ultrasensitivity , or threshold behaviour , where small changes in a stimulus near some threshold value produce a large change in a response , but large changes in the stimulus far from the threshold produce small changes in the response ( i.e. , a sigmoidal stimulus  response curve ) , is an important property of many biological systems koshland et al , 1982 ; ferrell , 1996 .
ultrasensitivity can function to dampen irrelevant fluctuations in the signal , yet allows a decisive response when needed .
prokaryotic transcription factors ( tfs ) often have ultrasensitive activating or repressing effects by virtue of cooperative binding of the tf to multiple dna sites .
although eukaryotic transcription is a highly synergistic process , involving the cooperative action of many different dna - associated proteins ( carey , 1998 ; naar et al , 2001 ; figure 1a ) , this synergy does not in itself provide a mechanism for ultrasensitive responses to changes in concentration or activity of a single tf .
such ultrasensitivity can be produced , as in bacteria , by direct cooperative binding of the tf due to favourable contacts between the tf protomers bound at different sites ( burz et al , 1998 ; figure 1a , tf1 ) .
tf protomers bound at different sites may be able to simultaneously make favourable contacts with some common target such as proteins of the transcriptional apparatus ( carey , 1998 ; figure 1a , tf2 ) .
alternatively , in collaborative competition , binding of the tf to one site competes with a nucleosome that can block binding of the tf to other sites ( miller and widom , 2003 ; figure 1a , tf3 ) .
however , it is not clear how common these cooperative binding mechanisms are in eukaryotes . here
, we propose a mechanism for generating high ultrasensitivity that does not require cooperative tf binding , but rather involves nucleosome modifications .
many eukaryotic tfs act , at least in part , by recruiting histone - modifying enzymes to the promoter region figure 1b .
these enzymes include histone acetyltransferases and deacetylases ( hats and hdacs ; struhl , 1998 ; naar et al , 2001 ; eberharter and becker , 2002 ) , methyltransferases ( hmts ; demers et al , 2007 ) and demethylases ( hdms ; swigut and wysocka , 2007 ) .
this recruitment can result in changes to the modification state of nearby nucleosomes ( vignali et al , 2000 ) . in turn , the histone modifications carried by the nucleosomes located in the vicinity of the promoter affect transcription , although in ways that are not yet well understood .
histone modifications change the dna - binding properties of the nucleosomes , affecting competition with other dna - binding proteins ( anderson et al , 2001 ) , and also can affect the binding of other proteins , including those of the transcription apparatus , to the nucleosomes themselves ( naar et al , 2001 ; taverna et al , 2007 ; vermeulen et al , 2007 ) .
again , recruitment of a particular histone - modifying enzyme should not in itself provide an ultrasensitive transcriptional response , because changes in the localized activity of the enzyme should be linear with respect to changes in the dna site occupancy of the tf that recruits it .
however , it has been proposed that nucleosome modifications can sustain positive feedback loops , where nucleosomes carrying a particular modification recruit enzymes capable of maintaining the same modification state among neighbouring nucleosomes ( grunstein , 1998 ; turner , 1998 ; schreiber and bernstein , 2002 ) .
we show that such feedback loops can convert linear changes in dna occupation by a tf into a highly ultrasensitive transcriptional response .
we previously introduced a simplified numerical model of positive feedback in nucleosome modification for investigation of epigenetic cell memory ( dodd et al , 2007 ) .
the model considers a patch of n nucleosomes carrying mutually exclusive modifications , for example , methylation or acetylation of a specific lysine residue ( such as lysine 9 of histone h3 ) , giving three types of nucleosome with regard to this modification : methylated ( m ) , unmodified ( u ) or acetylated ( a ) .
nucleosome interconversions can occur by a positive feedback process , where acetylated nucleosomes recruit specific hdms and hats , while methylated nucleosomes recruit hdacs and hmts that can act on any other nucleosome in the patch ( figure 2 ) . as well as
these local positive feedback reactions , non - feedback or  noisy ' interconversions can occur independently of the local nucleosomes , for example , through the action of non - recruited enzymes or enzymes bound to nucleosomes outside the patch .
the relative strengths of the feedback and noise processes is parameterized by  ( figure 2 ) , with the feedback - to - noise ratio , f defined as f=(1 ) .
we found that with increasing n and f , the patch of nucleosomes becomes increasingly bistable , existing in either a highly acetylated or a highly methylated state with rare transitions between the states .
bistability also required the positive feedback reactions leading to one modification to be of similar efficiency to those leading to the opposing modification ( dodd et al , 2007 ) . here , we make two changes to this model to investigate gene activation or repression by a tf that recruits a histone - modifying enzyme .
first , we allow the relative strengths of the positive feedback reactions in one direction ( e.g. towards a ) to be of a different strength to the reactions in the other direction ( e.g. towards m ) .
the balance of these reactions is expressed by the parameter  ( figure 2 ) . if  deviates significantly from 0.5 , indicating an imbalance , then the patch of nucleosomes is only stable in one modification state ( dodd et al , 2007 ) .
we imagine that for any particular mutually exclusive modification ( e.g. the m and a in figure 2 ) , the concentrations of modifying enzymes , their affinities for the recruiting nucleosomes and their enzymatic activities will often differ so that one modification will be favoured , possibly in a location - dependent manner , so that a particular patch of nucleosomes will tend to exist by default in a particular modification state .
second , we add a tf that binds to the dna within the nucleosome patch and recruits a histone - modifying enzyme , bringing the enzyme into proximity with the nucleosomes in the patch and facilitating their modification ( figure 1b ) .
we do not consider any other action of the tf ( e.g. direct interaction with the transcription complex ) .
the strength of the tf - stimulated reaction is given by  , which sets an additional rate for the u to a transition ; a larger  makes it proportionally more likely to change a u to an a anywhere in the system ( figure 2 ) .
we imagine that increases in  result from increases in the activity or concentration of the tf that , in turn , increase occupation of its dna - binding site or its ability to recruit the modifying enzyme . in figure 3 , we show the results of iterating the reaction steps of figure 2 when , by default , the patch contains a low proportion of a - type nucleosomes due to an imbalance in the feedback reactions due to =2/3 ( i.e. , m - favouring reactions are twice as likely as a - favouring reactions ) . as the strength of the tf - stimulated hat reaction
is increased (  increased from 0 to 1 ) , the average steady - state fraction of acetylated nucleosomes , a , increases to the point where a nucleosomes dominate the patch . depending on the parameters ,
the model is able to give a range of ultrasensitive responses of a to increases in  ( figure 3 , upper panels ) , as quantified by the hill coefficient h , with h>1 indicating ultrasensitivity ( koshland et al , 1982 ; ferrell , 1996 ) .
if promoter activity is taken to be proportional to a and  is taken to be proportional to tf activity , such that the only source of non - linearity in the system is the nucleosome modification reactions , then it is clear that this mechanism alone can generate an ultrasensitive response of promoter activity to tf activity .
when there is a high fraction of one nucleosome type , increases in the minority modification due to  are strongly resisted by the dominating activity of the majority type . however , as the threshold ( ma ) is approached , the resistance decreases , because the opposing feedback reactions are more balanced , allowing changes in a to occur easily . the strength of the positive feedback increases with f and n and thus these parameters control the ultrasensitivity ( figure 3 , top panels ) . in principle
, this could allow the ultrasensitivity of the response of a promoter to a tf to be tuned by adjusting the strength of the positive - feedback modifications relative to noise ( f ) or by alterations to the size of the nucleosome patch ( n ) .
although figure 3 shows activation by recruitment of a modifying enzyme that adds a modification ( figure 2 ) , recruitment of an enzyme that removes the opposing modification produces a similar response . of course , the system is not limited to activation by a tf .
if the modification favoured by the tf has an inhibitory effect on the promoter , then an ultrasensitive repressive response can be produced .
rossi et al ( 2000 ) found hill coefficients ( h ) of 1.63.2 for doxycycline regulation of promoters activated by tet repressor 
vp16 fusions , while ajo - franklin et al ( 2007 ) found h values of 2.73.6 for promoter control by vp16-derived activation domains fused to zifh or lexa dna - binding domains . because multiple adjacent activator dna - binding sites were used
, it is possible ( but was not tested ) that the ultrasensitivities were caused by cooperative dna binding .
however , the responses may also be explained by our model , as the vp16 activation domain has been shown to interact with hats , and when fused to the gal4 dna - binding domain can target hat activity to nucleosomes around the gal4-binding site ( vignali et al , 2000 ) . in these experiments ,
the two models could be tested by measuring ultrasensitivity with the same number of activator binding sites but placed at separate locations , which should disrupt cooperative binding , but not our mechanism .
the model can also explain the results of carey et al ( 1990 ) , in which an ultrasensitive promoter response was seen with respect to increasing copies ( in tandem ) of a binding site for a gal4vp16 activator . because all the binding sites were saturated with tf , carey and co - workers ruled out cooperative binding and proposed that the ultrasensitivity was due to 510 tf molecules simultaneously contacting the transcription machinery .
we find it sterically implausible that the transcription machinery has so many sites simultaneously available to adjacently bound vp16 moities , while the results are easily explained by vp16 at each saturated binding site recruiting additional hat activity , adding to . conditions in the model that give high ultrasensitivity also produce bistability .
the lower panels of figure 3 show the probability distributions of the fraction of a nucleosomes at various . in the high ultrasensitivity case ( left panel ) , there are some  values for which the promoter can exist with high probability in either a high- or a low - a state
. such bistability can not arise from cooperative tf binding , even at high ultrasensitivities , as without positive feedback there can be only one promoter activity for each tf activity .
bistability by our mechanism may help explain observations of bimodal gene expression , where in a population of cells , a promoter can be fully on in some cells and fully off in the others , despite the cells being in the same environment ( fiering et al , 2000 ; rossi et al , 2000 ) .
stable bimodal expression can result from the combination of high ultrasensitivity and differences in tf levels between cells , as long as these differences span the off  on transition region and are reasonably persistent .
in contrast , bimodal expression in our model can occur even when promoters are exposed to identical conditions , also explaining bimodal promoter activity within different nuclei in the one cell ( newlands et al , 1998 ) or even in the same nucleus ( riviere et al , 1998 ) .
we note also that the proposed mechanism provides a straightforward means by which the activities of many tfs can be integrated at the promoter .
if each tf recruits a histone - modifying enzyme that affects the balance between a particular pair of modifications ( i.e. , stimulates one of the four modification reactions ; figure 2 ) , then the activities of these tfs combine to produce a single output .
an analysis of the way in which these signals combine is beyond the scope of this report ; however , our preliminary investigations indicate that the integration is not simply additive and depends on which combination of reactions is stimulated . finally , the evolution of a system consisting of indirectly acting tf factors should be less constrained than for systems requiring direct contact with the transcription complex .
evolution of the promoter to respond to a new tf would simply require the formation of a binding site in the dna within the nucleosome patch .
in contrast , evolution of a new regulation by a tf that acts by recruiting or stabilizing rnap binding is likely to be limited by the stereochemical constraints of linking a dna - binding site and an available contact site on the transcriptional apparatus .
thus , positive - feedback loops in nucleosome modification can , in theory , provide not only a mechanism for long - term epigenetic memory ( dodd et al , 2007 ) , but also a powerful system for controlling the way in which a promoter responds to and integrates multiple signals .
our modelling highlights the importance of further experimental work to substantiate and characterize such feedback loops .
we previously introduced a simplified numerical model of positive feedback in nucleosome modification for investigation of epigenetic cell memory ( dodd et al , 2007 ) .
the model considers a patch of n nucleosomes carrying mutually exclusive modifications , for example , methylation or acetylation of a specific lysine residue ( such as lysine 9 of histone h3 ) , giving three types of nucleosome with regard to this modification : methylated ( m ) , unmodified ( u ) or acetylated ( a ) .
nucleosome interconversions can occur by a positive feedback process , where acetylated nucleosomes recruit specific hdms and hats , while methylated nucleosomes recruit hdacs and hmts that can act on any other nucleosome in the patch ( figure 2 ) . as well as
these local positive feedback reactions , non - feedback or  noisy ' interconversions can occur independently of the local nucleosomes , for example , through the action of non - recruited enzymes or enzymes bound to nucleosomes outside the patch .
the relative strengths of the feedback and noise processes is parameterized by  ( figure 2 ) , with the feedback - to - noise ratio , f defined as f=(1 ) .
we found that with increasing n and f , the patch of nucleosomes becomes increasingly bistable , existing in either a highly acetylated or a highly methylated state with rare transitions between the states .
bistability also required the positive feedback reactions leading to one modification to be of similar efficiency to those leading to the opposing modification ( dodd et al , 2007 ) . here , we make two changes to this model to investigate gene activation or repression by a tf that recruits a histone - modifying enzyme .
first , we allow the relative strengths of the positive feedback reactions in one direction ( e.g. towards a ) to be of a different strength to the reactions in the other direction ( e.g. towards m ) .
the balance of these reactions is expressed by the parameter  ( figure 2 ) . if  deviates significantly from 0.5 , indicating an imbalance , then the patch of nucleosomes is only stable in one modification state ( dodd et al , 2007 ) .
we imagine that for any particular mutually exclusive modification ( e.g. the m and a in figure 2 ) , the concentrations of modifying enzymes , their affinities for the recruiting nucleosomes and their enzymatic activities will often differ so that one modification will be favoured , possibly in a location - dependent manner , so that a particular patch of nucleosomes will tend to exist by default in a particular modification state .
second , we add a tf that binds to the dna within the nucleosome patch and recruits a histone - modifying enzyme , bringing the enzyme into proximity with the nucleosomes in the patch and facilitating their modification ( figure 1b ) .
we do not consider any other action of the tf ( e.g. direct interaction with the transcription complex ) .
the strength of the tf - stimulated reaction is given by  , which sets an additional rate for the u to a transition ; a larger  makes it proportionally more likely to change a u to an a anywhere in the system ( figure 2 ) .
we imagine that increases in  result from increases in the activity or concentration of the tf that , in turn , increase occupation of its dna - binding site or its ability to recruit the modifying enzyme .
in figure 3 , we show the results of iterating the reaction steps of figure 2 when , by default , the patch contains a low proportion of a - type nucleosomes due to an imbalance in the feedback reactions due to =2/3 ( i.e. , m - favouring reactions are twice as likely as a - favouring reactions ) .
as the strength of the tf - stimulated hat reaction is increased (  increased from 0 to 1 ) , the average steady - state fraction of acetylated nucleosomes , a , increases to the point where a nucleosomes dominate the patch . depending on the parameters ,
the model is able to give a range of ultrasensitive responses of a to increases in  ( figure 3 , upper panels ) , as quantified by the hill coefficient h , with h>1 indicating ultrasensitivity ( koshland et al , 1982 ; ferrell , 1996 ) .
if promoter activity is taken to be proportional to a and  is taken to be proportional to tf activity , such that the only source of non - linearity in the system is the nucleosome modification reactions , then it is clear that this mechanism alone can generate an ultrasensitive response of promoter activity to tf activity .
when there is a high fraction of one nucleosome type , increases in the minority modification due to  are strongly resisted by the dominating activity of the majority type . however , as the threshold ( ma ) is approached , the resistance decreases , because the opposing feedback reactions are more balanced , allowing changes in a to occur easily . the strength of the positive feedback increases with f and n and thus these parameters control the ultrasensitivity ( figure 3 , top panels ) . in principle
, this could allow the ultrasensitivity of the response of a promoter to a tf to be tuned by adjusting the strength of the positive - feedback modifications relative to noise ( f ) or by alterations to the size of the nucleosome patch ( n ) .
although figure 3 shows activation by recruitment of a modifying enzyme that adds a modification ( figure 2 ) , recruitment of an enzyme that removes the opposing modification produces a similar response .
if the modification favoured by the tf has an inhibitory effect on the promoter , then an ultrasensitive repressive response can be produced .
rossi et al ( 2000 ) found hill coefficients ( h ) of 1.63.2 for doxycycline regulation of promoters activated by tet repressor 
vp16 fusions , while ajo - franklin et al ( 2007 ) found h values of 2.73.6 for promoter control by vp16-derived activation domains fused to zifh or lexa dna - binding domains . because multiple adjacent activator dna - binding sites were used , it is possible ( but was not tested ) that the ultrasensitivities were caused by cooperative dna binding .
however , the responses may also be explained by our model , as the vp16 activation domain has been shown to interact with hats , and when fused to the gal4 dna - binding domain can target hat activity to nucleosomes around the gal4-binding site ( vignali et al , 2000 ) . in these experiments ,
the two models could be tested by measuring ultrasensitivity with the same number of activator binding sites but placed at separate locations , which should disrupt cooperative binding , but not our mechanism .
the model can also explain the results of carey et al ( 1990 ) , in which an ultrasensitive promoter response was seen with respect to increasing copies ( in tandem ) of a binding site for a gal4vp16 activator . because all the binding sites were saturated with tf , carey and co - workers ruled out cooperative binding and proposed that the ultrasensitivity was due to 510 tf molecules simultaneously contacting the transcription machinery .
we find it sterically implausible that the transcription machinery has so many sites simultaneously available to adjacently bound vp16 moities , while the results are easily explained by vp16 at each saturated binding site recruiting additional hat activity , adding to . conditions in the model that give high ultrasensitivity also produce bistability .
the lower panels of figure 3 show the probability distributions of the fraction of a nucleosomes at various . in the high ultrasensitivity case ( left panel ) , there are some  values for which the promoter can exist with high probability in either a high- or a low - a state .
such bistability can not arise from cooperative tf binding , even at high ultrasensitivities , as without positive feedback there can be only one promoter activity for each tf activity .
bistability by our mechanism may help explain observations of bimodal gene expression , where in a population of cells , a promoter can be fully on in some cells and fully off in the others , despite the cells being in the same environment ( fiering et al , 2000 ; rossi et al , 2000 ) .
stable bimodal expression can result from the combination of high ultrasensitivity and differences in tf levels between cells , as long as these differences span the off  on transition region and are reasonably persistent .
in contrast , bimodal expression in our model can occur even when promoters are exposed to identical conditions , also explaining bimodal promoter activity within different nuclei in the one cell ( newlands et al , 1998 ) or even in the same nucleus ( riviere et al , 1998 ) .
we note also that the proposed mechanism provides a straightforward means by which the activities of many tfs can be integrated at the promoter .
if each tf recruits a histone - modifying enzyme that affects the balance between a particular pair of modifications ( i.e. , stimulates one of the four modification reactions ; figure 2 ) , then the activities of these tfs combine to produce a single output .
an analysis of the way in which these signals combine is beyond the scope of this report ; however , our preliminary investigations indicate that the integration is not simply additive and depends on which combination of reactions is stimulated . finally , the evolution of a system consisting of indirectly acting tf factors should be less constrained than for systems requiring direct contact with the transcription complex .
evolution of the promoter to respond to a new tf would simply require the formation of a binding site in the dna within the nucleosome patch .
in contrast , evolution of a new regulation by a tf that acts by recruiting or stabilizing rnap binding is likely to be limited by the stereochemical constraints of linking a dna - binding site and an available contact site on the transcriptional apparatus .
thus , positive - feedback loops in nucleosome modification can , in theory , provide not only a mechanism for long - term epigenetic memory ( dodd et al , 2007 ) , but also a powerful system for controlling the way in which a promoter responds to and integrates multiple signals .
our modelling highlights the importance of further experimental work to substantiate and characterize such feedback loops .
the model outlined in figure 2 is simulated as an ongoing sequence of events ( time steps ) that take place in a system consisting of n nucleosomes ( sites ) .
each site can be in one of the three states : m , u or a. there are no intermediates between these states ; we disregard nucleosome heteromodification ( possible as any nucleosome has two copies of any specific histone residue ) .
furthermore , we only allow transitions between a and u , and u and m , that is , no direct transitions between a and m. furthermore , in the standard model , we disregard any positional effects , implying that the state of the system is completely characterized by the fraction of m sites ( m ) and the fraction of a sites ( a ) . given m and a the fraction of u sites is u = nam .
the dynamics of the system is determined by two intrinsic parameters , f=(1)>0 and   plus one external parameter  that quantifies the strength of the tf .
first , we consider the tf - independent move , extending the previous model ( dodd et al , 2007 ) to take into account asymmetry between the strength of the recruited enzymes facilitating more ms and the one facilitating more as .
this asymmetry is large when the parameter  is close to 0 or 1 . 
 at each time step ,
then : one selects a random number r from a uniform distribution over the interval . if r< , one attempts a move that simulates the feedback from the system .
this is done by selecting another random nucleosome n2 from anywhere in the system and letting this attempt to mediate a change in n1 : 
 if n2 is in the m state , one with probability  changes n1 one step towards m. this means that an n1 in the a state makes a transition to the u state , whereas an n1 in the u state makes a transition to the m state .
if n1 was already in the m state , no change is made.if n2 is in the a state , one with probability 1 changes n1 one step towards a. this means that an n1 in the m state makes a transition to the u state , whereas an n1 in the u state makes a transition to the a state .
if n1 was already in the a state , no change is made.if n2 is in the u state , no change is made.if r , one makes a noise move , where the change in n1 is independent of the rest of the system .
when n1 is in the m or a state , it is changed to u with probability 1/2 .
when n1 is in the u state , it is changed to m with probability 1/2 , or else it is changed to the a state . at each time step ,
then : one selects a random number r from a uniform distribution over the interval . if r< , one attempts a move that simulates the feedback from the system
this is done by selecting another random nucleosome n2 from anywhere in the system and letting this attempt to mediate a change in n1 : 
 if n2 is in the m state , one with probability  changes n1 one step towards m. this means that an n1 in the a state makes a transition to the u state , whereas an n1 in the u state makes a transition to the m state . if n1 was already in the m state , no change is made.if n2 is in the a state , one with probability 1 changes n1 one step towards a. this means that an n1 in the m state makes a transition to the u state , whereas an n1 in the u state makes a transition to the a state .
if n1 was already in the a state , no change is made.if n2 is in the u state , no change is made .
if n2 is in the m state , one with probability  changes n1 one step towards m. this means that an n1 in the a state makes a transition to the u state , whereas an n1 in the u state makes a transition to the m state . if n1 was already in the m state , no change is made .
if n2 is in the a state , one with probability 1 changes n1 one step towards a. this means that an n1 in the m state makes a transition to the u state , whereas an n1 in the u state makes a transition to the a state .
if r , one makes a noise move , where the change in n1 is independent of the rest of the system .
when n1 is in the m or a state , it is changed to u with probability 1/2 . when n1 is in the u state , it is changed to m with probability 1/2 , or else it is changed to the a state .
the above description of a single time - step involves some steps that take place with probability <1 and 1<1 . to simulate an event taking place with probability 
, we select a random number x from a uniform distribution between 0 and 1 , and do the change if and only if x<. to simulate the action of a tf that recruits a hat , we at each time - step supplement the above action and : 
 select a random number x. if x< , we randomly select one of the n nucleosomes in the system .
if this nucleosome is in the u state , it is changed to the a state .
select a random number x. if x< , we randomly select one of the n nucleosomes in the system .
if this nucleosome is in the u state , it is changed to the a state .
a small  simulates a situation where the tf is not bound to its operator very often , and therefore , only rarely influences the system .
a large  corresponds to the case where the tf often recruits a hat that makes an attempted acetylation somewhere in the system .
the model is sufficiently iterated over many time - steps to allow the system to explore all its possible states .
 eukaryotic transcription involves the synergistic interaction of many different proteins . 
 however , the question remains how eukaryotic promoters achieve ultrasensitive or threshold responses to changes in the concentration or activity of a single transcription factor ( tf ) . 
 we show theoretically that by recruiting a histone - modifying enzyme , a tf binding non - cooperatively to a single site can change the balance between opposing positive feedback loops in histone modification to produce a large change in gene expression in response to a small change in concentration of the tf . 
 this mechanism can also generate bistable promoter responses , allowing a gene to be on in some cells and off in others , despite the cells being in identical conditions . 
 in addition , the system provides a simple means by which the activities of many tfs could be integrated at a promoter . 
Introduction
Results and discussion
The model
Ultrasensitive responses
Comparison with experimental data
Adaptability to complex regulation
Materials and methods
Supplementary Material