Optimization Method
for Genetic Control Of Hematopoietic Stem Cell Frequency Is Mostly Cell Autonomous.

BLOOD 2000, 95:(7):2446-8

Introduction

Experimental biology and medicine is based on testing hypotheses. Frequently, this involves simple yes-no decisions, where the hypothesis is either correct or false. However, in complex systems, non-exclusive, overlapping hypotheses or models can sometimes best describe the data. Optimal Model Mixing (OMM) determines the optimal weights that quantify the relative contribution of each model to the experimental data. OMM is based on linear optimization and is not a statistical method. Thus, the size of the data set is irrelevant for the resolution and validity of the approach.

For a general approach, we considered 3 random variable, X,Y,Z.

Z represents the experimental data; data could be a single or a series of measurement(s). X and Y represent the numerical values predicted from the models that are tested. Of course, more than two models can be considered.

Taking the general approach, where we consider a series of measurements, we represent the models X and Y and the data Z by vectors:

To find the contribution of each model to the data Z, we introduce weights a and b, such that the new random variable W:= a X + bY approximates Z reasonably close. If we define

and

Then, obviously:

,

and

is a good approximation to Z in the sense that

where C < 1 is a small, positive constant. Here:

.

In other words, we measure the distance between W and Z by taking the root of the sum of squares of the vector's components. This approach is familiar from least square optimization.

Note that, since X, Y, and Z are vectors, the definitions of L, a, and b must be understood component-wise:

OMM has been applied to a problem in hematopoietic stem cell biology to estimate the contribution of the extrinsic and intrinsic models to the regulation of the size of the stem cell compartment. See Figure 1 for an outline of the approach.

Figure 1. Outline of the experimental design and hypothesis generation. To assess whether Scfr genes act stem cell intrinsically or extrinsic through the microenvironment, aggregation chimeras between D2 and B2 mice were analyzed for LTC-IC levels.

A hypothetical aggregation chimera that is comprised of exactly 50% of each parental strain is used as an example. For experimental mice, chimerism was determined in hematopoietic and non-hematopoietic organs. The chimerism in bone marrow was used as a baseline to calculate the expected ratios of LTC-IC of either genotype if the regulation would be either intrinsic or extrinsic. Bone marrow cells were then plated in limiting dilution. Four weeks later, individual colonies were harvested and each LTC-IC was genotyped by staining its progeny with mAb specific for H-2b (B6) and H-2d (D2). This provided a direct measurement of the extent of D2 <-> B6 chimerism in the LTC-IC compartment in individual aggregation chimeras. Comparison of the experimental values with the calculated values, allowed a quantitative assessment of the contribution of extrinsic and autonomous regulation of the size of the LTC-IC compartment.

OMM was used to estimate the contribution of the intrinsic and extrinsic models to the maintenance of the stem cell compartments in D2 <-> B6 chimeras. For each mouse, the LTC-IC frequency Z, was compared to the intrinsic model X and the extrinsic model Y. In this case, we have m =1 in the general case discussed above.

Therefore,

and

All calculations were performed with a program written in Mathematica.

The results from these calculations are shown in Table 1 of Muller-Sieburg et al., Blood 2000, 95: ___ and are summarized in Figure 2.

Figure 2. Optimal model mixing establishes the contribution of the extrinsic and autonomous models to the data. All values are expressed as the contribution (%) of B6 to the chimeric LTC-IC compartment. The numbers on the x-axis correspond to the mouse numbers in Table 1. Data (yellow) are the ratios measured in the chimeras, intrinsic (black) and extrinsic (red) values were calculated as detailed in the notes to Table 1. The new, best fit model (blue) was calculated by OMM to provide the optimal weights which minimize the difference of each model relative to the data.