WLS assumes that both x values and y values are absolute and tries to get the best fit between them whereas in fact there is uncertainty in both response (y) and calibrator concentration (x). Usually the uncertainty of the calibrator concentration is small compared to the calibration range and can be ignored but if the uncertainty in the responses is greater then this assumption of WLS may not hold.

Since x and y should be linearly correlated there should be no difference between 1/x^2 weighting and 1/y^2 weighting. If the assumpptions of WLS hold then either weighting will give equivalent results. Normally 1/x^2 is used as there is less variance in x but where the calibration gives a negative slope then it would be appropriate to use a 1/y^2 weighting.

It may be that the difference in weighting you observe is a result of increased uncertainty / spread of values in the brain homogenate, in which case use of the Deming regression should help. If there is a high degree of uncertainty in the calibrators then a Deming regression will not allow for this and you should try a Passing-Bablock regression. However most LCMS software uses WLS and you may need to change your calibration to reduce the uncertainty, for example by running additional calibrator replicates or using different concentrations.

You can also see which weighting should give the best fit by looking at the reproducibility of calibrator points. If all calibrators have equal variance then usually the curve should be unweighted. If the variance of the points is equal across the calibration range then 1/x weighting is likely to be most appropriate whereas if the standard deviation is equal (ie heteroscedastic data) then 1/x^2 is most likely, as you have found.