Paper ames

Ames test: a dose response approach Paolo Repeto Page 1 of 11 Table of Contents 1. Introduction ......................................................................................................................................... 3 2. Background .......................................................................................................................................... 3 3. Experimental design ............................................................................................................................ 3 4. Statistical Analysis................................................................................................................................ 4 4.1 4.2 Analysis of variance ............................................................................................................................. 4 Trend analysis ...................................................................................................................................... 4 5. Examples .............................................................................................................................................. 6 6. Parameter estimation ........................................................................................................................ 10 7. Conclusions ........................................................................................................................................ 11 8. References ......................................................................................................................................... 11 Page 2 of 11 1. Introduction The paper deals with different approaches one can adopt to analyze data from an Ames test. The main approaches are basically two: doses comparisons vs control group and dose response analysis. The first approach is just mentioned. The second one is described in details. Before tackling the subject, an introduction of the scientific background is presented 2. Background The Ames test traces back to Bruce Ames and his group in the early 1970s at the University of California, Berkeley. The Ames test is a widespread methodology to test the mutagenic potential of a test agent. It is a short-term bacterial test used routinely to screen industrial and agricultural chemicals. The test is a cheap and fast method and uses different strains of bacteria. The Ames test uses strains of bacteria, which are mutant and loss the ability to synthesize a certain amino acid (histidine). The histidine deficient strains are denoted as histidine minus (his-). These are auxotrophic mutants which loss the ability to synthesize histidine. In other words, these strains cannot grow without histidine. The main idea is that a chemical mutagen might cause a mutation in the histidine encoding gene and changes the his- gene to a his+ gene. Due to the mutation this bacterial strain can produce its own histidine to grow on plate. That means, the Ames test methodology is based on the detection of reversion-rates due to gene (point) mutations caused by the test compound in vitro. Quantifies the capability of the test chemical to induce mutations resulting in a reversion to a prototrophic status. In general, there are different strains to test for different kinds of mutations (base-substitutions, frameshift mutations,...). More revertants on the plates treated with the test chemical than in the control plates indicate a possible mutagenic effect, and further tests should follow. However some chemicals are non-mutagenic, but become mutagenic during the body metabolism. So a rat liver extract (S9 mix ) is added to mimic metabolism processes. In general, the aim is to compare data of different dose groups with vehicle data per each strain separately. 3. Experimental design For each treatment group (dose ) this experimental setting is performed Soft agar Petri dish with solid agar Revertants Treatment Strain of bacteria S9 mix or buffer mix Incubation at 37° C for 48 h – 72 h Preincubation at 37° C for 20 min Page 3 of 11 Often are used auxotrophic histidine-deficient mutants of different strains of Salmonella typhimurium (e.g. TA 1535, TA 100, TA 1537, TA 98, TA 102). For each strain, the reversion-rate back to prototrophy is evaluated in a     4. • negative control, • ≥ 3 concentrations of the test substance (often 6-8 different doses), • positive control. There are usually 3 petri dishes per group. The tests are performed both without and with an S9 mix. The treatment duration takes (usually) 48 hours (up to 72 hours). Colonies are counted using an automatic colony counter. Statistical Analysis Two kinds of data analysis can be performed on the number of revertants: 1. Analysis of Variance 2. Trend analysis 4.1 Analysis of variance  Dose comparison : each dose can be compared with negative control and with the other ones     4.2 all pairwise comparisons with Bonferroni correction Dunnett’s test in case of all comparisons vs negative control William’s test in case of all comparisons vs negative control in the presence of a monotonic trend Non parametric analysis Trend analysis Doses comparisons considers each dose effect separately. Sometime may be interesting to assess the presence of a dose response trend. The shape of the dose response curve can provide information about the mutagenic effect of the compound. In particular the presence of a downturn at higher doses can be a signal that a toxicity effect alongside with a mutagenic effect can affect the response. In particular if the higher dose induces a response more than two times that of negative control a quadratic model is applied. The result of a mutagenic effect should be an increasing dose response . The simplest function describing an increasing dose response relationship is the linear one given by y=0+1x , where y is the number of revertants , x the dose of the test compound, 1 represents the mutagenic effect, whereas 0 is the spontaneous number of revertants due to control group. Since the presence of a departure from linearity may be the symptom of a toxic effect, to assess for this departure the following model could be used y=0+1x+ 2x , with 2 as the 2 departure from linearity. Starting from this baseline the following strategy may be implemented: Page 4 of 11 1) Using the model y=0+1x+ 2x to assess the departure from linearity 2 2) If the coefficient 2 is not statistically significant ( p>=0.05) then all data are used to fit the linear trend by the model y=0+1x and determine the mutagenic effect given by . 3) If 2 is statistically significant (p<0.05) remove the higher dose and fit y=0+1x+ 2x again with the dose 2 removed. If 2 is still statistically significant remove the higher dose, otherwise fit the linear trend as previously. 4) Repeat step 3 while 2 is statistically significant and the number of remaining dose is >=3. 5) If the number of remaining dose is 3, stop the procedure and fit the model y=0+1x 2 times negative control 2 times negative control 0 18 24 36 2 times negative control 0 18 24 36 48 60 48 60 Dose µL 2 times negative control Dose µL 0 18 24 36 48 60 Dose µL Instead of removing the doses which generate a departure from linearity , one can include the toxic effect into a model in order to describe mutagenicity and toxicity as well. The main idea is to have a model of the form y=M(x)*T(x), where M(x) is an increasing function describing mutagenic effect and T(x) is a decreasing function explaining the toxic effect. The assumption is that a dose x would induce a number of revertants M(x) , while the same dose reduced the M(x) by a proportion T(x) in such a way to generate M(x)*T(x) revertants. There are a great variety of functions M and F available. The simplest function for M is the linear one y=M(x)=0+x . In order to describe a decay we can adopt the function T(x)=exp(-x) with >0. So the equation to be used is y=(0+x)exp(-x). In case of =0 the model is a linear one with no toxic effect. In the following chart there are some examples of different curves obtained by fixing =2.00 and varying  . Page 5 of 11 It can be noted that the curvature increases as  (toxic effect) increases. 180 160 140 =0.00 =2 120 =0.005 =2 100 =0.0 =2 80 =0.02 =2 60 40 0 20 40 60 80 From the model we can calculate the first derivative dy/dx=  exp(-x)- (0+x) exp(-x)= exp(-x) {-(0+x)}. To find the dose inducing the maximal response we set dy/dx =exp(-x) {- (0+x)} =0 then xmax= 1/-0 This dose represents the dose level at which the function y has a maximum. At a dose level greater than xmax the toxicity overwhelms the mutagenicity. Such a dose within the experimental dose range might suggest the presence of relevant toxicity. In order to evaluate the mutagenic effect adjusted for toxic effect, the first derivative at x=0 may be considered that is dy/dx=m0= - 0 . This represents the slope of the curve in the neighbor of 0 where the toxic effect might be negligible. The coefficient  would be the mutagenic effect in the absence of toxicity, whereas the term 0 is the adjustment due to the toxic effect. If 0 =0 then m0=. It can happen when =0 but also when o=0. In the latter case there is no spontaneous revertants in the neighbor of 0, so since there are no cells to be killed the toxic effect cannot be experienced. This is a theoretical approach that corresponds to the procedure outlined previously, hen the highest doses were removed to find the linear part of the curve . 5. Examples In the following some examples will be presented to compare the two approaches outlined before. Let’s start with the first example Example 1 Dose (µL) 0 12 24 36 48 60 rep1 49 79 86 88 95 122 rep2 53 99 106 108 101 102 rep3 43 87 99 106 114 105 rep4 44 93 88 111 98 98 rep5 31 79 99 100 120 103 rep6 57 95 104 103 104 98 Mean 46.2 88.7 97.0 102.7 105.3 104.7 sd 9.1 8.4 8.2 8.1 9.7 8.9 Page 6 of 11 Below the trend (mean value vs dose) 120.0 100.0 80.0 60.0 40.0 20.0 0.0 0 10 20 30 40 50 60 70 There are three doses that exhibit a departure from linearity. The first approach provides the following results: the highest doses were removed since the departure from linearity was statistically significant Dose Range used 0-24 0 1 52.47 2.06 Using the nonlinear model Dose Range used 0-60 *p<0.05 0   49.41 4.52 0.019* m0 3.57 xmax 41.03 Comparing m0 with  it looks like the theoretical slope around 0 is greater than the slope determined removing the highest doses. As one can see, the sequential approach tends to overestimate the value of the response at dose=0. The toxicity is statistically significant in the range 0-60. This is consistent with the evidence provided by xmax that lies within the same dose range. Below the charts of linear and nonlinear fitting. It can be noticed that even after removing the highest doses there are some evidence of a nonlinear trend. revertants=(Beta0+Beta1*dose)*EXP(-Tau*dose) 130 120 110 100 revertants 90 80 70 60 50 40 30 0 10 20 30 40 50 60 dose) PLOT revertants ypred Page 7 of 11 Example 2 Dose (µL) 0 12 24 36 48 60 rep1 64 60 70 92 106 111 rep2 59 71 81 83 101 111 rep3 42 62 101 87 105 90 rep4 44 73 88 96 93 104 rep5 48 60 88 93 84 84 rep6 35 81 75 100 100 78 mean 48.7 67.8 83.8 91.8 98.2 96.3 sd 10.9 8.6 11.0 6.1 8.3 14.3 120.0 100.0 80.0 60.0 40.0 20.0 0.0 0 10 20 30 40 50 60 70 Doses 48 and 60 cause a departure from linearity Dose Range used 0-36 0 1 5189 1.14 Using the nonlinear model Dose Range used 0-60 *p<0.05 0   47.94 2.65 0.012* m0 2.06 xmax 62.93 As previously m0 > , the toxicity is statistically significant. The xmax lies slightly outside the dose range. The toxicity looks smaller than that determined in the previous case. The empirical approach (see chart on the left) shows some departure from linearity revertants=(Beta0+Beta1*dose)*EXP(-Tau*dose) 120 110 100 revertants 90 80 70 60 50 40 30 0 10 20 30 40 50 60 dose) PLOT revertants ypred Page 8 of 11 Example 3 120.0 100.0 80.0 60.0 40.0 20.0 0.0 0 10 20 30 40 50 60 70 No linear departure was detected following the empirical approach all the doses were used to calculate the parameters.. Dose Range used 0-60 0 1 49.50 1.10 Using nonlinear model Dose Range used 0   0-60 45.90 1.73 0.0005 As it can be seen, the xmas is far from the dose range m0 1.51 xmax 180 revertants=(Beta0+Beta1*dose)*EXP(-Tau*dose) revertants=(Beta0+Beta1*dose) 130 130 120 120 110 110 100 100 90 revertants revertants 90 80 80 70 70 60 60 50 50 40 40 30 0 30 0 10 20 30 40 revertants 10 20 30 40 50 dose) 60 PLOT dose) PLOT 50 revertants ypred ypred Page 9 of 11 60 6. Parameter estimation The parameter estimation of the nonlinear model in the above examples was performed by Iteratively Reweighted Least Squares (IRLS). Generally speaking, based on the model 𝑦 = 𝑓(𝑥, 𝜃)+∈ the weighted least squares methods minimizes the sum 𝑆𝑆 = ∑(𝑦𝑖𝑗 − 𝑓(𝑥𝑖 , 𝜃)2 𝜔𝑖𝑗 = ∑ 𝑒𝑖𝑗2 𝜔𝑖𝑗 𝑖𝑗 Where xi is the dose i-th and j is the replicate j-th within the dose i-th. Usually the weights Ij do not depend on the parameter . When the weight variable depends on the model parameters, the estimation technique is known as iteratively reweighted least squares (IRLS). The weights Ij used in the code depends on eij and then on the model parameters. In order to provide an initial weight estimation a weighted regression with I =1 is performed . The residuals obtained are used to (𝑘) (𝑘−1) perform a further weighed regression with Ij = (eij). At step k the weight are such that 𝜔𝑖𝑗 = 𝜓(𝑒𝑖𝑗 ). The stopping rule of the procedure is based on a convergence criterion. In order to reduce the influence of extremes values of y, a Robust Regression is performed. To do this a standardized residual is defined as follows 𝑠𝑡𝑒𝑖𝑗 = 𝑒𝑖𝑗 𝑠𝑖 where si is the standard deviation of the y values at the i-th dose , A function Pi is defined as follows: Pij= = -A*Ind(steij<-A)+steij *Ind(-A<=steij <=A)+A*Ind((steij >A) Where Ind(condition) is 1 if the condition is true, 0 otherwise. Finally the weight is calculated as: ω𝑖𝑗 = 𝑝𝑖𝑗 2 𝑠𝑖 𝑠𝑡𝑒𝑖𝑗 A is a constant that defines the robust regression. As a results three cases may occur: 1. 2. 3. steij<-A then ω𝑖𝑗 = 𝐴 𝑠𝑖2 |𝑠𝑡𝑒𝑖𝑗 | (-A<=steij <=A) then then ω𝑖𝑗 = ω𝑖𝑗 = 𝐴 𝐴 𝑠𝑖2 𝑠𝑖2 |𝑠𝑡𝑒𝑖𝑗 | Recalling the sum 𝑆 = ∑𝑖𝑗 (𝑦𝑖𝑗 − 𝑓(𝑥𝑖 , 𝜃)2 𝜔𝑖𝑗 = ∑ 𝑒𝑖𝑗2 𝜔𝑖𝑗 , in case 1 and 3 the single addend of the sum is 𝑒𝑖𝑗2 ω𝑖𝑗 = |𝑒𝑖𝑗 | 𝑠𝑖 otherwise 𝑒𝑖2 ω𝑖 = 𝑒𝑖𝑗 2 𝑠𝑖 2 . The use of absolute value attenuates the effect of the outliers. The Page 10 of 11 constant A can assume different values. For low value of A the terms 𝑒𝑖𝑗2 ω𝑖𝑗 = the terms 𝑒𝑖𝑗2 ω𝑖𝑗 = 𝑒𝑖𝑗 2 𝑠𝑖 2 |𝑒𝑖𝑗 | 𝑠𝑖 are privileged, otherwise prevail. To apply a robust regression A=1.50 is suggested, whereas for a classical 2 regression A=1000 could be appropriate. The variance si is the observed variance of the replicate at the dose ith. Instead using the observed variance, the following linear model can be used: log(𝑠𝑖2 ) = 𝑎 + 𝑏 ∗ log(𝑥̅𝑖 ) where 𝑥̅𝑖 is the mean value of the replicates ad i-th dose. In case of good fitting this variance can be used otherwise the observed one is recommended. 7. Conclusions There are two approached to determine the mutagenic effect. One is empirical the other is based on a theoretical model. The empirical one tries to remove the nonlinearity ( a symptom of a possible toxic effect) in order to determine the mutagenic effect. The other approach relies on a theoretical model whose parameters are used to determine the mutagenic effect. The empirical approach has the advantage that no assumption is made about the trend. In the presence of a departure from linearity less data points are used to estimate the mutagenic effect with the consequence of a less efficiency in the parameter estimation. Using a theoretical model it is easier to describe the behaviour of the trend in the neighbour of zero relying on all the dose range an so to calculate the mutagenic effect without removing doses. On the other hand this approach requests the validity of the model adopted. So if there is valid (theoretical ) reason to use this model this approach should be adopted. 8. References Lawrence E. Myers, Nancy H. Sexton, Leslie I. Southerland and Thomas J Wolff. (1981): “Regression Analysis of Ames Test Data”, Enviromental Mutagenesis, 3, pp 575–586. Leslie Bernstein, John Kaldor, Joyce McCann and Malcolm C. Pike. (1982): “Ana empirical approach to the ststistical analysis of mutagenesis data from the Salmonella test”, Mutation research,97, pp 267–281. Kim B. and Margolin B. (1999): “Statistical methods for the Ames Salmonella assay: a review”, Mutation Research 436, pp 113–122. Page 11 of 11