Linear technology power supply, question about a transistors
Q5 takes a regulated 8 mA constant current from the output of the LT3081


The base of Q5 is at 0 volts and this means the emitter is 0.7 volts lower at -0.7 volts. This means that the voltage across the 549 ohm resistor is about 4.3 volts. 4.3 volts / 549 ohms = 7.83 mA.

In other words the ~8 mA is a constant "drag" from the LT3081 and allows it to operate down to 0 volts. You could use a simple resistor to pull 8 mA from that point but consider what happens when the LT3081 is outputting (say) 24 volts. A simple resistor that takes 8 mA from (ostensibly) 0 volts to - 5 volts would have a value of 625 ohms BUT that would take a current of (24 + 5) volts / 625 ohms (=46.4 mA) when the LT3081 output is at 24 volts and this is deemed to be either too inefficient or just plain "too much".

It's less clear to me what Q4 does other than to put INT_Vcc onto the RUN pin of the LTC3632. Maybe someone else can spot the reason.