Modern cars are fuel injected, therefore there is no choke, instead the fuel delivery is computed by the ECU on receipt of inputs from various sensors.

One of these is the CTS (Coolant Temperature Sensor) this as the name suggests informs the ECU of the temperature of the coolant, it therefore will then deliver a richer mixture for a cold start, for example, on your engine this will be between two of the spark plugs, and it actually measures the temperature of the head itself, opposed to the coolant.

Being these are electronic components they do go out of range after a while and give duff information, therefore the engine will run badly, it may be worth replacing this and if this is no good, scanning the car for fault codes, this can then isolate the problem to a particular sensor etc.

Martin