantly (See Figure 4G,H) Moreover, the amplitude of iPNFL oscillation (shown as a blue line, in the middle panel) could be extended as large as we want by maintaining an increase and balance

in positive and negative feedback strength (See Figure 6C) More importantly, in doing that, the amplitude of iPNFL oscillation ccan then be controlled freely, independent of the frequency of iPNFL oscillation (shown as a blue line in the bottom panel) which is almost the same

3.5 The frequency of iPNFL oscillation could be controlled freely and independently from amplitude by using a time delay

The time delay embedded in the negative feedback which provides time lag

τ0 between output P and its inhibitor X (Figure 5B) might influence the oscillation frequency Once might expect that increasing time delay τ would increase time lag τ0

between P and X causing a larger period (at smaller frequency) Indeed, changing time delay τdoes not alter the nullcline structures of the system (See Figure 5A and C) but does alter time lag τ0 between P and X (Figure 5 and D) The oscillation period is elongated but the amplitude is not heightened

The bifurcation diagram of output P with respect to time delay τ shows such behavior (See Figure 6D) By varying time delay τ, the frequency of the relaxation oscillation (shown as a blue line in the bottom panel) could be varied in a broad range More importantly, the variation in amplitude of iPNFL oscillation is very small in comparison to the variation range of time delay τ