Heat Shock Response in CHO Mammalian Cells Is Controlled by a Nonlinear Stochastic Process
Complex biological systems are built out of a huge number of components ., These components are diverse: DNA sequence elements , mRNA , transcription factors , etc ., The concentration of each component changes over time ., One way to understand the functions of a complex biological system is to construct a quantitative model of the interactions present in the system ., These interactions are usually nonlinear in terms of the concentrations of the components that participate in the interaction process ., For example , the concentration of a dimer is proportional to the product of the concentrations of the molecules that dimerise ., Besides being nonlinear , the interactions are also stochastic ., The production process of a molecule is not deterministic , and it is governed by a probability rate of production ., In what follows , a nonlinear stochastic model for the response to heat shocks in CHO mammalian cells will be developed ., Heat stress is just one example of the many ways a molecular system can be perturbed ., From a general perspective , the structure of a molecular system is uncovered by imposing different perturbations ( input signals ) on the system under study , and then the responses of the system ( output signals ) are measured ., From the experimental collection of pairs of input–output signals , laws that describe the system can be uncovered ., This is the fundamental idea in Systems and Synthetic Biology 1–5 and has long proved to be successful in the field of electronics ., The input signals are applied through the use of signal generators 6–8 ., An input signal that is easy to manipulate is a heat pulse , the parameters to change being the pulse temperature and its duration ., Members of the stress protein family such as the heat shock protein 70 ( HSP70 ) are highly responsive to temperature variations ., This protein is a molecular chaperone and is a critical component of a complex genetic network that enables the organism to respond to deleterious effects of stress 9–11 ., Since Hsp70 is thus an important regulator in a complex system , our goal was to find if it is possible to develop a mathematical model of the regulation of its expression in mammalian cells exposed to heat shock ., Our specific objectives were, 1 ) determine an equation representing the average expression of Hsp70 over time in a cell population after an initial heat shock ,, 2 ) determine how the physical parameters of heat shock ( temperature and duration ) influence the parameters of this equation , and, 3 ) determine the mathematical model that describes the expression of Hsp70 at the single-cell level ., We first describe the process of inferring the mathematical model from the experimental data ., Then a mathematical study of the model will follow ., To acquire the experimental data , we elected to use a system using a reporter gene where the expression of the green fluorescent protein ( GFP ) is under the control of the promoter region of the mouse Hsp70 gene ., The GFP reporter proved useful for quantitative analysis 12 and was used before in connection with Hsp70 in different biological systems 13–17 ., The Hsp70-GFP fusion gene was integrated into a plasmid and transfected in Chinese hamster ovary ( CHO ) cells ., Stable transfectants were selected for their low level of basal expression of GFP and their capacity to upregulate GFP effectively and homogenously after exposure to heat shock ., Flow cytometry was used to make precise quantitative measurements of the fluorescence of a large cell population ., Since the quality of the experimental data was critical to the feasibility of the mathematical analysis , steps were taken to minimize sample-to-sample and experiment-to-experiment variability and to maintain the experimental noise to a minimum ., To that effect , temperature and time were tightly controlled for heat shocks , the cells were treated as a batch in a single tube for each condition ( combination of temperature and time ) , and aliquots were taken at each time point ., All samples were fixed for at least 24 h before analysis by flow cytometry so that changes of fluorescence due to fixation would not be a factor , and all the samples from the same experiment were analyzed at the same time ., Flow cytometry was chosen for analysis because it allows a very accurate quantitative measurement of the fluorescence of a large number of events , independently of the actual size of the sample ., Within the same experiment and between experiments , the same instruments settings were used for the flow cytometer , and at least 1 × 104 cells were analyzed per sample ., Detailed protocols and experimental conditions are available in the Materials and Methods section ., First , we will follow a description of the time course of the mean response to a heat shock ., At elevated temperatures ( 39 °C to 47 °C ) , the heat shock promoter HSP70 is active and GFP starts to be synthesized ., The input signals were chosen in the form of a pulse at a temperature ( T ) and duration in time ( D ) ( Figure 1A ) ., In the first experiment , the dynamic response of GFP after a heat pulse at 42 °C for 30 min was monitored by taking samples each 30 min for 18 h ., Before and immediately after the heat shock , the GFP intensity remains at approximately the same level; this phenomenon was observed in all subsequent experiments ., The fold induction of GFP with respect to a reference ( GFP0 ) was then determined:, The reference is the first measured sample away from the end of the heat shock ( 30 min after the shock in Figure 1A ) ., Our finding is that the logarithm of the fold induction of GFP follows an exponential saturation trajectory ( Figure 1B ) , with tight confidence bounds for the estimated parameters and tight prediction bounds for nonsimultaneous observations ., The tight prediction bounds appear even when almost half of the data is not used during fitting ( Figure 1B ) ., The time t is measured relative to the reference time t0 . The initial fold induction at t = 0 ( or equivalent t0 after the end of heat shock ) is 1 ., This value of 1 for the initial fold induction is consistent with the entire time evolution if a fit with the expression, will give a value for parameter, very close to the value for parameter a ., Theoretically ,, must be equal with a to have a fold induction of 1 at t = 0 ., The result of the fit ( Figure 1B ) shows this consistency ., From now on we will take, ., The empirical law for the response of the cells to the heat pulse can be thus cast into the form:, The same law appeared in repeated measurements of pulses at 42 °C for 30 min duration ( unpublished data ) ., Parameter b describes the quickness of the response ., As b increases , the saturation value of the response is reached in less time ., Parameter a specifies the saturation value of the response ., The plateau reached by the fold induction is ea and thus grows exponentially with parameter a ., These findings suggest that the same law is valid for other heat shock pulses , parameters a and b being dependent on the heat pulse height T and its duration D ( Figure 1C ) ., To find the range of validity for the empirical law , measurements were taken for the responses to heat shocks at various heat pulse parameters T and D in a series of three experiments that partially overlapped Figure 2 ., The law was again present in all responses for temperatures between 41 . 5 °C and 42 . 5 °C , ( examples selected in Figure 3A , fit 3 , 4 ) ., For lower temperature ( 39 . 5 °C to 40 . 5 °C ) , the law was valid , but with poor 95% confidence intervals for estimated parameters a and b , as in Figure 3A , fit 1 , 2 ( the activity of the Hsp70 promoter was low ) ., At high temperatures or long durations ( Figure 3B ) , the double exponential law still explains the main characteristic of the stress response and is valid after a few hours from the end of the heat shock ., In the following , a theoretical model will be developed to explain the experimentally discovered law ., The exponential accumulation of the GFP shows that the derivative with respect to time of the mean GFP is proportional with itself:, There must be thus a molecular process , described by the exponential term abe−bt , which controls the heat shock response ., This theoretical suggestion is confirmed by previous studies of the heat shock system which revealed that the accumulation and subsequent degradation of the heat shock transcription factor 1 ( HSF1 ) regulates Hsp70 18–22 ., Experimental results 18 show that HSF1 activation is characterized by a rapid and transient increase in hsp70 transcription which parallels the kinetics of HSF1–DNA binding and inducible phosphorylation ., This rapid increase in HSF1–DNA binding activity reaches a maximal level and thereafter attenuates to a low level ., This rapid increase in activity followed by attenuation will form the starting point for our theoretical model ., An activation–accumulation two-component model will be developed as a minimal theoretical description of the empirical law ., The “activation” variable ( X1 ) represents the first phase of the heat shock response and includes components like HSF1–DNA binding activity ., X1 will increase during the duration of the heat shock and then , after the shock , will decrease with a lifetime proportional to parameter b ( Figure 4 ) ., The “accumulation” variable ( X2 ) includes the products of transcription and translation ., This second variable , at low levels before the shock , will gain momentum after the shock ., To connect the model with the experimental data , the GFP will be considered to be proportional with X2 ., The speed of accumulation of X2 , that is , dX2/dt , will be proportional to the product X1X2 ., Immediately after the shock , X1 has a big value ( the activation is high ) , and thus the speed of X2 is high ( the accumulation is in full thrust ) ., This will trigger an initial fast accumulation of GFP , which is proportional with X2 ., Later on , the activity X1 disappears , nullifying the product X1X2 and thus the speed of X2 ., The process is then terminated ( the accumulation stops ) ( Figure 4 ) ., The empirical law follows directly as a solution of the activation–accumulation system of equations:, with b and c as some constants ., Indeed , given the initial conditions X1 ( 0 ) and X2 ( 0 ) at a zero time reference t0 = 0 , the solution to this system of differential equations is, With the notation, the empirical law follows from X2 ( t ) :, The theoretical model contains two parameters: b and c ., Parameter b is directly accessible to experimental measurements , whereas parameter c is not; however , the product cX1 ( 0 ) which equals the product of a and b can be measured ., It is interesting to notice that the above time evolution can be re-expressed as a conservation law which is independent of any reference time ., For any two time points t1 and t2 , the following holds, At this point , there is no more information in the activation–accumulation description above than is in the empirical law ., However , one can search for more information hidden in the above two-component description by turning attention to the full data available , not only to the mean value of GFP ., For each sampled time , the full data available consists of measured GFP levels for at least 10 , 000 single cells ., These 10 , 000 single-cell measurements are typically distributed as in Figure 5 . There is a long tail at high values of GFP ., This biological variation in response to the stress is explained by turning the deterministic two-component system into a stochastic two-component system 6 , 7 ., The stochastic description must be completely enforced by ideas behind the deterministic two-component system ., The stochastic model is simple ., X1 is the mean value of a stochastic activation variable which will be denoted by q1 , X1 = 〈q1〉 ., After the heat shock , q1 will decrease with a probabilistic transition rate bq1 ., The activation–accumulation stochastic model is based on the same relation as before ( compare bq1 with bX1 ) , but now it describes the probabilistic transition rate and not a deterministic speed of attenuation ., By the same token , X2 is the mean value of q2 and its probabilistic accumulation rate is cq1q2 ., One notices that the transition probability rate cq1q2 is nonlinear in the variables q1 and q2 ., The stochastic two-component description is thus a mirror image of the deterministic two-component system ., However , the probabilistic system is more powerful as it predicts that the histograms of GFP ( proportional with q2 ) obtained from the flow cytometry measurements follow a Gamma distribution, with GFP ≡ x ., This prediction is confirmed experimentally ( Figure 5 ) ., The fact that the levels of proteins in gene networks tend to follow a Gamma distribution , which is a continuum version of a discrete negative-binomial distribution , was presented in 23 , 24 ., The papers 23 , 24 develop theoretical models describing the steady-state distribution of protein concentration in live cells ., Our interest lies in the non–steady-state behavior of these distributions ., Namely , the aim is to find the time evolution of the parameters that characterize these distributions ., The entire time evolution of the distributions is presented in Figure 6 . The distributions become wider as time passes ., The experimental data reveal that parameter ρ remains constant in time and only θ changes ., These experimental findings are theoretically explained in detail in the section Analysis of the Theoretical Model ., What follows summarizes the theoretical conclusions that are useful in understanding the experimental results of Figures 5 and 6 . The probability distribution for the discrete molecule number q2 , predicted by the stochastic activation–accumulation model , is the negative-binomial distribution ., This distribution appeared in earlier theoretical studies of genetic networks 23 , 24 and in physics 25 , 26 ., The GFP intensity is proportional with q2 and appears in measurements as a decimal number and not as a pure integer ., Thus , to describe the probability distribution of the GFP intensity , a continuous version of the discrete negative-binomial distribution is necessary ., This continuous version is the Gamma distribution observed experimentally in Figures 5 and 6 . The physical interpretation of parameter ρ will now be discussed ., At initial time t0 , immediately after the heat shock , there will be at least one cell from the entire cell population which contains the minimum number of molecules q2 ., Denote this number by N0 ., As the time passes , the molecule number q2 will grow , following the described stochastic process ., However , there is a nonzero probability , though extremely small , that the process of accumulation in one cell does not start even after 24 h ., This can happen in one of those cells that contain the minimum number of molecules q2 at the initial time t0 ., Thus , at any later time t > t0 , the lowest possible number of molecules q2 in a cell is N0 as it was at the initial time t0 ., It can be shown ( see the section Analysis of the Theoretical Model ) that ρ = N0 ., This explains the time independence of the experimental values of ρ; it also gives a physical meaning to ρ as being proportional to the minimum number of GFP molecules in a cell ., Parameter θ contains the time evolution of the stochastic accumulation of the GFP molecules ., This evolution can be again expressed as a time conservation property, valid between any two time points t1 and t2 ., The above relation Equation 10 contains parameters a and b and can be used to check the consistency of the model ., Using the data from Figure 6 , it follows that a = 3 . 159 with a 95% confidence interval ( 3 . 074 , 3 . 244 ) and b = 0 . 2572 with a 95% confidence interval ( 0 . 2358 , 0 . 2785 ) ., From the mean value for GFP , it results that a = 2 . 423 with a 95% confidence interval ( 2 . 351 , 2 . 496 ) and b = 0 . 2579 with a 95% confidence interval ( 0 . 2344 , 0 . 2814 ) ., Parameter a is sensitive to the estimation procedure , a phenomenon connected with the fact that parameter ρ is not perfectly constant but decreases a bit with time ., The mean value of the Gamma distribution is ρθ ., For a perfectly constant ρ , the estimated value for a would be the same using either the θ values or the ρθ data ., Contrary to parameter a , parameter b is independent of the way it is estimated , and the estimation is highly reliable ., To further check the reality of the Gamma distribution for heat shock response , a comparison of the Gamma fit with the lognormal fit is presented in Figure 7 . The lognormal was chosen because it can be viewed as a result of many random multiplicative biological processes ., A loglikelihood ratio less than 1 favors the Gamma distribution against the lognormal ., Moreover , at 37 °C the Gamma distribution is not a good fit ( loglikelihood ratio is bigger than 1 ) as it should be because the promoter is not active ., The law, is useful in making predictions for the fold induction to many other heat shock pulses ., For a heat pulse of a given temperature and duration , parameters a and b can be read out from Figure 8 . The constant level contours were inferred from the experimental data ., The level patterns differ; parameter a increases monotonically with the temperature and duration of the heat pulse ( Figure 8A ) , while the levels of parameter b form an unstable saddle shape pattern ( Figure 8B ) ., The conclusion of this section will be rephrased using a control theory perspective ., The end result of this paper is an input–output relation for the response of the CHO cells to heat shocks , together with a theoretical model that explains it ., The input signals are pulses of a precise time duration D and temperature height T . The output measured signals are the GFP intensity ., The input–output relation is given by the time-dependent probability density for GFP intensity, with, Parameters a and b are functions of the input signal , that is a = a ( T , D ) and b = b ( T , D ) ., The dependence of parameters a and b on temperature T and duration D is given by the contour plots of Figure 8 . The functional forms of a = a ( T , D ) and b = b ( T , D ) is a consequence of biological phenomena that take place during the heat shock ., We do not have a theoretical model for the phenomena that take place during the heat shock ., To explain the time evolution of the output variable ( GFP intensity ) , we developed a coarse-grained model for the heat shock response ., This coarse-grained model is valid for the biological phenomena that takes place after the end of the heat shock ., The model predicts the existence of a molecular factor that controls the GFP accumulation ( variable q1 ) ., We associated this theoretical factor with the heat shock factor HSF1-DNA binding activity ., The theoretical model is based on an activation variable q1 and an accumulation variable q2 ., The state of this two-component model is thus ( q1 , q2 ) , and any pair of positive integer numbers can be a possible state ., The main goal is to find the mean value and standard deviation for the activation and accumulation variable , respectively ., These quantities will be obtained from the equation for the probability P ( q1 , q2 , t ) that the system is in the state ( q1 , q2 ) at the time t ., The equation for P ( q1 , q2 , t ) depends on the multitude of transitions which can change a state ( q1 , q2 ) ., The experimental results suggest that two possible transitions change the state ( q1 , q2 ) ., One transition represents the decreasing of the activation variable from q1 to q1 − 1 ., On the state ( q1 , q2 ) , this attenuation appears as ( q1 , q2 ) → ( q1 − 1 , q2 ) , with an unaffected accumulation variable q2 ., The second transition will describe the accumulation of the accumulation variable from q2 to q2 + 1 ., On the state ( q1 , q2 ) , this accumulation appears as ( q1 , q2 ) → ( q1 , q2 + 1 ) , with the activation variable q1 now being unaffected ., A notation for the transition direction can be introduced: ɛ−1 = ( −1 , 0 ) ., The degradation transition can thus be written as ( q1 , q2 ) → ( q1 , q2 ) + ɛ−1 ., The negative sign in the index −1 is just a reminder of the fact that the transition reduces the number of molecules; the 1 in the subscript tells us that the transition is on the first variable ., Likewise , the accumulation transition can be expressed as ( q1 , q2 ) → ( q1 , q2 ) + ɛ2 and ɛ2 = ( 0 , 1 ) ., The index 2 is positive ( accumulation ) and is associated with the second component ., To find the probability P ( q1 , q2 , t ) , the transition probabilities per unit time are needed ., The experiment suggests we use, as the transition probability rate for the attenuation of the activation component , and, as the transition probability rate for the increasing of the accumulation component ., The stochastic model can be represented with the help of a molecular diagram 7 ( Figure 9 ) ., The components q1 and q2 are represented by ovals and the transitions by squares ., The lines that start from the center of a transition square represent the sign of that transition and point to the component on which the transition acts ., The transition ɛ−1 is negative , so the line ends in a bar and acts on q1 ., The transition ɛ2 is positive and so the line ends with an arrow; it acts on q2 ., The lines that stop on the edges of the transition squares represent the transition probability rates ., The line that starts from q1 and ends on ɛ−1 represents the transition probability rate bq1 ., In other words , the transition ɛ−1 is controlled by q1 ., The lines that start on q1 and q2 and merge together to end on ɛ2 represent the product cq1q2 , ( the merging point represents the mathematical operation of taking the product ) ., At this point , the theoretical model is fixed and what comes next is a sequence of computations to extract information out of it ., This information will be compared with the experimental results ., Given the transition rates , the equation for the probability P ( q1 , q2 , t ) is given by the following equation 7 , 25 ., The above equation for P ( q1 , q2 , t ) is not easy to solve ., We will use the method outlined in 6 , 7 and work with the function X ( z1 , z2 , t ) defined by, The equation for the function X ( z1 , z2 , t ) is a consequence of the equation for P ( q1 , q2 , t ) :, The goal is to find the time variation of the mean value and standard deviation for the activation and accumulation variable: 〈q1〉 , 〈q2〉 ,, ,, , 〈q1 , q2〉 , etc ., Here 〈〉 is a notation for the mean value with respect to the probability distribution P ( q1 , q2 , t ) ., From X ( z1 , z2 , t ) , the above mean values can be obtained by taking partial derivatives of X ( z1 , z2 , t ) at z1 =1 , z2 = 1 ., These partial derivatives are actually the factorial cumulants of the probability distribution P ( q1 , q2 , t ) ., In what follows , the sign =: means that the right side is introduced as a notation ., The equations for X1 ( t ) , X2 ( t ) , X11 ( t ) , and higher factorial cumulants result from the equation for X ( z1 , z2 , t ) :, The activation–accumulation model being nonlinear , the equations for the factorial cumulants cannot be reduced to a finite system of equations , unless some approximation technique is employed ., All third-order cumulants were discarded to obtain the above system of equations ., In 7 it was shown , using simulations , that the effect of discarding higher-order factorial cumulants is negligible ., The finite system thus obtained contains X1 , X2 , X12 , X11 , and X22 as variables ., Although it can be solved for X1 and X2 , we found that the influence of the correlation term X12 is small and cannot be experimentally detected in the GFP response ., Taken thus , X12 = 0 , and the system of equations is reduced to:, The solution to X22 from the four-equation system is, with k a constant determined from the initial value X2 ( t0 ) at some time t0 after the heat shock ., The solution can be restated in terms of the variance , Var , of the variable q2 ., The transformation from the factorial cumulants to Var is, And , thus , remembering that the mean value of q2 is X22 , it follows that, Such a relation between Var and Mean is satisfied by the negative-binomial distribution , a point to which we will return later ., Employing the general procedure , we continue to solve the system of equations for X1 , X2 , X11 , and X22 ., However , for the case of negligible X12 , the stochastic process is decoupled in two stochastic processes , each of which is exactly solvable ., It is thus useful to solve directly for the probability distribution of q2 at this point ., The transition probability rate for the first stochastic process ( for the activation component q1 ) is the same as before:, ., For the second one , it changes from, to, ( the coupling between q1 and q2 is through the mean value of q1 now ) ., This simplifies the problem of finding the distribution of q2 ., Denote the mean value of cq1 with g ( t ) , which acts actually like a signal generator on q2 6 , 7 ., The time variation of g ( t ) from the first equation in Equation 20 is, so the stochastic process for q2 now has an accumulation transition rate, The origin of time , t = 0 , is taken at the end of the heat shock , so X1 ( 0 ) represents the mean value of the activation variable at the end of the heat shock ., The probability P ( q2 , t ) to have q2 number of molecules at time t can be found from the master equation for this process, To find the solution , an initial condition P ( q2 , t0 ) must be specified ., The time t = t0 is some time taken after the heat shock pulse ( t0 > 0 ) , when the effects of the shock start to be detectable; it can be , for example , 30 min or 2 h after the pulse ., The probability distribution P ( q2 , t0 ) can be obtained , in principle , from the experimental values of GFP since GFP = fq2 ., There is an obstacle though: the proportionality factor f is unknown ., The factor f converts the number of molecules q2 into the laser intensity which is the output of the flow cytometry machine ., The conversion from the molecule numbers to the laser intensity can be more complicated than the proportionality relation GFP = fq2 ., For example , a background B can change the relation into: GFP = fq2 + B . We measured the GFP in regular CHO cells ( no HFP70-GFP construct ) and found that the background B is about 50 times less than the minimum intensity of GFP in the transfected CHO cells ., The settings of the flow cytometry instrument were set in a linear response range , and thus we will use the scaling relation GFP = fq2 to connect the flow cytometry readings with the number of molecules ., To conclude this initial condition discussion , in a perfect setting we would know the scaling factor f and then get P ( q2 , t0 ) from the measured data ., Because the scaling factor f is unknown , the problem will be solved in two steps ., The first step in choosing P ( q2 , t0 ) is based on a simple assumption: all cells have the same number of molecules q2 = N at the time t = t0 ., That is P ( q2 , t0 ) = δ ( q2 , N ) where δ is the Kronecker delta function ., The solution to Equation 26 with this initial condition is, Here q2 can take only values greater than N , q2 = N , N + 1 , ··· ., This distribution appeared in the study of cosmic rays 26 , and in the context of protein production was presented in 23 ., In terms of the variable x = q2 − N , it is known as the negative-binomial distribution , with interpretations that are not connected with the present problem ., The number N also represents the minimum possible number of molecules q2 in any cell ., This physical interpretation of N will be helpful in what follows ., The variable p ( t ) in the distribution is time-dependent , since the signal generator g ( t ) acts on q2:, The mean and variance for q2 are, from which follows Equation 23, Although the assumption that all the cells contain the same number of molecules at t = t0 is unreal , it produces a valuable outcome ., The negative-binomial distribution implies a Gamma distribution for the GFP intensity ( through the scaling relation GFP = fq2 ) , a fact to be discussed shortly ., Because the Gamma distribution is a good fit for the experimental data , we conclude that the negative-binomial is the correct solution for the distribution of the accumulation variable q2 ., The second step in choosing the probability distribution P ( q2 , t0 ) will be guided by the experimental results ., The experimental results show that the biological system passes through a chain of events from an unknown distribution of GFP before the heat shock , to a Gamma distribution at some time t0 after the heat shock ( 2 h , for example ) ., Also , the experiment shows that the distribution of GFP is Gamma at later times t > t0 ., In other words , the distribution of q2 becomes a negative-binomial at some time t0 after the heat shock and then afterward remains negative-binomial ., These experimental observations are mathematically explained by showing that a solution to Equation 26 with a negative-binomial distribution at t0 remains negative-binomial for all later times t > t0 ., Indeed , the solution to Equation 26 with a negative-binomial initial condition, is, which is a negative-binomial at all times t > t0 ., The number N0 is the minimum number of molecules q2 to be found in a cell at t0 and also at all later times t > t0 ( because q2 cannot decrease ) ., The time evolution of the mean 〈q2〉 is, and represents , using Equation 24 , the same empirical law ( Equation 8 ) as before ., To conclude , the dynamical system is such that once the cells enter into a negative-binomial distribution at some time after the heat shock , the distribution remains negative-binomial at later times ., As the time passes , all the distributions will have the same parameter N0 but different parameters p ( t ) ., To connect the theory with the experimental results , the probability distribution for the GFP intensity is needed ., This distribution is the continuum limit of the distribution for q2 ., It is a well-known fact that the continuum limit of a negative-binomial distribution is the Gamma distribution ., This continuum limit is presented here in order to find parameters ρ and θ , which can be experimentally measured ., The change from the integer variable q2 to the real variable fq2 is simple if advantage is taken of the fact that the common parameter N0 is a small number ., Parameter N0 is less than any possible molecule number q2 present in the system after the time t0 , q2 ≫ N0 ., Then , writing for simplicity p ( t ) as p ,, In the last step , we used the approximation 1 − y ≅ e−y for small values of y ., To go from the discrete variable q2 to the continuous variable GFP , we write the above relation as an equation for the probability density, with Δq2 = 1; then scale to GFP , ( GFP = f q2 ) ., The probability density, P℘ for GFP is then, This is a Gamma distribution for GFP ≡ x, with, From Equation 24 and 28 , we get, The mean value of the Gamma distribution is ρθ from which the empirical law Equation 8 follows ., The way the material is organized and presented in this paper is an outcome of a series of guiding principles imposed upon the project ., These guiding principles were formulated to keep in balance the experimental data with both the mathematical and biological models ., The guiding principles are:, 1 ) start from experimental measurements and discover an empirical law from data using signal generators as input into the system;, 2 ) build a simple mathematical model with as few parameters as possible to explain the empirical law;, 3 ) check the mathematical model using additional experimental information;, 4 ) use a general math
In many biological systems , the interactions that describe the coupling between different units in a genetic network are nonlinear and stochastic ., We study the interplay between stochasticity and nonlinearity using the responses of Chinese hamster ovary ( CHO ) mammalian cells to different temperature shocks ., The experimental data show that the mean value response of a cell population can be described by a mathematical expression ( empirical law ) which is valid for a large range of heat shock conditions ., A nonlinear stochastic theoretical model was developed that explains the empirical law for the mean response ., Moreover , the theoretical model predicts a specific biological probability distribution of responses for a cell population ., The prediction was experimentally confirmed by measurements at the single-cell level ., The computational approach can be used to study other nonlinear stochastic biological phenomena .
The structure of an unknown biological system is uncovered by experimentally perturbing the system with a series of input signals ., The response to these perturbations is measured as output signals ., Then , the mathematical relation between the input and the output signals constitutes a model for the system ., As a result , a classification of biological molecular networks can be devised using their input–output functional relation ., This article studies the input–output functional form for the response to heat shocks in mammalian cells ., The Chinese hamster ovary ( CHO ) mammalian cells were perturbed with a series of heat pulses of precise duration and temperature ., The experimental data , taken at the single-cell level , revealed a simple and precise mathematical law for the time evolution of the heat shock response ., Parameters of the mathematical law can be experimentally measured and can be used by heat shock biologists to classify the heat shock response in different experimental conditions ., Since the response to heat shock is the outcome of a transcriptional factor control , it is highly probable that the empirical law is valid for other biological systems ., The mathematical model explains not only the mean value of the response but also the time evolution of its probability distribution in a cell population .
mus (mouse), computational biology