: How does losing fat and gaining weight work? Can gaining weight whilst losing fat mass be a plausible thing to do and if so, how does it work?

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Because the only variables that adults are typically working with are muscle mass and fat mass, to increase total mass while decreasing one means that you have to increase the other. Because of this you have to consider the rate at which you are capable of building muscle and reconcile that with your energy balance over time. Let's take a hypothetical new lifter who fits the bill of Lyle McDonald's projected rates of muscle growth.

Two pounds of month is the maximum that he can gain in a month. This will require him consuming at least 230 calories above maintenance every day just to have gained two pounds total. To reach two pounds of muscle, that surplus might need to be raised by 50-100% or more. Which would see him gaining 3-4 pounds with 2 of that being muscle.
The further you stray from this optimal anabolic environment, the slower your body's capacity to build muscle becomes. There's no way to know for sure, but let's say that in maintaining weight (neutral energy balance) a person is capable of building half of their maximum potential each month. This is referred to as "body recomposition".
If you increase calories ever so slightly (less than 50 above maintenance per day) you might expect to gain a little under half a pound in a month. If you were capable of gaining 1 lb of muscle per month while maintaining weight, it makes sense that the half pound gained will have been entirely muscle while having lost fat.

However, if you are more advanced than a new lifter (or less genetically gifted than Lyle presumes) and you are only maximally capable of gaining half a pound of muscle per month and you only can gain a quarter pound of muscle while maintaining weight and you gain about half a pound, then you will almost definitely have also gained fat. Your rate of weight gain cannot exceed your body's capacity to build muscle for this hypothetical scenario to work. This means being in a super tiny surplus. We're talking about slowly gaining 5 lbs in a year (or less) for most people.

How does it work biologically? Your fat stores are broken down to provide energy for muscle growth. Fat is not stored because you are so close to maintenance calories and fat is only being broken down because the training stimulus demands that muscle be built. It works exactly the same as a body recomposition except that you're having a few more calories.

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Yes—gaining muscle mass whilst losing fat is entirely possible given the right conditions. The subject is discussed at greater length here. But in the simplest terms, any diet/training regimen in which energy is in deficit, in which the diet contains adequate protein, and in which we are training strength, will lead to simultaneous muscle gain and fat loss. It is possible, also, when energy is in surplus, provided that overfeeding comes as a consequence of excessive protein, rather than excessive carbohydrate or fat.

The mechanics of the two processes—lipolysis and the citric acid cycle, and hypertrophy—are complex, but it is important to note that they are distinct and independent. This is why the common belief that the two cannot co-exist is erroneous: it assumes that the processes amount to nothing more than a transfer of energy, and that the two substrates are functionally equivalent. They are not.

The primary function of fat is energy storage. And although muscle requires significant energy to be built, and hence also represents a store of energy, its primary function is to perform mechanical work. Thus, provided that conditions simultaneously favour fat metabolism (aerobic lipolysis) and muscle cell hypertrophy, the two processes can coincide.

I hope that helps.