Abstract
Cells respond to changes in the internal and external environment by a complex regulatory system whose end-point is the activation of transcription factors controlling the expression of a pool of ad-hoc genes. Recent experiments have shown that certain stimuli may trigger oscillations in the concentration of transcription factors such as NF-
B and p53 influencing the final outcome of the genetic response. In this study we investigate the role of oscillations in the case of three different well known gene regulatory mechanisms using mathematical models based on ordinary differential equations and numerical simulations. We considered the cases of direct regulation, two-step regulation and feed-forward loops, and characterized their response to oscillatory input signals both analytically and numerically. We show that in the case of indirect two-step regulation the expression of genes can be turned on or off in a frequency dependent manner, and that feed-forward loops are also able to selectively respond to the temporal profile of oscillating transcription factors.
Introduction
Cells are dynamic environments constantly adapting themselves to internal and external stimuli. The response to such stimuli is a tightly controlled multi-step process from sensing the stimulus, usually by means of receptors present in the external and internal membrane, transmission of the signal across the cell by a cascade of protein modifications and protein-protein interactions, that activates specific transcription factors which, in turn, regulate the expression of target genes. Fine tuning regulations, e.g. post-translational and post-transcriptional modifications, take place at every step in process providing robustness against noise, specificity to the triggering stimulus and insulation between the different pathways.
Recent discoveries have revealed that transcriptional regulation itself is a very complex process and genes are not just activated or deactivated by transcription factors. Rather transcription factors activate a pool of genes [1] that share a high level of connectivity forming transcriptional networks in which the expression of one gene controls in turn the expression of others generating temporal expression programs. Determining the dynamics of the genetic response from the topology of transcriptional networks is not always straightforward therefore it is important to develop new theoretical and experimental approaches to better understand the mechanisms responsible for regulating gene expression.
Some insights have been gained from identifying so called network motifs. Network motifs are patterns of connectivity that are present in a much higher frequency than in a network of similar dimensions but whose links between its nodes are generated randomly [1]. As the network motifs recur in different organisms, and have been selected by evolution over other possible configurations, they are thought to have special relevance in biological systems, and certain features linked to their topology have been identified [2], [3]. For example, negative auto-regulation, occurring when a gene promotes its own inhibition, has been shown both theoretically and experimentally to be used by cells to speed up the response of gene expression and to promote robustness to fluctuations in production rates [4]. On the other hand positive auto-regulation slows down the response [2], and can lead to bistability [5]–[7] keeping the gene active or inactive even after the stimulus is turned off. The role of certain network motifs in selectively responding to signals depending on their temporal structure has also been studied [8].
Among the network motifs feed-forward loops have been widely investigated both theoretically and experimentally and many of their properties have been described, such as persistence detection, protecting against transient loss of signals [9], generating pulses of expression [10], e.g. playing a role in the temporal organization of the cell cycle [11], speeding up the response [12], detecting fold over basal expression [13], [14], or generating non monotonic response functions [15]. In most previous studies the response of the target genes was studied in the case of a persistent step-like on/off stimulus. However it is becoming more and more evident that more complex temporal patterns in protein concentrations and sequential activation by oscillatory signals can play an important role in determining the outcome of gene expression.
Oscillations have been observed for a long time in the most varied biological systems e.g. cell cycle [11], neuronal firing, heart beat arising as an emergent property of thousands of cells, embryogenesis [16], calcium oscillations associated with differential activation of transcription factors [17], frequency modulated calcium dependent gene expression of Crz1 [18], [19], and in the concentration of transcription factors such as p53 [20], [21], HES-1 [22] and NF-
B [23]–[25].
For transcription factors the functional role of oscillations is not well understood. A number of studies provide supporting evidence that the oscillatory temporal dynamics of nuclear NF-
B may encode information about the required genetic response [25]–[27]. Moreover it has recently been shown that for cells stimulated by TNF-
, oscillations in the dynamics of gene expression are a widespread phenomenon [28], [29] occurring in almost 15% of the human genome. These oscillations occur not only in genes targeted by NF-
B, suggesting that other oscillatory transcription factors may exist and that the oscillations may propagate to other pathways through the transcriptional regulatory network, for example TNF-
stimulated cells also show oscillations induced in MAP kinase activity [30].
In this work we theoretically and numerically investigate how the transcriptional activity of genes regulated by simple network motifs is affected by oscillations in the concentration of transcription factors. First we study and characterize quantitatively the properties of direct regulation. We then use and extend these results to understand the behavior of indirect two-step regulation and feed-forward loops, driven by oscillating transcription factors with varying period and temporal profile. The specific aim is to analyze how various characteristics of the oscillatory input signal (e.g. frequency and shape) can control differential expression of genes, that is not possible in the case of steady state responses. A better understanding of such mechanisms based on theoretical models can help identifying the functional role of experimentally observed oscillations in the expression of various genes. We focus on the genetic response produced by synthetic oscillatory input signals, where we can directly control the different characteristics of the signal.
Methods
In the following we present and analyze differential equation based models that link the temporal dynamics of a transcription factor 
to the expression of the targeted genes. We investigate the effects of changing the oscillation period and the shape of the temporal profile of the concentration 
while its average value remains the same. We assume that the concentration of the transcription factor is normalized so that 
varies between 0 and 1. We choose the temporal profiles of the input signal 
such that it is above the value 0.1 for 75% of the time and above the value 0.75 for 25% of the time. We have considered the three cases shown in Figure 1, in which 
builds up rapidly and decreases slowly (blue curve), the symmetric case in which 
goes up and down in the same amount of time (green curve) and the case in which 
increases slowly and decreases quickly (red curve). The temporal profiles have been obtained by spline interpolation across the selected points over the time interval [0, 1] (see caption of Figure 1), and then stretching and repeating them so that 
produces a periodically oscillating signal.
Figure 1. Constuction of the input signal 
with different shapes.
The plot shows the 
signal skewed to left (blue), the symmetric one (green) and the one skewed to right (red), with a period of 1 h. The shape of the signals have been chosen so that all of the three signals are above the value 0.1 for 75% of the time and above the value 0.75 for 25% of the time. The shape of the blue curve has been obtained interpolating the points (0,0) (0.05, 0.1) (0.1,0.75) (0.1333,1.0000) (0.35,0.75) (0.8, 0.1) (1, 0); the green curve interpolating the points (0,0) (0.125,0.1) (0.375, 0.75) (0.5,1) (0.625, 0.75) (0.875, 0.1) (1, 0); the red curve interpolating the points (0, 0) (0.2, 0.1) (0.65, 0.75) (0.8667, 1) (0.9, 0.75) (0.95, 0.1) (1,0).
Analytical solutions for the components of the considered mechanisms have been derived (see Results section) and have been used to run the simulations presented in this work.
Results
Direct gene regulation
We first studied the effects of oscillations on the average expression of a gene 
when its transcription is directly regulated by the transcription factor 
. We assume that 
is synthesized at a rate 
when the concentration of 
is below a certain threshold 
and at a rate 
when 
. Thus, the analog signal 
is converted into the digital signal 
that is 1 when 
and 0 otherwise. If 
then 
is an activator for the gene Y, else it is an inhibitor. For the sake of clarity in the following we assume that 
is an activator, but analogous results can be obtained for inhibitors. We assume that 
is degraded following mass action kinetics with decay rate 
. Thus the expression of 
can be described by the differential equation
 |
(1) |
where 
is the step function
 |
A similar formulation of the model could be given by assuming a Hill rate function for the up-regulation of the synthesis of 
by 
as:
 |
that becomes equivalent to step-function above in the limit when 
. We will use the form with the step function as a simple approximation for the gene activation, since that somewhat simplifies the analysis of the models and can help understanding of the basic mechanisms governing gene responses [2]. Although this simplification may slightly modify the dynamics of the expression level of 
, the qualitative behavior remains the same (see Supporting Information S1).
The solution 
of the ODE (1) is a piecewise function of the form:
 |
(2) |
where 
is the 
-th intersection of the signal with the threshold for 
, i.e. 
, 
, and we assume that 
. Under the action of a transcription factor 
increases exponentially towards 
when 
is above the threshold of activation and otherwise decreases exponentially towards the value 
(Figure 2).
Figure 2. Example of direct regulation.
The Figure shows the temporal dynamics of a transcription factor 
(blue), and the response of the directly regulated gene 
(green) at stationary regime. As 
increases and decreases it crosses the threshold of activation 
(dashed line) determining a sequence of intervals 
such that 
is synthesized when 
(
odd), and it is degraded when 
(
even). The dotted line represents the digital signal 
. For the plot the following values have been used: 
, 
, 
.
Although the specific solution 
depends on the temporal profile of 
that determines the sequence of on and off times 
, the response of the gene can be characterized by its mean value over a longer time period 
. This may also be appropriate for interpreting experimental data from cell populations in which the individual traces of gene expression of single cells are not known and only the population average is measured. When 
is periodic it can be shown that after a transient time 
also becomes periodic in time. Moreover in the stationary regime the average value of 
is fully determined by the proportion of time spent by 
over the activation threshold of gene 
(see Supporting Information S1):
 |
(3) |
where 
is the time-average of the digital function 
. The formula (3) shows that the average value of expression of 
is a weighted average of the equilibrium values that would be attained with no stimulation at all or with constant stimulation. For a signal of given shape, varying the period of oscillation does not change the fraction of time spent over any given threshold, therefore from (3) automatically follows that the average value of 
is independent of the period of oscillation of 
. This type of response is described, for example, in Ref. [18] where the expression of genes targeted by Crz1 has the same profile as the frequency of bursts of nuclear Crz1 varying in response to Ca+.
When the concentration of the oscillatory transcription factor crosses the threshold of activation back and forth only once in each cycle of oscillation (as is typically the case, e.g. NF-
B [27]), it is possible to determine the maximum and minimum values of 
in the stationary regime as:
 |
(4) |
 |
(5) |
where we defined 
the non-dimensional oscillation period of 
measured relative to the degradation time of 
. 
is an increasing function of the period of oscillation and 
is decreasing, and they both tend to the average value 
as 
. Thus, for a gene that is directly controlled by a single oscillatory transcription factor, although variations in the minimum and maximum level of expression occur (see Figure 3), its average value does not respond to changes in the frequency of oscillations (see Figure 4) or in the shape of the periodic signal as long as the overall percentage time of gene activation remains the same.
Figure 3. Direct regulation time course dynamics varying the period of 
.
The plots show the response of a gene 
(red curve) and its average value at stationary regime (red dashed) in the case of direct regulation by a symmetric oscillating transcription factor 
(black curve) having a period of 3 hrs (A) 1 hr (B) 0.5 hr (C). 
oscillates with varying amplitude depending on the period of oscillation of 
. As the frequency of oscillation of 
increases, the time 
has to adjust decreases, leading to smaller amplitude of its oscillations. The parameters are: 
, 
, 
, 
(black dashed, the value has been chosen to activate the production of 
for 25% of time).
Figure 4. Direct regulation.
The minimum (blue) maximum (red) and average (green) values of a transcription factor 
controlled by direct regulation at stationary regime, corresponding to different values of the period of oscillation of 
. Simulations have been run using 
, 
= 
= 1.5, 
.
Two-step regulation
The simplest extension of the direct regulation model is the case in which 
directly regulates 
that in turn regulates a third gene 
. Similarly to the direct regulation case, we assume that 
changes the synthesis rate of 
from 
to 
when its concentration is above the threshold 
, and that 
is degraded following a mass-action law with coefficient 
.
In the case of direct regulation we have shown that the period of oscillation of 
influences the minimum and maximum concentration of 
(4–5). Therefore the proportion of time spent by 
above or below the threshold 
in each cycle of oscillation, i.e. the proportion of time when the expression of 
is activated, varies as well in response to changes in the period of oscillation of 
(Figure 3). Since changing the period of 
has opposite effects on the minimum and maximum expression levels of 
(Figure 4) we can have two types of period dependent responses in the two-step regulation system, depending on the value of the threshold 
. When the threshold is higher than the average concentration of 
, 
, the expression of 
is sensitive to the maximum value of 
, that decreases when the period is shortened and eventually 
can no longer activate 
. Thus, in this case the average concentration of 
decreases as the period of the input signal 
is reduced, and its expression is switched off completely below a certain oscillation period. Conversely if 
, the expression of 
is controlled by the minimum value of 
, and as the oscillation period of 
is decreased 
spends more and more time over the value 
till eventually 
is fully expressed (Figure 5). Thus, in the two-step gene regulatory system, changing the frequency of the input signal can have opposite effects on the expression of genes with different activation thresholds.
Figure 5. Two-step regulation time course dynamics varying the period of 
.
The plots show the expression of a gene 
(blue) controlled by the symmetric signal 
(black) for the two-step model. The left (right) column shows the case in which the value of 
(green dashed line) is above (below) the average value at stationary regime of the transcription factor 
(red curve). The first three rows show the time-course dynamics for 
, 
and 
for three different oscillation periods, whereas the last row shows the average value of 
at stationary regime as a function of the period of oscillation of 
. 
is turned off in the case 
as the period of oscillation decreases (left column) while it is increasingly expressed in the case 
(right column). The parameters used in the simulations are: 
, 
, 
, 
(this value has been chosen so that 
) and 
for the left column and 
for the right column.
The delay occurring between the activation of gene 
by the transcription factor 
and the activation of 
by 
can be evaluated from the time-dependent concentration profile of 
in the increasing branch of (2), i.e. when 
is odd, by finding the value 
such that 
. After non-dimensionalization, using again the characteristic lifetime of 
as time unit, we obtain
 |
(6) |
The time between the inactivation of the 
and 
genes can also be obtained with similar calculations, and combining these expressions together the fraction of time spent by the transcription factor 
over the threshold 
can be evaluated as:
 |
(7) |
This expression is valid provided that 
is such that 
. Otherwise, either 
activates the gene 
all the time so that 
and 
, or 
is never activated, i.e. 
and 
. From (7) we can see that depending on the sign of the two logarithmic terms, that can be either positive or negative, the activation time of the target gene 
can be either longer or shorter than the time of activation of the intermediate transcription factor 
. Figure 6 shows how the delay and the duration of the activation change depending on 
and the period 
for the same set of parameters as the example given in Figure 5. For a fixed period increasing the activation threshold 
reduces the activation time of 
. When the period is varied for a given threshold, the activation time, and consequently the average concentration of 
changes monotonously with 
either increasing, when 
, or decreasing otherwise.
Figure 6. Y expression versus normalized time.
The top plot shows the percentage delay in 
signal with respect to the 
signal, for varying period of oscillation 
and threshold value 
, the bottom plot shows the percentage duration of the 
signal. For values of the threshold 
lower than the average value of 
, decreasing the period of oscillation causes a faster response (lower delay) and a higher duration of 
(that eventually is active all the time when the period is small enough). For values of the threshold 
higher than the average value of 
, 
is usually delayed and of short duration, decreasing the period of oscillation causes the maximum value of 
to fall below 
leading to no activation of 
. For the plots the parameters 
, 
, 
have been used.
To conclude, in the two-step regulation the average expression of the gene 
is dependent on the period of the oscillatory input signal 
. Oscillatory signals with different shapes activating 
for the same fraction of time, produce a 
signal that oscillates between the same minimum and maximum values, therefore have no influence on the activation time of 
. Thus, the average concentration of 
is determined by 
, the fraction of time when the input signal is above the activation threshold of the directly regulated gene 
and the period of the input signal.
Feed-forward loops
In a FFL the transcription factor 
regulates the target gene 
both directly and indirectly through an intermediate transcription factor 
that in turn regulates the transcription of 
. Each of the interactions between the transcription factors 
, 
and 
can be either activating or inhibitory, so there are eight different possible combinations as shown in Figure 7. These can be split into two categories: in Coherent Feed Forward Loops (CFFLs) 
regulates 
in the same way both directly and indirectly (that is 
activates or inhibits 
to some extent, through both branches) and Incoherent Feed Forward Loops (IFFLs) in which 
activates 
through one branch and inhibits it through the other. The transcription of gene 
controlled by the FFL is activated by a logic gate, that encapsulates various processes such as DNA binding, RNA polymerase recruitment and so on [2], combining the concentrations of the transcription factors 
and 
into the expression of 
. For example, an AND gate in the case of CFFL-1 activates the expression of 
when the concentrations of both 
and 
are higher then their separate activation thresholds for 
, 
and 
. In the case of an IFFL-1 an AND gate allows the transcription of 
only when the concentration of the activating factor 
is above its direct regulatory threshold 
, and the inhibitor concentration 
is below the threshold 
. The OR gate activates the expression of 
when at least one of the two branches are activated. Several properties of such FFLs have been characterized and tested [1], [31] mostly in the case of step-function type stimulus, here we investigate the properties of FFL motifs when the input signal is an oscillatory transcription factor using the methodology first introduced by Alon in [2, Chap. 4]. In the following we discuss two representative types of FFLs, the CFFL-1 and the IFFL-1, both with AND gates. The other types produce qualitatively similar behavior, just the conditions corresponding to different regimes are interchanged according to the type of interactions between the components. The case of OR gate is also similar and is discussed in Supporting Information S1 using CFFL-1 and IFFL-1 as prototypes for our analysis.
Figure 7. Different types of feed forward loops.
CFFL-1
The activation and inactivation of a CFFL-1 with a step-function on-off stimulus is shown in Figure 8. At time 
crosses both thresholds 
and 
and activates the expression of 
that starts accumulating, and when its concentration reaches 
the condition 
AND 
is satisfied and the expression of 
is activated. When the input signal 
is switched off (
), 
falls below the threshold 
and although the concentration of 
is still higher than 
the transcription of 
stops instantly. Thus, this type of FFL produces a sign-sensitive delay, i.e. the response is delayed at the on signal but there is no delay when the stimulus is switched off.
Figure 8. Response of CFFL-1 with AND gate to a step-like stimulation.
The figure shows the typical response of a CFFL-1 with AND gate to a transient transcription factor 
. A step-like transcription factor 
activates simultaneously both branches of the FFL (A). Under the stimulus of 
(signal 
), 
starts accumulating (B), but the transcription of 
is delayed until 
reaches the value 
, and the signal 
turns on (C). 
accumulates, but its synthesis is immediately turned off when 
is removed (C). The following parameters have been used: 
, 
, 
, 
, 
.
In the case of an oscillating transcription factor, one branch acts in the same way as explained in the two-step regulation model, but the expression of 
is also influenced by 
directly on the other branch. Figure 9 shows how the activation of the two branches, combined together by the logic gate, regulates the expression of 
. The relative values of 
and 
determine which of the two branches is activated first in each cycle of oscillation and the times that 
spends above the thresholds 
and 
, respectively. In the case shown in Figure 9
, so that 
first crosses the threshold 
activating the signal 
and after some time it goes above 
activating also the signal 
. As 
decreases the two branches are deactivated in opposite order, hence 
activates the direct branch for a longer time than the indirect one.
Figure 9. Oscillating transcription factor controlling the expression of gene 
by a CFFL-1 with AND gate.
The plot shows the response of a gene expressed under the stimulus of the symmetric signal 
oscillating with period 
. The thresholds 
and 
split the signal 
into the two digital signals 
and 
(A). 
controls the expression of 
that when over the value 
generates the digital signal 
(B). The digital signals 
and 
are combined by the logic gate to finally control the expression of the gene 
, in the case of an AND gate the logic gate only allows the expression of 
when both 
AND 
are active (C). The parameters used in the simulation are: 
, 
, 
, 
, 
, 
, 
.
From the analysis of the two-step regulation model we know that the duration of activation of 
by 
varies with the period of oscillation of 
. In the case of a CFFL-1 this means that the signal 
, and consequently the average value of 
, also depend on the period of oscillation of 
. The value 
determines whether the duration of activation of 
by 
increases (
) or decreases (
) with the period of oscillation. As in the two-step regulation case if 
the average value of 
is eventually switched off as the frequency of oscillation increases (Figure 10, left column). If 
the average value of 
increases as the frequency of oscillation increases. However in contrast with the two-step regulation 
is never fully expressed. Even when the minimum of 
is above the threshold of activation of 
, the expression of 
is still limited by the activation of the direct branch. The value of 
, that determines the fraction of time when 
directly activates 
, sets a limit to the maximum average concentration of 
(Figure 10, right column).
Figure 10. Time course simulation of CFFL-1.
The plots show the dynamics of 
controlled by a CFFL-1 with AND gate stimulated by the symmetric signal 
with varying period when 
. The left column illustrate the case 
, the right column the case 
. As explained in the main text in the case 
the duration of the 
signal diminishes causing the gene 
to be inhibited, while when 
the duration of 
increases causing an increase in the average value of 
. The response is different from the two-step regulation response because the presence of the direct branch limits the maximum duration of the expression of 
. The parameters 
(corresponding to 
), 
, 
, 
have been used for all the plots. For the left column 
; for the right column 
.
The period of oscillation and the temporal profile of 
also influence the delay between the signals 
and 
. This suggests that the shape of the signal 
can play a role in controlling the expression of 
. If the oscillations of 
are skewed to the left, i.e. steep increase followed by slower decay, then 
and 
are activated almost simultaneously, but one of them is deactivated well before the other, conversely if 
is skewed to the right, one of the two signals is activated before the other and they are deactivated almost at the same time. The delay of 
with respect to 
due to the time required for the accumulation of 
, can results in out of phase activation of 
and 
, reducing the duration of activation of the gene 
. The effects of the shape of the signals is discussed further below.
Varying the period of the input signal we have identified four different classes of responses, depending on the relative values of the thresholds 
, 
, and on whether the indirect regulation of 
is controlled by the low or high values of the intermediate transcription factor 
, i.e. 
or 
. Figure 11 shows the average concentration of 
at stationary regime obtained stimulating the CFFL-1 (AND gate) with oscillatory signals of different shapes and varying the oscillation period:
Figure 11. CFFL-1 AND gate, average response of the gene 
at stationary regime for various configurations of the parameters.
For all the plots the values 
have been used. The thresholds of activation for the various cases are: (A) 
, 
; (B) 
, 
; (C) 
, 
; (D) 
, 
.
A)
. The average level of 
is not affected by the period of oscillation of 
as in the case of direct regulation by a single transcription factor. Since 
, the signal 
contains 
and the accumulation of 
starts before 
directly activates 
. Since 
is low, 
is activated before 
, and it is deactivated after 
is switched off. The result is that 
so that 
is expressed as if only directly regulated by 
, and is independent of the indirect branch. Therefore the average concentration of 
does not change with the frequency, and the shape of the input signal 
has no effect on the final outcome.
B)
. The expression of 
is switched off at high frequency oscillations, then increases with the period and saturates at a value corresponding to the direct activation of 
by 
. Similarly to the two-step regulation the switch is controlled by the maximum value of 
that for high frequency oscillations falls below the threshold 
, that switches off the expression of 
. The delay between 
and 
also influences the response and shows a gradual switch between activation and inactivation when 
activates 
and 
simultaneously, and a sharp transition when the activation of 
is delayed with respect to 
with a delay.
C)
. In this case 
is activated after 
. However since 
decreasing the period of oscillation causes an increase in the duration of 
, so that 
and 
overlap for longer time and the average expression of 
increases. In this case as well, the shape of the input signal influences the response since as the period decreases the signal 
lasts longer causing the duration of the overlap to vary smoothly or abruptly depending on the delay between 
and 
.
D)
. The expression of 
is controlled by the indirect branch through 
, therefore this case is equivalent to the two-step regulation case presented earlier. Since 
the duration of 
decreases with the decreasing of the period of oscillation until eventually 
is no longer active. The shape of the signals does not affect the response because the duration of 
is short compared to 
and therefore the variations in the delay do not cause any significant change in the average value of expression.
IFFL-1
As a representative of the IFFLs we illustrate the behavior of the IFFL-1 with an AND gate. In the IFFL-1 
directly promotes the expression of the gene 
and inhibits it indirectly by activating the expression of the repressor 
. The transcription of 
is activated when 
AND 
, i.e. following to the digital signal 
. In Figure 12 the activation and inactivation of the IFFL-1 by a constant step-like stimulus is shown. At the time 
the transcription factor crosses the thresholds 
and 
and since 
is initially not present, 
, the transcription of 
is activated. Meanwhile, 
starts accumulating and after a transient time reaches the threshold of inhibition 
that turns off the expression of 
. Thus, in the case of a step-like sustained stimulus the IFFL-1 is a pulse generator promoting the expression of the gene 
only for a limited time.
Figure 12. Response of IFFL-1 with AND gate to step-like stimulation.
The plots show that under a step-like stimulus an IFFL-1 generates a pulse-like response. When 
activates the FFL (A) the logic gate is activated by 
but is not inhibited by 
starting the expression of 
. Under the stimulation of 
, 
starts to build up, but it only inhibits the expression of 
after the delay required to reach the value 
(B). As a result 
starts decreasing when 
is still present. For the simulation the following parameters have been used: 
, 
, 
, 
, 
, 
, 
.
In the presence of an oscillating factor 
, however, the IFFL-1 acts as an oscillation detector, continuously activating and deactivating the expression of 
, so that the average amount of 
can be high in the presence of sustained oscillations. This is illustrated in Figure 13, using the same parameters as in Figure 12. The oscillating transcription factor 
periodically crosses the thresholds 
and 
turning on and off the two branches with a certain delay relative to each other. Under the direct regulation of 
, the repressor 
is expressed and degraded crossing back and forth the threshold 
, generating the oscillatory signal 
. This combined with the direct activation of 
leads to the periodic expression of 
and is not turned off completely after a transient time. Once again while the average amount of 
is independent of the period of oscillation of 
, its maximum and minimum values are not, and as a consequence, it affects the signal 
that controls the expression and the average concentration of 
. As explained before, depending on the value of 
, the time spent above the threshold 
by the concentration of 
can either increase or decrease with the period of oscillation. In the specific case of the IFFL-1 with an AND gate if 
as the period of oscillation of 
decreases, the time when the repressor 
is active decreases as well. As a consequence, the average concentration of 
increases when the period of oscillations is increased (Figure 14). Similarly to the CFFL-1 the values of the thresholds determine the relative delays between the activation of the different branches, so that IFFLs can be activated differently by transcription factors 
with different temporal profiles.
Figure 13. Oscillating transcription factor controlling the expression of gene 
by means of an IFFL-1 with AND gate.
The plot shows the response of a gene expressed under the stimulus of an oscillating transcription factor 
. The thresholds 
and 
split the signal 
into the two digital signals 
and 
(A). 
controls the expression of 
(B). The digital signals 
and 
are combined by the logic gate to finally control the expression of the gene 
(C). For the simulation the following parameters have been used: 
, 
, 
, 
, 
, 
, 
.
Figure 14. Time course simulation of IFFL-1.
The plots show the dynamics of 
stimulated by an oscillating transcription factor in the two different cases when 
(left column) and 
(right column), stimulated with oscillating transcription factors of varying period. As explained in the main text, in the case 
the duration of the 
signal diminishes as the frequency of oscillation increases; as a consequence the average value of the signal 
increases and so does the average value of 
. When 
, the duration of the 
signal increases with the frequency of oscillation and so the average value of 
decreases. The values 
, 
, 
, 
have been used for all the plots. For the left column 
, for the right column 
.
The response to changing the period of the oscillation of 
can be classified again into four different regimes as shown in Figure 15:
Figure 15. IFFL-1 AND gate, average response of the gene 
at stationary regime.
For all the plots the values 
have been used. The thresholds of activation for the various cases are: (A) 
, 
; (B) 
, 
; (C) 
, 
; (D) 
, 
.
A)
In this case the signal 
contains 
. Since the threshold of activation is low, the repressor 
is already active when 
activates 
. Since the activity of the repressor completely overlaps with the direct activation, the gene 
is not expressed regardless of the oscillation period.
B)
In this case the average expression of 
decreases as the period increases. Since 
the duration of the activity of the repressor decreases as the frequency of the oscillation increases, until eventually the maximum concentration of 
falls below the threshold 
and the repressor 
is completely switched off. The relative delay between the direct activation and the activation of the repressor in this case influences the final outcome since it determines whether 
has sufficient time to accumulate to reach the threshold of inhibition. This is shown by the distinct response functions obtained for the signals with different temporal profiles of the oscillatory input signals.
C)
In this configuration of the parameters the signal 
is contained in 
. Since the threshold 
is low, 
inhibits the transcription of 
almost immediately after its expression is activated by 
, limiting the transcription activated directly by 
. As the frequency of oscillation increases 
spends more and more time over the threshold inhibiting 
for longer times, until eventually completely switches off the expression of 
. The relative delay between 
and 
determines the sharpness of the transition.
D)
Since the threshold of inhibition 
is high, activation of the repressor 
needs some time. When the period of 
is large enough, 
reaches the threshold inhibiting 
, but as the period of oscillation decreases 
does not have time to accumulate and its inhibitory effect ceases. Also, since the threshold of activation is high, the variations in the delay and length of 
are small compared to the duration of 
and the shape of the signal does not influence the average synthesis rate of the gene 
.
Discussion
Oscillations are a widespread phenomenon arising in many biological systems [32]. Gene expression however has been mostly studied as a static phenomenon mainly focussing on the total amount of transcription factor activated by various types of stimuli, usually observed after a relatively long treatment. This approach allows to infer information about the processes ongoing in the cell at population level, but does not provide insight into the dynamics of the components involved and about their influence on the final outcome of gene expression. Nevertheless high-throughput experiments have started to unravel the complexities of temporal dynamics and have shown the importance of understanding the information encoded in the temporal dynamics of cellular processes.
Previous studies have investigated both theoretically and experimentally the properties of regulatory networks in relation to their topology. Alon and coworkers have demonstrated various properties of simple regulatory motifs like negative auto-regulation [4] and feed forward loops [2], [6], [9], [12]–[15]. In [33] the authors have shown by means of numerical simulations and linear analysis that the presence of common three and four node motifs could be beneficial for the robustness of biological networks to small perturbations and noise. The dynamic response to bursts of activation for common motifs like IFFL-1, diamond-motif and the interlocked negative loop was studied in [8] by characterizing the optimal duration of inter-pulse intervals that maximizes the time-averaged response.
Under adequate stimulation oscillations in gene expression may involve a large number of transcription factors, propagating across different pathways and occur at different cellular levels [28]–[30]. In principle, such background oscillations allow for refined context-dependent activation of pathways in response to different specific stimuli. For example, recent work on the NF-
B pathway has shown that the oscillations caused by the negative feedback loop through IkB family of proteins and A20, are tightly regulated and suggest a frequency as well as amplitude dependence of the transcription of targeted genes, although the mechanisms of differential response by means of oscillations has not been clarified yet. 
dependent bursts of nuclear Crz1 in yeast and bacteria has shown that oscillation in and out the nucleus can be advantageous for maintaining the relative amount of certain proteins constant in the cell [18].
In this work we studied the possibility of frequency dependent responses in simple gene regulatory schemes, that could be used in decoding information from time-dependent oscillatory signals, and to generate differential regulation of multiple genes controlled by the temporal dynamics of the same transcription factor.
In the case of direct regulation the key factor regulating the gene expression is the fraction of time when the transcription factor concentration is above the activation threshold of a certain gene. As a consequence, modifying the frequency of oscillation cannot modulate the expression of a gene. Varying the amplitude of oscillation though, may cause changes in the duration of the activity of transcription factors and could regulate the average level. Such a mechanism might be ideal to regulate those genes whose average level of expression in cells and tissues should not change when the cellular environment is perturbed by a stimulus that gives rise to oscillations.
For the two-step regulation the frequency of oscillation is capable of switching on or off the expression of the target genes. Increasing the frequency of oscillation of the regulating transcription factor causes the intermediate component to oscillate closer to its average value 
. As a consequence, depending on the threshold of activation of the target genes they could be up or down-regulated in a frequency dependent manner. However, since the input signal activates gene expression by crossing over a single threshold, this mechanism cannot distinguish between different temporal profiles of the transcription factor. This is possible for feed forward loops when the input signal activates two different genes with different activation thresholds.
Thus increasing the complexity of the gene regulatory network provides the cell with more refined mechanisms for decoding information from the temporal dynamics, that is not possible in the case of steady-state responses with no temporal dynamics. We have identified distinct types of response behaviors depending on the parameters, for example: on/off switching of the gene expression in a frequency dependent manner, maintenance of a constant average expression, frequency dependent switching of the expression level between two distinct regimes. Moreover we have shown that, as 
activates the two branches of a FFL at different times depending on the shape of the signal, the temporal profile of 
can affect the final average expression of the targeted gene. For our simulations we have used signals that vary between the same maxima and minima and have approximately the same average value, but yet the outcome on gene expression is different. Such a behavior could for example explain why in certain experiments involving cell population measurements, even if the amount of the considered transcription factor is the same in different samples the genetic response can be completely different.
Gene expression mediated by two-step regulation and FFLs could be advantageous in driving cell fate in those situations for which the transcription factor can regulate opposite cellular processes. NF-
B and p53 for example are known to regulate both apoptosis and cell proliferation. We have shown that different genes may respond differently to the same oscillatory signal depending on the parameters and the topology of the interaction networks. Thus, regulation of such different cell fates may be possible by encoding certain environmental information in the frequency of oscillations of NF-
B so that certain genes favoring one process or the other become activated.
Future extensions of this work could consider how combining together several of these regulatory mechanisms affects the ability to decode information from the temporal dynamics of transcription factors in transcriptional networks with more complex topology. Another interesting possibility would be to consider gene regulatory motifs controlled by oscillatory input signals that depend on multiple stimuli, to explore how multiple information can be transmitted and recovered from the temporal dynamics of a single transcription factor. The inputs influencing the dynamics of an oscillatory transcription factor typically would modify not just the frequency but also other characteristics of the signals, e.g. average expression rate, amplitude of the oscillations etc. Therefore the frequency dependent responses that we described may be combined with or dominated by other changes occurring simultaneously.
While in our models we focussed on the time-averaged response behavior of a stationary oscillating system, in many cases transient signaling and the timing of the gene expression is also important. Relevant information may also be encoded in the temporal profile of transient stimuli, that could lead to selective transient expression of different genes. Simple gene regulatory networks can also play a role in decoding such information as it was shown for example in the context of genes involved in cell cycle regulation [11].
Frequency dependent expression of genes regulated by NF-
B has been observed experimentally in [34]. In this work oscillations of NF-
B activity were triggered by stimulating the cells with pulses of the inflammatory stimulus, TNF-
, promoting waves of translocation of NF-
B into the nucleus resulting in differential gene expression, dependent on the period of the external stimulus. NF-
B regulates hundreds of genes whose expression is likely to be interconnected, and therefore this pathway could be a good candidate as a model system for validating our theoretical findings. This could be done for example by identifying groups of genes with qualitatively similar activation patterns in response to changes in the oscillation period, e.g triggered by different concentrations of TNF-
. Then the next step would be to find correlations between the different types of frequency-dependent responses with the characteristic gene interaction patterns. Mutant cells in which different forms of I
B have been suppressed leading to irregular period of oscillations could also be used to test the effects of oscillation period on a the final outcome of gene expression. Another potential candidate for such experimental work is the oscillatory transcription factor p53 that regulates hundreds of genes whose period of oscillation has been shown to be dependent on the cell type and varies in response to different stimuli [20].
Supporting Information
This supporting material contains the mathematical derivation of all the formulas presented in the main text. The numerical results obtained simulating Network Motifs with an OR gate stimulated by oscillatory transcription factors are shown and discussed extensively using the same framework introduced in the main text.
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Footnotes
Competing Interests: The authors have declared that no competing interests exist.
Funding: L.C. was supported by the Irish Research Council for Science, Engineering and Technology. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
This supporting material contains the mathematical derivation of all the formulas presented in the main text. The numerical results obtained simulating Network Motifs with an OR gate stimulated by oscillatory transcription factors are shown and discussed extensively using the same framework introduced in the main text.
(PDF)