BatchKi Reference Manual

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[Velocities]

This section of the program settings file controls how initial velocities are analyzed. The default values are listed below.

[Velocities]
  AdjustVoInitialEstimate = yes
  AutoBaseline = yes
  AutoBaselineCutoff = 0.02
  AverageReplicates = yes
  BackgroundTolerance = 0.2
  ConstantWeightRange = 2
  ConstantWeights = yes
  DeleteSingleOutlier = no
  DilutionRatioCV = 50
  DilutionRatioMax = 100
  DilutionRatioMin = 1.05
  ExcludeLargeSquares = yes
  ExcludeNegativeRates = no
  HuberTuningConstant = 1.345
  MaximumFractionOfOutliers = 0.5
  MinimumRobustWeight = 0.9
  MinimumWeightsRatio = 4
  Model = Morrison
  MultipleOfMedianSquare = 10
  NegativeBaselineLimit = 0.5
  OptimizeHillRatio = 0.5
  OptimizePositiveBackground = no
  RelativeWeights = yes
  RequireNegativeControl = yes
  RobustRegression = yes

In the following text, the control parameters are described in the logical rather than alphabetical order.

Parameter AdjustVoInitialEstimate

This control parameter determines whether or not the initial estimate of the enzyme activity observed in the absence of any inhibitor (``negative control'') should be adjusted for background activity. It has two possible values:

• AdjustVoInitialEstimate = yes
• AdjustVoInitialEstimate = no

The use of this parameter is best explained by using an example.

Let us assume that the in the absence of any inhibitors, the experimentally observed enzyme activity (e.g., the initial reaction rate in an enzyme assay) is 100 units. In the case of the Morrison equation as the fitting model, this is the (adjustable) parameter $V_0$ in Eqn (4.3). Normally, BatchKi will estimate the value of $V_0$ from the reaction rate (activity) experimetally observed at $[I] = 0$. Practical experience shows that in many cases it is beneficial to subtract from the initial estimate for $V_0$ the baseline activity $V_b$, otherwise the nonlinear fit may end up in a false minimum in the parameter space.

Let us further assume that the experimentally observed baseline activity (apparent enzyme activity observed either in the presence of a very large amount of highly potent inhibitor, or in the absence of any enzyme and/or substrate) is 10 units. In this particular example, if AdjustVoInitialEstimate = yes, then the initial estimate for $V_0$ made by BatchKi will be 100 - 10 = 90 units, instead of the usual 100 units of activity.

The default value is AdjustVoInitialEstimate = yes.

Parameter Model

The control parameter Model can have two different values:

• Model = Morrison
• Model = FourParameterLogistic

BatchKi can perform a least-squares or robust fit of initial velocities by using two different fitting models, namely, the modified Morrison equation 4.3 (Model = Morrison), or the four-parameter logistic equation 4.4 (Model = FourParameterLogistic).

$$v = v_b \, + \, v_0 \,\, \frac{[E] - [I] - K_i^{\rm app} + \sqrt{\left ( [E] - [I] - K_i^{\rm app} \right )^2 + 4 \, [E] \, K_i^{\rm app} }}{2\,[E]}$$ (4.3)

$$v = V_{\rm min} \, + \frac{V_{\rm max} - V_{\rm min}}{1 + \left ( [I]/IC_{50} \right )^{n_{\rm Hill}}}$$ (4.4)

In equation 4.3, $K_i^{\rm app}$ is the apparent inhibition constant [9], $v_b$ is the baseline velocity (usually set to zero), $v_0$ is the reaction velocity observed in the absence of inhibitors, and $[E]$, $[I]$ are concentrations of the enzyme and inhibitor, respectively.

In equation 4.4, $IC_{50}$ is the concentration of inhibitor that causes a half-maximum inhibitory effect, and $n_{\rm Hill}$ is the Hill exponent, having the typical value $n_{\rm Hill} = 1.0$.

Parameter ExcludeLargeSquares

The control parameter ExcludeLargeSquares can have two different values:

• ExcludeLargeSquares = yes
• ExcludeLargeSquares = no

When this parameter is set to `no', no action is taken with regard to possibly eliminating form further analysis reaction progress curves that were found suspiciously ``noisy'', as evidenced by the standard deviation of fit (typically to the linear or polynomial model).

When this parameter is set to `yes', the program might ignore certain reaction velocities in the compilation of dose-response curves. In particular, the standard deviation of fit is examined for each relevant progress curve. If the standard deviation of fit from fitting the progress curve (typically to a straight-line model or the polynomial model) exceeds a certain threshold value defined by the parameter MultipleMedianSquare (see below), the reaction velocity determined from this progress curve will be ignored.

The purpose of this procedure is to eliminate from analysis those reaction velocities that were determined from reaction progress curves that are obviously defective, as is evidenced by the fact that they are exceedingly ``noisy''.

Parameter MultipleOfMedianSquare

If the parameter ExcludeLargeSquares is set to `yes' (see above), some reaction progress curves might be excluded from further analysis if the associated standard deviation of fit is larger than a certain multiple of the median standard deviation for the entire plate-reader data set. In other words, the program executes the following algorithm:

1. Fit all reaction progress curves found in the plate reader data set.
2. Compute the standard deviation of fit for each progress curve.
3. Compute median standard deviation of fit for the entire plate.
4. Compare the standard deviation of fit for each progress curve with the median value for the entire plate.
5. If ExcludeLargeSquares is set to `yes', and if a particular standard deviation of fit for the given progress curves is larger then MultipleMedianSquare $\times$ the median standard deviation, ignore the particular initial reaction velocity in the compilation of dose-response curves.

NOTE

If ExcludeLargeSquare is set to `yes', it is especially important to include several replicate measurements of the control velocity, measured in the absence of any inhibitors. If only a single measurement of the control velocity were taken, this particular data point might be accidentally excluded from analysis because the progress curve might be too ``noisy''. If that happened, the analysis of initial reaction velocities could not proceed at all, because all dose-response curves normally must contain the control velocity (at zero inhibitor concentration).

Parameter ExcludeNegativeRates

If ExcludeNegativeRates is set to `yes', the software will exclude from analysis any well associated with an apparently negative reaction rate.

A special case arises when, in the normal, expected course of a continuous enzyme assay, the experimental signal (e.g., UV/VIS absorbance) decreases over time. In that case, the observed reaction velocity (change in absorbance per second) is negative by definition. The software will then treat as ``negative'' (meaning, anomalous) initial velocities from those wells that give rise to positive reaction rate.

If ExcludeNegativeRates is set to `no' (default setting), the software will include in the kinetic analysis all wells, regardless of the possibility that some of them will produce initial velocities with anomalous sign. Please note that it is possible to treat any apparently negative velocities by using the AutoBaseline control parameter (see above).

Parameter AverageReplicates

If AverageReplicates is set to `yes' (the default value), the software will compute an average and standard deviation from any replicated wells, and then use a single averaged value in the nonlinear least-square fit to the modified Morrison equation.

If AverageReplicates is set to `no', each individual measurement of initial velocity is considered separately (no averaging). This option is particularly useful when RobustRegresion is set to `yes' (see below). In that case, the software is more able to identify and possibly exclude from analysis a single grossly outlying data point.

Parameter ConstantWeights

If ConstantWeights is set to `yes' (the default value), the software will assign a unit weight ($w_i = 1.0$) to each data point (index $i$) in the computation of the weighted sum of squared deviation $S$ according to equation 4.5. In equation 4.5, $n_D$ is the number of initial velocity data points, $y_i$ is the experimentally observed reaction rate, and the $\hat{y}_i$ is the corresponding theoretically predicted value.

$$S = \sum_{i = 1}^{n_D} w_i (y_i - \hat{y}_i)^2.$$ (4.5)

Parameter RelativeWeights

If ConstantWeights (see above) is set to `yes', this control parameter is ignored. Otherwise, if and only if AverageReplicates is set to `yes', the software will attempt to perform weighted regression according to equation 4.5, with nonequal values of $w_i$. In particular, the weight of the $i$th data point, $w_i$, is computed as the reciprocal standard deviation, $1/\sigma_i$.

The parameter RelativeWeights controls whether the computed (nonequal) weights will be normalized to unity.

If RelativeWeights is set to `yes' (the default value), the software will first assign a unique weight to each data point as the reciprocal standard deviation from replicates ( $w_i = 1/\sigma_i$), and then it will normalize the resulting weights such that $\sum_{i = 1}^{n_D} = 1.0$. In the opposite case (when RelativeWeights is set to `no'), the normalization to unity is not performed.

The normalization of weights is important in interpreting the resulting values of (a) weighted sum of squares and (b) standard deviation of fit (see section 6.3.1).

Parameter RequireNegativeControl

This control parameter can have two different values:

• RequireNegativeControl = yes
• RequireNegativeControl = no

If RequireNegativeControl is set to `yes' (the default value), the software will expect that at least one well on the given plate is marked with the special inhibitor name CONTROL, meaning that only the enzyme and the substrate are present, but not the inhibitor. If no such wells are found, BatchKi will issue and error and will not process the plate.

If RequireNegativeControl is set to `no', the initial estimate of $v_0$ in the modified Morrison equation 4.3, or the initial estimate of $V_{\rm max}$ in the four-parameter logistic equation 4.4, is set to the reaction rate observed at the lowest inhibitor concentration found on the plate. In this case, wells marked as CONTROL are not required. However, when such wells are in fact present, BatchKi will use the corresponding reaction rates as initial estimates of $v_b$ or $V_{\rm max}$.

NOTE

If CONTROL wells are absent, it is very important to design the experiment such that the reaction velocity observed at the lowest used inhibitor concentration is not very much lower than approximately 90% of the true control rate.

Parameters related to optimized baseline rate

The following control parameters are related to the handling of $v_b$ in Equation 4.3 or, alternately, $V_{\rm min}$ in Equation 4.4.

1. AutoBaseline : Decide whether baseline rate should be treated as a fixed constant (equal to zero) or as an optimized parameter.
2. AutoBaselineCutoff : Tolerance parameter used when AutoBaseline is set to `yes'.
3. NegativeBaselineLimit : Controls the lower limit on optimized negative baseline rate.
4. OptimizePositiveBackground : Decide whether a positive baseline rate should be treated as a fixed constant (equal to zero) or as an optimized parameter.
5. BackgroundTolerance : Tolerance parameter used when OptimizePositiveBackground is set to `yes'.

NOTE

Related to the handling of $v_b$ in Equation 4.3 is the XML attribute <Data background="..."> (see section 5.6.1). Also important is the presence or absence of any inhibitor wells identified by using the special label BACKGROUND (see section 5.4.3).

Parameter AutoBaseline

The control parameter AutoBaseline can have two different values:

• AutoBaseline = yes
• AutoBaseline = no

The interpretation of these two choices depends on the fitting model.

Fit to the modified Morrison equation

When AutoBaseline is set to `no', the program will fit the initial reaction velocities to the simple Morrison equation 4.6, in which the baseline velocity $v_b$ is absent.

$$v = v_0 \,\, \frac{[E] - [I] - K_i^{\rm app} + \sqrt{\left (... ...i^{\rm app} \right )^2 + 4 \, [E] \, K_i^{\rm app} }} {2\,[E]}$$ (4.6)

When the parameter AutoBaseline is set to `yes', the program will examine the lowest reaction velocity found in the experimental data set. If the lowest experimental reaction velocity is negative, the program will check its absolute value (see parameter AutoBaselineCutoff below). If this absolute value is larger then a certain fraction of the average control velocity $v_0$, the program will then consider the baseline velocity $v_b$ as an optimized or adjustable parameter. In this case the fitting model is the modified Morrison equation 4.3.

Fit to the four-parameter logistic equation

When AutoBaseline is set to `no', the program will fit the initial reaction velocities to the simplified logistic equation containing only three adjustable parameters, equation 4.7.

$$v = \frac{V_{\rm max}}{1 + \left ( [I]/IC_{50} \right )^{n_{\rm Hill}}}$$ (4.7)

When the parameter AutoBaseline is set to `yes', the program will examine the lowest reaction velocity found in the experimental data set. If the lowest experimental reaction velocity is negative, the program will check its absolute value (see parameter AutoBaselineCutoff below). If this absolute value is larger then a certain fraction of the average control velocity $V_{\rm max}$, the program will then consider the baseline velocity $V_{\rm min}$ as an optimized or adjustable parameter. In this case the fitting model is the full four-parameter model equation 4.4.

Parameter AutoBaselineCutoff

This parameter is used to determine whether a (negative) baseline velocity should be considered as an optimized parameter in equation 4.3. A typical, useful value is 0.01, 0.02, 0.05, or a similar number close to zero.

The decision is made on the basis of the following rule. The experimental control velocity, $v_0$, is first calculated as an average of all initial reaction rates observed in the absence of inhibitors. Then the remaining reaction rates are compared with the experimental value of $v_0$.

Fit to the modified Morrison equation

If any reaction velocity is negative and at the same time larger in absolute value than AutobaselineCutoff $\times$ $v_0$, the model equation 4.3 is chosen in the nonlinear least-squares regression. Otherwise, the model equation 4.6 is used.

Fit to the four-parameter logistic equation

If any reaction velocity is negative and at the same time larger in absolute value than AutobaselineCutoff $\times$ $v_{\rm max}$, the model equation 4.4 is chosen in the nonlinear least-squares regression. Otherwise, the model equation 4.7 is used.

Parameter NegativeBaselineLimit

This parameter is used in the rare instances where the baseline velocity, observed at very high concentrations of very potent inhibitors, is apparently negative. The control parameter is utilized when two prerequisites are satisfied: (a) AutoBaseline (see above) is set to `yes', and (b) NegativeBaselineLimit is greater than zero (NegativeBaselineLimit = 0 signals to BatchKi that the control parameter should be ignored).

The role of this control parameter is to assure that $v_b$ in equation 4.3 does not become too negative in least-squares (or robust) fit. In particular, the fixed lower limit on $v_b$ is computed by using the following formula:

$$\mbox{lower limit for } v_b = - \mbox{\tt NegativeBaselineLi... ...t ( V_{\rm max}^{(obs)} - V_{\rm min}^{(obs)} \right ) \quad ,$$

where $V_{\rm max}^{(obs)}$ and $V_{\rm min}^{(obs)}$ are the largest and smallest, respectively, values observed in the given dose-response curve. The recommended value is NegativeBaselineLimit = 0.5.

Parameter OptimizePositiveBackground

This control parameter can have two different values:

• OptimizePositiveBackground = yes
• OptimizePositiveBackground = no

The control parameter is useful for those enzyme assays, in which a nominally positive signal (reaction rate) is observed even at zero enzyme concentration. This situation arises in rotamase assays (uncatalyzed background rate) or in end-point kinase assays using radioactive substrates (nonspecific binding).

When OptimizePositiveBackground is set to `no', the parameters $v_b$ in equation 4.3 or $V_{\rm min}$ in equation 4.4 are set to zero and held constant during least-squares optimization. When OptimizePositiveBackground is set to `yes', BatchKi's behavior subtly depends on the fitting model.

In the fit to the modified Morrison equation 4.3, the handling of $v_b$ can be represented as shown in Algorithm 1.

On the other hand, in the fit to the four-parameter logistic equation 4.4, the handling of $V_{\rm min}$ is shown in Algorithm 2.

Parameter BackgroundTolerance

The role of this control parameter is explained in the preceding paragraph (see OptimizePositiveBackground).

Parameters related to robust regression analysis

The following control parameters are related to robust fit, as opposed to least-squares fit. For theoretical details, see ref. [5].

• RobustRegression
• HuberTuningConstant
• DeleteSingleOutlier
• MaximumFractionOfOutliers
• MinimumRobustWeight
• MinimumWeightsRatio
• ConstantWeightRange

If RobustRegression is set to `yes' (the default value), the program will perform robust regression analysis of initial reaction rates, according to the Huber Minimax method described elsewhere [5]. If RobustRegression is set to `no', the software will perform simple nonlinear least-squares regression.

The HuberTuningConstant parameter will control the degree to which the regression analysis will resemble least-squares fit (Huber tuning constant smaller than one), or the least-absolute deviation fit (Huber tuning constant significantly greater than one). The theoretically optimum value is of HuberTuningConstant = 1.345. For further details and an illustrative example consult reference [5].

The function and mutual interactions of control parameters DeleteSingleOutlier, MinimumRobustWeight, MinimumWeightsRatio, and MaximumFractionOfOutliers is described in Algorithm 3.

The control parameter ConstantWeightRange is used to control the special treatment of negative controls, that is, initial rates observed at zero inhibitor concentration. Empirically it has been observed that under certain special circumstances, the weight for the negative control data point should be kept constant at unity, even as the remaining data points are subjected to Huber's Minimax reweighting scheme. The involvement of ConstantWeightRange is described by in Algorithm 4.

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Petr Kuzmic | Jul 12 2008